Noise, Detectability, and Implications for System Design

Task-Based Imaging Performance in 3D X-Ray Tomography: Noise, Detectability, and Implications for System Design

Jianan (Grace) Gang

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Institute of Biomaterials and Biomedical Engineering University of Toronto

Task-Based Imaging Performance in 3D X-Ray Tomography: Noise, Detectability, and Implications for System Design

Jianan (Grace) Gang

Doctor of Philosophy

Institute of Biomaterials and Biomedical Engineering University of Toronto

2014 Abstract Quantifying imaging performance is an important aspect in the development, optimization, and assessment of medical imaging systems. This thesis addresses new challenges in the characterization of imaging performance for advanced x-ray tomographic imaging technologies.

Central to the work is a task-based cascaded systems analysis framework that encompasses aspects of system geometry, x-ray beam characteristics, dose, detector design, background anatomy, model observers, and the imaging task. The metrology throughout includes Fourier domain descriptors of spatial resolution (modulation transfer function, MTF), noise (noise-power spectrum, NPS), noise-equivalent quanta (NEQ), and task-based detectability. Central elements and advances of the work include: a task-based model for 3D imaging performance in tomosynthesis and cone-beam CT (CBCT); generalization of imaging performance metrics to include the influence of anatomical background clutter; validation of the model in comparison to human observer performance; extension to dual-energy (DE) tomographic imaging; analysis of non-stationary (i.e., spatially varying) signal and noise characteristics; and extension to model- based statistical image reconstruction. In each case, the analytical framework demonstrates the

ii importance of task-based assessment and the capability for system optimization in a fairly broad scope of clinical applications ranging from breast to abdominal and musculoskeletal imaging.

The validity of the framework in describing “local” signal and noise characteristics is demonstrated under conditions of strong nonstationarity, ranging from simple phantoms to complex anthropomorphic scenes. In addition to providing a framework for system design and optimization, the analysis opens potential new opportunities in task-based imaging and statistical reconstruction, with examples demonstrated in the design of optimal regularization in iterative reconstruction.

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Acknowledgments

First and foremost, I would like to thank my supervisor, Dr. Jeff Siewerdsen. He introduced me to the exciting world of medical physics when I was just a clueless summer student and fostered my interest in science and engineering into a potential career. He had not only offered his invaluable expertise in medical imaging, but provided me with tremendous support and encouragement. This work would not be possible without his guidance. His unwavering passion for science will always be an inspiration to me. I would also like to thank the members of my supervisory committee, Drs. Krsity Brock and Mike Joy, for their support and helpful comments. Members of my exam committee, Drs. David Jaffray, Martin Yaffe, and John Boone are gratefully acknowledged.

I am deeply grateful for the collaboration and support from my colleagues who helped to make this work possible - chronologically, Dr. Samuel Richard for teaching me what NPS, MTF, and IPA are; Daniel Tward for sharing with me his insights in image science and many interesting math puzzles; Dr. Junghoon Lee for remotely working with me on the observer study which materialized into Chapter 4 of this thesis; Dr. Wojtek Zbijewski for his valuable assistance with the dual-energy study in Chapter 5; and last but not least, Dr. Web Stayman for his mentorship in the last leg of my studies and for sharing with me his expertise in statistical reconstruction.

I am lucky to have worked in two great labs – the IGTx lab at the University of Toronto and the ISTAR lab at Johns Hopkins University. Staff and students in both labs have made my journey through graduate school an enjoyable one: Hany, Carlos, Thao, Nate, Angela, Mike, Harley, Jenny, Carolyn, Jordan, Steve, Greg, Nick, Sun-mo, and Noor from Toronto; Ja, Adam, Ali, Yoshi, Sebastian, Saj, Muhit, Yifu, Paul, Dan, Prakhar, Hao, Jen and Zhe from Hopkins.

I am forever grateful to my dearest friends, old and new, for being part of my life: Tina, Neil, and Sheryl, for always being there for me and willing to lend an ear/shoulder; Chunjing, Zhong, Zhu, Siyu, and Christine, for friendship through the years and hopefully for many more to come; the EngSci crew, especially Yinming, John, Luke, and Chunpo, for all the fun and games and making Toronto feel like home; Huihui, for the ever-so-effective shopping therapy; Yuxuan, Carmen, and Claire, for inspiring me with their dedication towards work and good food, and for making life in Baltimore fun and exciting.

Last but not least, I would like to thank my family, especially my parents, to whom I am infinitely indebted. Words cannot express my gratitude for their love and kindness, for allowing me to pursue my dreams in a faraway land, and for supporting me to be an educated, independent woman. Thank you!

Table of Contents

ACKNOWLEDGMENTS ...... IV

TABLE OF CONTENTS ...... VI

LIST OF TABLES ...... XII

LIST OF FIGURES ...... XIII

LIST OF APPENDICES ...... XVII

LIST OF NOTATIONS ...... XVIII

CHAPTER 1 ...... 1

INTRODUCTION...... 1

CHAPTER 2 ...... 5

THEORETICAL BACKGROUND ...... 5

2.1. FOURIER DOMAIN IMAGING PERFORMANCE METRICS ...... 5

2.1.1. Linear and Shift-Invariant (LSI) Systems ...... 5

2.1.2. Stationarity ...... 6

2.1.3. Modulation Transfer Function (MTF) ...... 7

2.1.4. Noise-Power Spectrum (NPS) ...... 10

2.1.5. Detective Quantum Efficiency (DQE) and Noise Equivalent Quanta (NEQ) ...... 12

2.2. CASCADED SYSTEM ANALYSIS ...... 14

2.2.1. Signal and Noise Transfer Characteristics ...... 14 2.2.1.1. Gain Stages ...... 15 2.2.1.2. Deterministic Spreading Stage ...... 16 2.2.1.3. Stochastic Spreading Stage ...... 17

2.2.1.4. Sampling Stage ...... 18

2.2.2. Cascaded Systems Analysis ...... 19 2.2.2.1. Stage 0: Incident quanta ...... 21 2.2.2.2. Stage 1: Quantum Detection Efficiency...... 22 2.2.2.3. Stage 2: Conversion from X-Rays to Optical Photons ...... 23 2.2.2.4. Stage 3: Spreading of Secondary Quanta in the X-Ray Converter ...... 29 2.2.2.5. Stage 4: Coupling of Secondary Quanta to Detector Apertures...... 31 2.2.2.6. Stage 5: Integration of secondary quanta by detector pixel aperture ...... 31 2.2.2.7. Stage 6: Sampling of detector signal ...... 33 2.2.2.8. Stage 7: Readout with Additive Electronic Noise ...... 34 2.2.2.9. Stage 8: Post-readout Binning / Sampling ...... 34 2.2.2.10. Stage 9: Log-Normalization ...... 36 2.2.2.11. Stage 10: Ramp Filter ...... 36 2.2.2.12. Stage 11: Apodization filter ...... 37 2.2.2.13. Stage 12: Interpolation of Filtered Projection Data ...... 38 2.2.2.14. Stage 13: 3D Backprojection ...... 38 2.2.2.15. Stage 14: 3D sampling ...... 41 2.2.2.16. Stage 15: Post-Reconstruction Binning and Sampling ...... 42

2.3. THE RESULTING 3D FOURIER DOMAIN PERFORMANCE METRICS ...... 43

2.4. TASK-BASED IMAGE QUALITY METRIC - DETECTABILITY INDEX ...... 46

APPENDIX.2A. EXTENSIONS TO THE BASIC MODEL ...... 50

2A.1. Focal Spot Blur ...... 50

2A.2. X-Ray Scatter ...... 50

2A.3. Cone-Beam Artifact ...... 53

CHAPTER 3 ...... 55

ANATOMICAL BACKGROUND AND GENERALIZED DETECTABILITY IN TOMOSYNTHESIS AND CONE-BEAM CT ...... 55

3.1. INTRODUCTION ...... 55

3.2. METHODS ...... 57

3.2.1. Analytical Basis and Design of a “Clutter” Phantom for Power-Law Noise ...... 58

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3.2.2. Image Acquisition and Reconstruction ...... 62

3.2.3. Measurements of Signal, Noise, and Power Spectral Density ...... 64 3.2.3.1. Signal-Difference-to-Noise Ratio (SDNR)...... 64 3.2.3.2. Measurement of Background Power Spectra ...... 65 3.2.3.3. Power-Law Noise Parameters ...... 68 3.2.3.4. Detectability Index in Tomosynthesis and CBCT ...... 70

3.3. RESULTS ...... 72

3.3.1. Signal-Difference-to-Noise Ratio (SDNR) ...... 72

3.3.2. Background Power Spectra ...... 74

3.3.3. Power-Law Parameters ...... 79

3.3.4. Detectablity Index ...... 83

3.3.5. Imaging performance optimization for specific imaging tasks ...... 87

3.4. DISCUSSION AND CONCLUSIONS ...... 88

APPENDIX 3A. POWER-LAW NOISE IN PROJECTIONS, TOMOSYNTHESIS, AND CBCT .. 92

3A.1 The Clutter Power Spectrum in the 3D Object and the 3D CBCT Image ...... 92

3A.2 The Clutter Power Spectrum in a 2D Projection ...... 93

3A.3 The Clutter Power Spectrum in a 2D Slice of the 3D CBCT Volume ...... 93

3A.4 The Clutter Power Spectrum in 3D Tomosynthesis ...... 96

APPENDIX 3B. CALCULATION OF THE POWER SPECTRUM AS A DISCRETE SUM OVER (FEW) PROJECTIONS ...... 99

CHAPTER 4 ...... 102

ANALYSIS OF FOURIER-DOMAIN TASK-BASED DETECTABILITY INDEX IN TOMOSYNTHESIS AND CONE-BEAM CT IN RELATION TO HUMAN OBSERVER PERFORMANCE ...... 102

4.1. INTRODUCTION ...... 102

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4.2. METHODS ...... 104

4.2.1. Generalized Detectability Index ...... 104 4.2.2.1. “Slice” Detectability vs. “3D” Detectability ...... 106

4.2.2. Imaging Tasks and Physical Imaging Phantoms ...... 108 4.2.2.1. Detection in Uniform Background ...... 108 4.2.2.2. Detection / Discrimination in Background Clutter ...... 109 4.2.2.3. Imaging Tasks ...... 109

4.2.3. Imaging Bench and Acquisition Parameters ...... 115

4.2.4. Experimental Validation: Human Observer Performance ...... 115 4.2.4.1. Human Observer study ...... 115 4.2.4.2. Comparison of Theoretical and Experimental Results ...... 117

4.3. RESULTS ...... 119

4.3.1. Comparison of Theoretical Detectability and Human Observer Performance ...... 119

4.3.2. Number of Projections and Orbital Extent: Effect on Quantum Noise and View Aliasing ...... 123

4.3.3. Implications for Task-Based System Design in Tomosynthesis and CBCT ...... 126

4.4. DISCUSSION AND CONCLUSIONS ...... 129

CHAPTER 5 ...... 133

CASCADED SYSTEMS ANALYSIS OF NOISE AND DETECTABILITY IN DUAL-ENERGY CONE-BEAM CT ...... 133

5.1. INTRODUCTION ...... 133

5.2. METHODS ...... 136

5.2.1. Reconstruction-Based Dual-Energy Image Decomposition ...... 136

5.2.2. Cascaded Systems Analysis of DE-CBCT ...... 137

5.2.3. Experimental Validation of 3D Dual-Energy NPS and NEQ ...... 139

5.2.4. Detectability Index as an Objective Function ...... 141

5.2.5. Optimization of DE-CBCT Imaging Techniques ...... 143

5.3. RESULTS ...... 147

5.3.1. Comparison of Theoretical and Experimental NPS and NEQ ...... 147

5.3.2. Detectability Index ...... 150

5.4. DISCUSSION AND CONCLUSIONS ...... 156

CHAPTER 6 ...... 159

NON-STATIONARY NOISE AND DETECTABILITY IN FILTERED BACKPROJECTION AND PENALIZED LIKELIHOOD ESTIMATION ...... 159

6.1. INTRODUCTION ...... 159

6.2. METHODS ...... 162

6.2.1. The Spatially-Varying NPS and MTF for FBP ...... 162

6.2.2. The Spatially Varying NPS and MTF for Statistical Image Reconstruction ...... 169

6.2.3. Digital Phantoms and Image Simulation ...... 173

6.2.4. Analysis of Noise and Resolution ...... 176

6.2.5. Optimization of Reconstruction Parameters in PL ...... 179

6.3. RESULTS ...... 181

6.3.1. Spatially Varying NPS and MTF...... 181

6.3.2. Optimization of Reconstruction Parameters ...... 188

6.3.3. Design of an “Optimal” Regularization Map ...... 193

6.4. DISCUSSION AND CONCLUSIONS ...... 195

CHAPTER 7 ...... 200

SUMMARY AND FUTURE WORK ...... 200

7.1. SUMMARY ...... 200

7.2. FUTURE WORK ...... 203

REFERENCES ...... 207

List of Tables

Table 2.1. Summary of terms and notation in the cascaded systems analysis model ...... 20

Table 3.1. Experimental conditions under the constant-Δθ and constant-Nproj cases...... 63

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List of Figures

Figure 2.1. Illustration of modulation transfer of a sinusoidal signal, f(x), through a system with transfer function H(f). The modulation of the output and input signal is related by the MTF...... 9 Figure 2.2. Theoretical relationships between PSF, LSF, ESF, and MTF...... 10 Figure 2.3. Illustration of the cascaded systems model for 2D image formation in a flat-panel detector followed by 3D filtered backprojection for CBCT...... 43 Figure 3.1. Experimental setup...... 61 Figure 3.2. Coronal images of the clutter phantom including a conspicuous stimulus (nylon cylinder) for illustration of clutter rejection and measurement of SDNR...... 73 Figure 3.3. Coronal and axial slices of the clutter phantom in tomosynthesis and CBCT...... 75 Figure 3.4. Coronal and axial slices of the normalized 3D power spectra of clutter phantom images...... 76

Figure 3.5. Normalized power spectra measurements along fx and fz axes extracted from the 3D power

spectrum measurements for the constant Δθ [(a), (b)] and constant Nproj [(c), (d)] schemes...... 78 Figure 3.6. Measurements of  and  obtained from linear regression of the normalized power

spectra in Fig. 3.5...... 80

Figure 3.7. Detectablity index (d’) as a function of (a)  obj and (b)  obj, for four orbital extents (sphere

detection task). (c) Plot of d’ vs. θtot for different exposure levels (exposure to detector after attenuation by 10 cm of water), illustrating different levels of quantum noise relative to anatomical

background noise (Gaussian detection task). (d) Plot of d’ vs. θtot for different total number of

projections (Nproj), demonstrating effects of view-aliasing at low Nproj (high θtot) and electronic noise

at very high Nproj (sphere detection task)...... 86

Figure 3.8. Surface plots of d’ versus Nproj and θtot for (a) delta-function detection task (uniform weighting of spatial frequencies), (b) a 3 mm sphere detection task (emphasizes low and middle frequencies), and (c) Gaussian detection task (a low-frequency task)...... 88 Figure 3.9. Three regions of the Fourier domain relevant to computing detectability for few projections covering a wide source-detector arc are identified, and the boundary frequencies are labeled...... 99 Figure 4.1. Fourier-domain task functions (left column) and coronal image ROIs (x-z) (images at right) for

varying angular extent under the constant- and constant-Nproj cases: (a) sphere detection on uniform background; (b) large sphere detection in clutter; (c) small sphere detection in clutter; (d)

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cube vs. sphere discrimination in clutter; and (e) encapsulated sphere vs. solid sphere discrimination in clutter...... 114 Figure 4.2. Comparison of theoretical and measured performance for five imaging tasks in the constant- acquisition scheme...... 119 Figure 4.3. Comparison of theoretical and measured performance for five imaging tasks in the constant-

Nprojacquisition scheme...... 121

Figure 4.4. Illustration of Fourier domain projection distribution on the axial (fx-fy) plane according to the central slice theorem...... 125

Figure 4.5. Surface plots of d’slice for a wide range of tot and Nproj for 4 imaging tasks: a low-frequency task corresponding to detection of a 3.6 mm Gaussian on (a) uniform and (b) cluttered backgrounds; and higher-frequency tasks corresponding to discrimination of two Gaussians of size (c) 3.1 and 3.7 mm and (d) 1.3 and 1.8 mm, each on a cluttered background...... 127 Figure 5.1. Schematic diagram for cascaded systems analysis of reconstruction-based DE-CBCT...... 138 Figure 5.2. Comparison of theoretical and measured DE NPS and NEQ from the [60, 130] kV DE image...... 148 Figure 5.3. Comparison of theoretical and experimental DE NPS for a range of experimental conditions. (a) Fixed high-energy beam, and varied low-energy beam, also for a constant tissue cancelation factor w = 0.5. (b) Fixed low-energy beam, and varied high-energy beam for a constant tissue cancelation factor w = 0.5. (c) Varying levels of w at fixed low- and high-energy beams (60 and 130 kV)...... 150 Figure 5.4. Optimization results for a kidney stone discrimination task and an iodine vs. bone discrimination task ...... 151 Figure 5.5. Detectability index computed as a function of w and A for a low-frequency detection task at (a) 15 mGy, (b) 1.5 mGy, and (c) 0.15 mGy, and for (e) a mid-frequency task and (f) a high-frequency task at 15 mGy. The integral of the quantum noise NEQ [Eq. (5.5)] at 15 mGy is plotted in (d). Superimposed on (a) are plots of the absolute values of dual-energy contrast between a breast tumor and fibroglandular tissue and the dual-energy anatomical "clutter" magnitude DE...... 154 Figure 5.6. Detectability index and integral of the NEQ computed for a low-frequency task at 15 mGy [(a) and (b)], a low-frequency task at 1.5 mGy [(c) and (d)], and a mid-frequency task at 15 mGy [(e) and (f)]. Dual-energy contrast between the tumor and fibroglandular tissue in the DE image is plotted in (g). The optimal kV selection demonstrates a combined effect of NEQ and DE contrast, and is affected by total dose more than the frequency content of the task...... 156

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Figure 6.1. Illustration of line integrals at different view angles transmitting variable fluence to the detector, thereby imparting non-stationarities in detector signal and noise and the resulting image reconstruction...... 163 Figure 6.2. Three digital phantoms investigated in this work. (a) A uniform circular object. (b) An elliptical object composed of three materials approximating water, bone, and fat. (c) A thorax approximating a realistic distribution of tissue heterogeneities. The numeric symbols (1-4) mark four locations at which the local MTF, NPS, NEQ, and detectability index were analyzed specifically...... 174 Figure 6.3. Vertical and horizontal line-pair patterns and associated task functions...... 176 Figure 6.4. Variance maps, 2(x,y), calculated from multiple realizations of FBP (top) and PL (bottom) reconstructions of the three phantoms shown in Fig. 6.2...... 182 Figure 6.5. Measured and theoretical NPS and MTF at four locations within the uniform Circle phantom, illustrating the non-stationary, anisotropic characteristics of each for FBP and PL...... 186 Figure 6.6. The same as Fig. 6.5 but for the Ellipse phantom...... 186 Figure 6.7. The same as Fig.6.5 but for the Thorax phantom...... 187 Figure 6.8. Detectability index at location 1, 2 and 4 in the Ellipse (Fig. 6.2) as a function of regularization parameter  for the vertical line pair detection (top) and horizontal line pair detection (bottom). Example ROIs in reconstructions at select values of  are shown on the right. Both the trends and magnitude in d' agree qualitatively with visual assessment of the images...... 189 Figure 6.9. Detectability index computed as a function of smoothing parameters for FBP [(a),(c)] and PL [(b),(d)] reconstructions for a mid-frequency task [(a),(b)] and an all-frequency task [(c),(d)]...... 191 Figure 6.10. Detectability maps, d'(x, y), computed for FBP and PL reconstructions in each phantom for the mid-frequency task...... 193 Figure 6.11. Spatially varying regularization. (a-c) Spatially varying  maps designed to maximize d’ at each point in the image. (d-f) The resulting detectability maps. (g-i) Comparison of the global average resulting from the spatially varying  map (straight horizontal line) plotted in comparison to achieved with a constant value of  (plotted as a function of  for the latter). A slight improvement in detectability is achieved with the spatially varying map. (j-l) Ratio of detectability index achieved with a spatially varying and constant , showing improvement up to ~10%, particularly in more heavily attenuating regions of the phantom...... 194

Figure 6.12. Detectability index for the vertical and horizontal line-pair tasks computed as a function of  for the PW and NPWEi observer models. Each case corresponds to Location 1 in the Ellipse phantom...... 199

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List of Appendices

Appendix 2A. Extensions to the Basic Model...... 50 Appendix 3A. Power-Law Noise in Projections, Tomosynthesis, and CBCT...... 92 Appendix 3B. Calculation of the Power Spectrum as a Discrete Sum Over (Few) Projections...... 99

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List of Notations AFC Multiple-alternative forced-choice CBCT Cone-beam CT CT Computed Tomography DE Dual-energy DFT Discrete Fourier Transform DQE Detective Quantum Efficiency FBP Filtered-backprojection reconstruction FPD Flat panel detector LSI Linear and shift-invariant MAFC multiple-alternative forced-choice MTF Modulation transfer function NEQ Noise Equivalent Quanta NPS Noise Power Spectrum NPW Non-prewhitening observer NPWE Non-prewhitening observer with eye filter NPWEi Non-prewhitening observer with eye filter and internal noise PL Penalized-likelihood PSF Point spread function PW Prewhitening observer PWE Prewhitening observer with eye filter ROC Receiver operating characteristic ROI Region of interest SDNR Signal difference to noise ratio SNR Signal to noise ratio SSS Strict sense stationary VOI Volume of interest WSS Wide sense stationary

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Chapter 1 Introduction

The importance of quantifying medical image quality is difficult to overstate. It not only provides guidance for early stages of system design and development but also governs the optimization of imaging systems to improve performance and minimize dose. With the proliferation of new medical imaging technologies in recent years – particularly in digital x-ray imaging and computed tomography (CT) – a general image quality framework would be invaluable in assessing and comparing performance across systems and modalities, which is key in improving efficacy in diagnostic imaging.

The definition of image quality, however, has not always been straightforward. Despite the abundance of physical metrics including the pixel signal-to-noise ratio (SNR), modulation transfer function (MTF), noise-power spectrum (NPS), etc., which describe fundamental properties of the imaging systems and the images they produce, such metrics in themselves do not relate to clinical performance. To address these problems, Fellgett and Linfoot1 first proposed image quality as a measure of fidelity to the object. However, an imaging system can only represent a fraction of the information of a real object; therefore, different objects may result in the same mean image.2 In addition, some distortion of the object may even be desirable in the image (e.g., edge enhancement).3

A central paradigm in the modern theory of image quality was established but 40 years ago.

Through his monumental work in 1972, Bob Wagner asserted that image quality "must be defined in terms of the task the image is destined to perform".4 For example, comparing the imaging tasks of micro-calcification detection and soft tissue tumor detection in mammography,

1 the former requires an image of high spatial resolution while the latter is best performed on an image with a low noise level to enhance sensitivity to small contrast differences. The task-based paradigm has since inspired a wealth of research which founded a generalized, rigorous theory of image quality rooted in statistical decision theory with applications across a wide range of imaging modalities. Such theory has been increasingly accepted as the standard for quantitative assessment of image quality, with generalizations and extensions to new technologies still an active area of research in the imaging community.

This thesis is a collection of work that addresses new challenges in image quality assessment pertinent to the development of advanced x-ray imaging technologies. Central to the thesis is a task-based image quality model which, through this work, has been extended to new modalities and applied in the design and optimization of such imaging systems. Apart from the obvious utility in system development, this work can also be appreciated from an image science perspective, with much effort devoted to examining the fundamental performance limits in these technologies and understanding the complex tradeoffs among different factors affecting image quality.

This thesis is divided into seven chapters: Chapter 1 broadly introduces the motivation and significance of this work; Chapter 2 provides a theoretical background on established research in the field of image quality, including the primary analytical framework used in this dissertation - a cascaded systems analysis model for cone-beam CT (CBCT) and prevalent Fourier-domain image quality metrics. The subsequent four chapters correspond to studies that extend the cascaded systems analysis framework in Chapter 2 and address challenges in image quality assessment in relation to various advanced x-ray imaging modalities and algorithms: Chapter 3 focuses on the characterization of anatomical background noise in the context of tomosynthesis

2 and CBCT and understanding the noise sources limiting performance under different imaging conditions; Chapter 4 provides validation of theoretical predictions of image quality with human observers in the 2D-3D continuum of tomosynthesis and CBCT; Chapter 5 extends the cascaded systems analysis model to dual-energy-CBCT (DE-CBCT) and demonstrates utility of the model in the optimization of example DE imaging applications; and Chapter 6 addresses the issue of noise nonstationary in CBCT images and extends the task-based imaging performance framework to statistical image reconstruction algorithms. Finally, Chapter 7 summarizes results of this thesis and highlights areas of future work.

The main contributions of the author include work presented in Chapters 3-6, specifically: (1) the design of a physical phantom that presents power-law noise based on principles of fractal self- similarity and characterization of NPS under various imaging conditions in tomosynthesis and

CBCT; (2) an analysis of the effect of such power-law noise on task-based detectability; (3) an observer study designed to measure the level of agreement between theoretical predictions and the performance of human observers in volumetric imaging and thereby validate the applicability of such theoretical models to system design and optimization; (4) extension of the cascaded systems analysis model to reconstruction-based dual-energy CT/CBCT (DE-CBCT) and for the first time, and apply a task-based framework to optimizing DE imaging techniques; (5) extension of the cascaded systems model to describe nonstationary noise and resolution in filtered- backprojection (FBP) reconstruction and application of the task-based framework in penalty design for model-based statistical image reconstruction.

Specific motivation for each element of the thesis is provided in the introduction of each chapter, with the work overall fitting broadly within the development of digital x-ray imaging systems for tomosynthesis,5-8 cone-beam computed tomography (CBCT),9-14 and dual-energy (DE)

3 tomographic imaging 15-19. The broad scope of technologies described by the approach comes in part from the generality of the theoretical model – i.e., in terms of the system geometry, x-ray beam characteristics, dose, detector design, background anatomy, and imaging task. Similarly, the potential scope of specific clinical applications is fairly broad – applicable to, for example: single-energy or dual-energy tomosynthesis and/or CBCT for breast imaging,5-8, 20-22 musculoskeletal extremity imaging,23, 24 image-guided radiotherapy,25-27 and image-guided surgery28-33. Application of the theoretical framework within such application-specific contexts is demonstrated at various points in the chapters that follow, but the focus and emphasis of the dissertation as a whole is on the more fundamental aspects of the analytical framework in general terms that can, in turn, be directed to specific questions of system design or clinical application.

Chapter 2 Theoretical Background

This chapter presents the theoretical background essential for the development for this thesis.

The first section introduces Fourier domain imaging performance metrics, and the second section elaborates on the cascaded systems analysis model for CBCT.

2.1. Fourier Domain Imaging Performance Metrics 34-37; The sections below describe a number of basic assumptions and general metrology associated with characterization of imaging performance. Given the focus of this dissertation on tomography (i.e., 3D imaging), notation and specific forms below are shown in three spatial dimensions (x, y, z) and corresponding spatial frequency dimensions (fx, fy, fz), and reduction to

2D forms (or extension to 4D or beyond) is straightforward - see, for example, the general multidimensional notation in Siewerdsen et al.38

2.1.1. Linear and Shift-Invariant (LSI) Systems The basic assumption for application of Fourier analysis is that the system is linear and shift- invariant (LSI). A system is linear when the output from a weighted sum of input signals is equal to the weighted sum of the output from each individual input. If a system is denoted S{} with output S { f (x,y)} arising from input f(x,y) linearity requires that:

KK S fkkx,,,, y z  S f  x y z . (2.1) kk11

A system is shift-invariant if a translation in the input signal produces an identical translation of output , i.e.,

if g x,,,, y z  Sf k  x y z, then

gxxyyzz 0,,,,  0  0 Sf k  xxyyzz  0  0  0  . (2.2)

This implies that the point spread function of the system is the same for all locations (x,y,z) in the image.

Few physical imaging systems strictly obey such requirements; rather, the assumptions are invoked with respect to a range in signal or spatial extent that is "local" (i.e., small signal differences and/or limited region of support). For flat-panel detectors (FPDs) considered in this work, linearity (quantified by the mean signal response of detector elements to incident radiation) holds within an exposure range below pixel saturation (e.g., ~50% of pixel saturation). The discrete nature of pixelated digital imagers, however, inherently violates the shift-invariance assumption. Early research has argued that: (1) digital systems are cyclically invariant, i.e., shift- invariant to shifts equal to pixel spacing, which is sufficient for the application of Fourier analysis;39 (2) the degree to which shift-invariance is violated is fairly small under a wide range of imaging conditions;40 and (3) the spatial extent over which shift-invariance holds should be limited due to oblique incidence of x-rays.41 Recognizing the conditions under which the LSI assumptions hold, and recognizing that they hold under conditions that are "local" with respect to both signal size and spatial extent, Fourier analysis of imaging performance has been widely adopted and proven useful in many applications.42

2.1.2. Stationarity The stationarity of a random process refers to the extent to which its statistical distribution (e.g., mean signal, variance, etc.) are constant in space. There are varying degrees of stationarity: a process is strict sense stationary (SSS) if its joint probability density function (or probability

6 mass function for discrete random processes, or cumulative distribution function - i.e., all moments of its statistical distribution) does not change when shifted in space. For purposes of power spectral analysis, it is sufficient for a process to be wide-sense stationary (WSS), i.e., the mean and autocorrelation (i.e., the first and second-order moments of the statistical distribution) are invariant to shifts in space. A WSS process is sometimes also referred to as "weakly" stationary. Autocorrelation for such processes is therefore completely specified by the displacement between pixels, denoted as x,,  y  z , i.e.,

RxyzE,,,,,,   ff xyz  xxyyz       z . (2.3)

The condition is not strictly satisfied for real objects with finite support, but may be useful within a certain spatial extent. Yet another degree of stationarity is "cyclo-stationarity," meaning that the (typically, first and second-order) statistics do not change under periodic displacements in space,39 related to the assumption of cyclic invariance for digital detectors noted in the previous section. As a useful property, an LSI system with a WSS process as the input will yield a WSS output.

2.1.3. Modulation Transfer Function (MTF) Modulation transfer function (MTF), denoted T, is a common Fourier metric that describes the frequency-dependent mean signal response of the imaging system. The MTF has the following mathematical form:

Hf  T  , (2.4) H 0

7 where H(f) is the transfer function of the system defined as the Fourier transform of the point spread function (PSF), and f is the Fourier domain coordinate (e.g., radial frequency). The term

"modulation" in this context refers to the contrast of a periodic signal relative to its average value,

i.e., for a 1-D input function f( x ) A B sin(2 f0 x ) , the modulation m f is given by:

ffmax min B mf . (2.5) fmax f min A

As illustrated in Fig.2.1, the output from the system is g( x ) AH (0) B H ( f00 ) sin(2 f x ) ,

with its modulation mg given by:

ggmax min B H() f0 mg . (2.6) gmax g min AH(0)

The above form of MTF describes a frequency-dependent degradation in contrast imparted by the imaging system, from the ratio of the output to input modulation:

m Hf() T g 0 , (2.7) mHf (0) which can also be interpreted as a description of the frequency response of the system.

Figure 2.1. Illustration of modulation transfer of a sinusoidal signal, f(x), through a system with transfer function H(f). The modulation of the output and input signal is related by the MTF.

The MTF is commonly used to describe spatial resolution of the imaging system. In radiographic systems, characterization of the MTF involves measurement of the PSF, the line-spread function

(LSF), edge-spread function (ESF), or other function describing the response to a known input.

(In fact, to the extent that the input distribution is known, the MTF could be characterized from the response to any input.). In CT systems, MTF is frequently characterized from 2D axial slices

(yielding the 2D axial MTF) following similar methods. Measuring the PSF directly involves the imaging of a point-like object (e.g., a pinhole on the detector for radiography, or a tiny sphere in

CT). An alternative method to measure spatial resolution is from the LSF (e.g., from the image of a narrow slit). It can be shown that the Fourier transform of the LSF is equal to one radial spoke of the Fourier transform of the PSF, i.e., the Fourier transform of the LSF of a vertical line

at the origin, Lf x  , is related to the transfer function by:

L fxx  H f ,0. (2.8)

When the PSF is isotropic, the LSF is sufficient to characterize the full transfer characteristics of the system. The MTF can also be determined form the ESF (e.g., from a flat, sharp plate on the detector for radiography, or the surface of a block or sphere in CT), the derivative of which gives the line spread function. Recent research has proposed measurement of the 3D MTF from surfaces of spheres.43, 44 A summary of relationships among the various spatial domain and

Fourier domain descriptors is shown in Fig.2.2.

Figure 2.2. Theoretical relationships between PSF, LSF, ESF, and MTF.

2.1.4. Noise-Power Spectrum (NPS) X-ray quanta arriving at the detector follow a random process well described by Poisson

2 statistics, with the variance in the number of incident quanta (per unit area)  0 equal to the mean

incident fluence, q0 . Apart from the Poisson noise intrinsic to the input signal, various stages of the imaging chain can increase or decrease the noise and change its spatial frequency content

(e.g., noise added by the readout electronics). Noise in the system is frequently quantified by the

NPS, which describes both the magnitude and the spatial-frequency content of noise in a stationary random process.

The noise component alone is described by the autocovariance which has the signal mean removed from autocorrelation:

 KxyzE,,,,,,  ff xyz   xxyyzz   . (2.9)

The NPS is defined as the Fourier transform of the autocovariance function:

S fx,,,, f y f z   FT K  x  y  z . (2.10)

An alternative (and sometimes equivalent) definition of power spectra is the expected periodogram:

2 1 1 1 x  y  z 2i ( f x  f y  f z ) S f, f , f limf  x , y , z ex y z dxdydz  x y z  x   y   z x,,  y  z  2x 2  y 2  z

(2.11)

where indicate an ensemble average over multiple realizations of f x,, y z , where

in Eq. (2.11) denotes an image of image fluctuations only – i.e., an image from which the mean signal and any deterministic structure in the image have been removed. This definition can be applied to both stationary and nonstationary random processes. When stationarity holds, the two definitions are equivalent.

Writing Eq. (2.11) in the discrete domain yields the form of NPS with normalization factors frequently used in experimental measurements:

2 N N y N 1 1 1 x z 2i ( x f  y f  z f ) Sfff, , limf xyze , , nx x n y y n z z aaa  x y z     nx n y n z  x y z NNNx,, y z  Nx a x N y a y N z a z nx1 n y  1 n z  1

2 N N y N a a a x z 2i ( x f  y f  z f )  limx y z f x , y , z e nx x n y y n z z     nx n y n z  NNNx,, y z  NNNx y z nx1 n y  1 n z  1

(2.12)

11 where a and N denote the voxel size and number of voxels, respectively, in each region contributing to the ensemble.

A useful relationship between NPS and variance is that the area under the NPS is equal to the variance, i.e.,

K(0,0,0) 2 Sfffdfdfdf , , . (2.13)   x y z x y z

2.1.5. Detective Quantum Efficiency (DQE) and Noise Equivalent Quanta (NEQ) The DQE has several equivalent definitions. Conceptually, it describes the fidelity with which the system transfers the signal-to-noise ratio (SNR) from its output to its input:

2 SNR DQE  out , (2.14) SNRin i.e., an ideal system has DQE = 1, and a non-ideal system (e.g., one that adds noise or suffers other losses in fidelity) has a DQE less than 1 and would require 1/DQE times the number of incident quanta to achieve the same SNR as an ideal system. The term SNRin is the square root of

the mean fluence at the detector, q0 , following Poisson statistics of incident x-rays. The signal

at the output can be written as Gq0 T where G is the system gain and T is the system MTF. Noise at the output is characterized by the NPS, as detailed in the previous section and denoted S.

Combining these terms, the DQE becomes:

2 Gq T / S TT22 DQE 0   (2.15) qq S/ Gq 2 q S 0000  norm

2 where Snorm represents the NPS normalized by the square of the mean signal Gq0  . Since all quantities in Eq.(2.15) are measurable, this is the form of DQE often used in relation to experimental measurements.

An equivalent interpretation of DQE is given by the "stochastic" form posed by Cunningham45 as the fraction of incident photons an ideal system would require to produce the same NPS at each spatial frequency:

Sideal f x,, f y f z  DQE fx,, f y f z   . (2.16) Sout f x,, f y f z 

In Eq. (2.16), the term Sout is the NPS of the real system (either measured or predicted by an accurate system model), and Sideal is the NPS of a system that adds no noise or blur to image, but in other ways is identical to the real system. The stochastic form is useful in theoretical analysis as described in the following section.

The NEQ is related to the DQE by the total incident fluence from all projections:

NEQ fx,,,, f y f z  q tot DQE f x f y f z  . (2.17)

The interpretation of NEQ directly follows that of the DQE, as the effective number of incident photons contributing to the image - i.e., the NEQ for a non-ideal system is lower than the actual number of incident quanta, with the effective loss in quanta (at each spatial frequency) given by the DQE.

2.2. Cascaded System Analysis Cascaded systems analysis has been widely used in modeling detector performance in 2D radiography.46-54 Such modeling provides a theoretical analysis of various aspects of the imaging chain in relation to Fourier image performance metrics introduced in Sec.2.1. Such analysis provides valuable insight into the factors limiting imaging performance and offers guidance in detector design and optimization. With recent proliferation of new x-ray imaging technologies, the cascaded systems analysis model has seen exciting development into advanced modalities, including dual-energy radiography,55, 56 cone-beam CT,57, 58 tomosynthesis,59, 60 dual-energy

CBCT,61 photon-counting detectors,62, 63 and phase contrast CT64, 65.

2.2.1. Signal and Noise Transfer Characteristics The cascaded systems analysis model describes signal and noise transfer characteristics through discrete stages corresponding to physical or mathematical processes in the imaging chain. Signal in the imaging system is described by a spatial distribution of quanta, denoted q(u,v). Incident x- ray quanta are assumed to follow a Poisson distribution. The noise in the imaging system is described by the NPS. The transfer characteristics in the imaging chain are modeled according to four types of stages - gain, deterministic spreading, stochastic spreading, and sampling - as described below. Because many of these stages relate to processes in formation of the 2D projection image, notation in following subsections is primarily two-dimensional [with spatial coordinates in the projection domain (u,v) and corresponding spatial frequency coordinates

(fu,fv)], and extension to 3D in specific cases is noted when appropriate.

2.2.1.1. Gain Stages A gain stage corresponds to either the amplification or loss of the mean number of quanta. Signal

transfer for stage i is related to the mean input fluence, qi1  u, v, through the mean gain, gi , according to:

qi u,v giqi1u,v. (2.18)

Noise transfer for a gain stage i is given by:

S  f , f  g 2S  f , f  2 q  S  f , f  (2.19) i u v i i1 u v gi i1 addi u v where S f, f is the NPS at the input to stage i,  2 is the variance of the gain and S i1  u v  gi addi represents additive noise. The variance in the gain can be described by the "Poisson" excess,  , gi defined as the relative amount the variance exceeds that of a Poisson distribution:

 2  gi 1. (2.20) gi gi

A related quantity is the Swank factor of the output quanta, I, which is a statistical quantity defined as the ratio between the square of the first moment (denoted M1) to the product of the zeroth moment and the second moment. If the input quanta follows a Poisson distribution, the

Swank factor can be simplified as follows:

2 M 2 gq  1 I 1 ii1  . (2.21) 2 2 2 2 MM02 q g q  g i11 i qii1 g i  i 1 2 gi

If the gain factor is deterministic,  2  0 ,  1, and I 1. If the stage corresponds to a gi gi binomial selection process (e.g., only two possible outcomes, as in the absorption of quanta),

 2 gg1 , g , and Ig . If the gain follows Poisson statistics,  2  g ,   0 , gi i i  gii i gii gi

and I gii/ (1 g ) .

2.2.1.2. Deterministic Spreading Stage A deterministic spreading stage is so called, because it relocates quanta according to an absolutely deterministic behavior - as opposed to, for example, a random relocation ("scattering") of quanta (treated below under “stochastic” spreading). An example process where a deterministic spreading model is appropriate is an integrating aperture: all quanta falling within the aperture are deterministically relocated to a single point (e.g., the center of the aperture).

A deterministic spreading stage blurs the signal according to the PSF of the stage, denoted

pi  u, v :

qi u,v  qi1u,v**pi u,v, (2.22) where ** denotes a two-dimensional convolution:

 quv ,  q uvpuuvvdudv ', '   ',  ' ' '. (2.23) i  i1 i

For the example of a deterministic aperture, the PSF is the aperture itself (e.g., a 2D rect function for rectangular pixels).

Noise is filtered by the square of the transfer function of a deterministic spreading stage:

2 Si  fu , fv   Si1  fu , fv Ti  fu , fv  (2.24)

For the example of a rectangular aperture, the MTF in Eq. (2.24) is a sinc function. Note that for a deterministic spreading stage, the incident NPS (in its entirety, including correlated and uncorrelated components therein) is modulated by the square of the MTF. Such a process effectively treats signal (squared) and NPS equivalently in terms of the imparted blur - a behavior that is distinct from that of "stochastic" spreading (scattering) processes described below.

2.2.1.3. Stochastic Spreading Stage A stochastic spreading process relocates quanta randomly in space, with the distance of

relocation governed by a probability density distribution pi  u, v . Mathematically, this process can be represented as a (two-dimensional) stochastic convolution:66

qi u,v  qi1u,v**s pi u,v (2.24)

where the subscript "s" denotes a stochastic convolution operator.

If the input fluence is represented as a distribution of Dirac delta functions

N q(,),, u v  u v  uii v  , where N is the number of quanta and ri is the site of interaction i

N th of the i quanta, then q(,),, u v  u v  uii   u v   v , where uv,  is a random i variable following the probability density function p(u,v). As such, a conventional convolution blurs or smears the quanta, while the stochastic convolution simply relocates them.

Noise transfer through such stages follows the relationship derived by Rabbani et. al 67

2 Si  fu , fv   Si1 fu , fv  qi1 Ti  fu , fv  qi1 (2.25)

This equation reveals that the correlated component of the input noise, Si11 f u, f v  q i , is modulated by the transfer function, while the uncorrelated ("white") component of the input

noise, qi1 , is not and passes through the stochastic spreading stage unaffected. Put simply: uncorrelated noise remains uncorrelated, but correlated noise is modulated by the MTF (squared).

2.2.1.4. Sampling Stage Signal transfer through sampling stages corresponds to multiplication of the input signal with a

train of impulse functions spaced according to the sampling distance au and av , which can in turn be written in relation to the comb function, III , as follows:

uv q u,,, v q u v III ii  1   . (2.26) aauv

The corresponding NPS transfer follows a convolution with the Fourier transform of the impulse train, which is another impulse train:

Sffiuv , = S i1  ff uv , ** aaIIIfafa uv uuvv , . (2.27)

As detailed below, such a process could arise at several distinct stages in the analysis of tomographic systems - both in relation to the formation of 2D projections and 3D reconstructions.

Signal and noise transfer of sampling stages in 3D follow straightforward extensions of Eqs.

(2.26) and (2.27) to the reconstruction domain:

x y z qxyz,,,,,, q xyzIII ii  1   , (2.28) ax a y a z

Sfff,,,,***,, S fff aaaIIIfafafa and ixyz  i1  xyz xyz xxyyzz  (2.29)

18 where *** denotes 3D convolution.

The most important effect associated with a sampling stage is the potential for increase in NPS associated with “aliasing” – i.e., the presentation of noise from spatial frequencies above the

Nyquist frequency. Aliasing results from sampling when the Shannon sampling criterion is not satisfied, resulting in increased NPS and nontrivial effects on the NEQ. The importance of aliasing has been discussed at length in relation to 2D detectors (e.g., indirect-detection and direct-detection FPDs)49, 68, 69 as well as 3D image reconstruction.70, 71

2.2.2. Cascaded Systems Analysis The sections below present the cascaded system analysis model for 2D projection imaging and

3D image reconstruction developed and validated through the work of Siewerdsen,47, 57

Richard,55 and Tward58. With respect to the novel contributions in this dissertation, the cascaded systems framework summarized below is advanced in later chapters to include the effects of anatomical background noise and dual-energy CBCT. A glossary of notation is provided in Table

2.1, along with nominal values for each of the parameters in the imaging chain for three example

CBCT systems: (1) a dedicated musculoskeletal (MSK) CT scanner with a PaxScan 4030CB flat-panel detector (Varian Imaging Products, Palo Alto, CA) with a 250mg/cm2 CsI:Tl converter, nominally operating at 90 kV with 4 mm Al + 0.3 mm Cu added filtration;24 (2) a C-arm system with a PaxScan 4030CB detector (Varian Imaging Products, Palo Alto, CA) with CsI:Tl converter, nominally operating at 100 kV with 6.52 mm Al + 0.5 mm Cu added filtration;72 and

(3) a CBCT system integrated with a medical linear accelerator (Elekta, SL-20, Elekta Oncology

Systems, Atlanta, GA) for image-guided radiotherapy, using a RID-1640 AL1 detector

(PerkinElmer Optoelectronics, Wiesbaden, Germany) with a Gd2O2S:Tb converter, nominally operating at 120 kV and 0.625 mAs, with 2 mm Al + 0.127 mm Cu added filtration.25

Term Definition Nominal Value MSK C-arm Linac

Tj Transfer function for stage j - - -

Sj Noise-power spectrum at stage j - - - 5 5 5 qX0 / Fluence per unit exposure 2.67×10 2.69×10 2.69×10 -2 -2 -2 qo Mean fluence incident on the detector per projection 960 mm 373 mm 47 mm g1 Mean fraction of incident photons interacting with 0.78 0.79 0.43 the detector g 2 Mean number of optical photons produced per 1905 1844 948 incident photon, including k-shell interactions

Pk Transfer function associated with k-fluorescence - - -

T3 Transfer function due to spread of optical photons - - - g 4 Coupling efficiency of photodiode 0.6 0.6 0.55 apd Width of (square) photodiode 0.16 mm 0.16 mm 0.36 mm

T5 Transfer function due to photodiode aperture - - - ai Detector sampling interval in direction i 0.194 mm 0.194 mm 0.400 mm III Sampling function (comb, or Shaw function) - - -

add Additive electronics noise 2000 e/pixel 2000 e/pixel 2000 e/pixel

Ai Width of binning aperture (in pixels) in direction i 1, 2 1, 2 1

Bi Downsampling factor in direction i 1, 2 1, 2 1

T8 Transfer function due to 2D binning aperture - - -

T10 Ramp filter - - -

T11 Apodization filter - - - hwin Smoothing parameter 0.5–1.0 0.5-1.0 0.5-1.0

T12 Interpolation filter - - - m Number of projections acquired across circular orbit 480 200 330 o o o tot Total angular extent of acquisition 240 178 360 M Magnification factor, SDD/SAD 1.28 1.97 1.55

T13 Transfer function associated with backprojection - - -

Θ13 Transfer function associated with backprojection of - - - noise bi 3D sampling interval in direction i 0.15 mm 0.25 mm 0.25 mm Table. 2.1. Summary of terms and notation in the cascaded systems analysis model. Nominal values are provided for three example imaging systems: (1) a dedicated musculoskeletal (MSK) CT scanner operating at 90 kV and 0.1 mAs (per projection), with 4 mm Al + 0.3 mm Cu added filtration, attenuated by 10 cm of water;24 (2) a C-arm scanner at 100 kV and 2.7 mAs (per projection), with 6.52 mm Al + 0.5 mm Cu added filtration, attenuated by 16 cm of water;72 and (3) a CBCT system integrated with Elekta

Linac for image-guided radiotherapy, operating at 120 kV and 0.625 mAs (per projection), with 2 mm Al

+ 0.127 mm Cu added filtration, attenuated by 32 cm of water .25

2.2.2.1. Stage 0: Incident quanta

The incident x-ray spectrum is represented by the fluence qE0 () , where E is the energy in eV.

The spectrum is assumed to be uniformly distributed (stationary with respect to mean, variance, and energy distribution) and uncorrelated in space, so the dependence on detector location (u,v) is dropped. Since x-rays incident on the detector follow a Poisson distribution, the NPS at stage 0

is given by S00 fuv, f  q . The input to the model is the energy-normalized spectrum:

qE0 () qEnorm ()  (2.30) q() E dE 0 0 which can be interpreted as the probability density function of an incident quantum as a function of energy.

The mean number of quanta (per unit area) is related directly to the exposure, X (defined as charge generated by ionizing radiation in air, with units milli-Roentgen, mR), calculated according to:

q k q() E 01 norm dE (2.31) XEE0  ()/  ab air

8 where k1 is a constant equal to 5.45×10 eV/g/mR derived from the definition of the Roentgen, and ()/E is the energy absorption coefficient of air.  ab air

Results reported in subsequent chapters computed the incident spectrum using the Spektr

q toolkit,73 which is based on the TASMIP model of Boone and Seibert.74 Example values of 0 X are given in Table 2.1.

2.2.2.2. Stage 1: Quantum Detection Efficiency Stage 1 is a gain stage accounting for the interaction of incident x-rays with the x-ray converter –

e.g., a scintillator in the case of an indirect-detection FPD. The gain factor gE1()is the mean fraction of x-rays depositing energy in the scintillator:

  Es g( E ) 1 e  (2.32) 1

 E where  is the mass attenuation coefficient of the scintillator material and s is the surface density with units mg/cm2.

The mean gain (also called the quantum detection efficiency, QDE) is calculated as gE1() weighted by the normalized spectrum from Stage 0:

 g g()() E q E dE (2.33) 110 norm

This stage is a binomial selection stage in which an x-ray is either absorbed or not, so the signal and noise transfer characteristics are:

q1 q 0 g 1 (2.34) and

S(,), f f g22 S f f q 1u v 1 0 u v gi i 1 g2 q  g(1  g ) q 1 0 1 1i 1 (2.35)  qg01

2.2.2.3. Stage 2: Conversion from X-Rays to Optical Photons Stage 2 is a stochastic gain stage describing the conversion of x-rays into secondary quanta

(specifically, optical photons produced in the scintillator). The process was described using a parallel cascade model following the work of Yao and Cunningham75 and Richard55, including three possible scenarios of energy absorption in the scintillator: (a) Branch A: photons interact and deposit energy locally without producing K x-rays; (b) Branch B: a K x-ray is produced and the remaining energy of the incident photon is deposited locally and liberates optical photons; and (c) Branch C: a K x-ray is reabsorbed and produce optical photons at a remote location.

Details of each branch and the combination of all three are shown as follows:

(a) Branch A: local deposition of energy without K-shell interaction

The production of K x-ray corresponds to a Bernoulli process where the probability of a K x-ray being produced is given by the product of the probability of a photon undergoing a K-shell interaction,  , and the fluorescent yield (the probability that a K x-ray is produced when a K- shell interaction occurs),  . For CsI, the parameters , , and the K shell binding energy, EK, are computed as a weighted average based on the fractional weight of each element, yielding

  0.834 ,   0.870 and EK  35 keV . All photons with energies above the K edge have probability  to produce a K x-ray, while those below the K edge have zero probability of doing so, i.e.,;

0 if E

The probability of Branch A is therefore 1E   . The mean number of optical photons produced per keV absorbed is denoted W and taken to be 56 photons/keV in CsI. The fraction of

optical photons that escape the scintillator, denoted gesc , is assumed to be independent of energy and empirically determined to be 0.55. The gain factor of Branch A is therefore:

g21AE  W  E  q E 1   E g esc , (2.37a) with the effective (energy-averaged) gain given by:

WE  q1  E 1   E d E gg  . (2.37b) 2 AqdE 1 E E esc  1    

Following signal transfer of gain stages governed by Eq.(2.18), the fluence with Branch A is given by:

q2AA q 0 g 11  g 2 . (2.38)

2 Noise transfer follows directly from Eq.(2.19). Expressing gain variance,  gA2 , in terms of the

Poisson excess,  gA2 , [Eq.(2.20)], the NPS of Branch A is:

S u,, v g22 S u v q 2A  2 A 1  g 2 A 1 (2.39) q0 g 111  g 2A g 2 A    g 2 A 

The Poisson excess of the gain distribution is in turn determined from its Swank factor:

g 2 I  2 A (2.40) 2 A MEEE/EE q  d q  d EE2A 1 1

The second moment, ME2 A   is given by:

g 2 E M E  2 A (2.41) 2 A I E

where I(E) is the Swank factor computed from moments of the absorbed energy distribution

(AED)76. The Swank factor is finally related back to Poisson excess through the following

relationship:

1  g22 Ag A  11  , (2.42) I2 A

allowing the calculation of NPS according to Eq.(2.39).

(b) Branch B: local deposition of energy after K-shell interaction

The probability of Branch B first includes the probability of K x-ray production given by  E

following Eq. (2.36). The gain factor of Branch B is similar to that of Branch A, except that the

energy carried away by the K x-rays (EK) needs to be accounted for. For convenience, a fraction

 is defined as the ratio of the energy lost to the total energy, i.e.,   E/EK , such that the

remaining energy deposited locally is 1E  .

The energy-dependent and mean gain of Branch B are therefore, respectively:

g21BE  W  1  E  q E  E g esc , (2.43a)

W1  E  q1  E   E d E and gg  (2.43b) 2BqdEEE esc  1    

The signal and noise of Branch B are:

q2BB q 0 g 1 g 2 (2.44)

and S2B u,1 v  q 0 g 1 g 2 B g 2 B    g 2 B  , (2.45) where the Poisson excess can be calculated in the same way as Branch A.

The probability of Branch C first needs to include the probability of K-shell interaction,  E , given by Eq. (2.36). The probability of a K x ray being reabsorbed is described by the energy-

77 dependent probability fK(E) determined from previous work by Vyborny. An effective fK was calculated as a weighted average from the normalized spectrum:

fEEEE q    d E K 1 f  (2.46) K qdEEE   E 1

The K x-rays then undergo a stochastic spreading stage where they are relocated and reabsorbed at a remote location. The radial reabsorption distance, r, is a random variable described by the

78 point spread function, PSFK  r,E . Following the work of Que, the PSF is given by:

(E) L K (E)(E)(E)   L    e PSFK r,E  C 1  e  e 1  e 22 E K PE (E K ) d (2.47) 0 r 

26 where  is the linear attenuation coefficient of the scintillator, PE is the linear attenuation coefficient due to photoelectric interactions, L is the thickness of the scintillator,  is the thickness of an infinitesimal layer of the scintillator, and C is the normalization constant that ensures unit area under the PSF. The Fourier domain transfer function, TK, is obtained from the

Fourier transform of PSFK  r,E .

The distribution of K x-rays follows a Poisson distribution with the mean given by:

q2CKK , q 0 g 1 f (2.48)

The NPS of the K x-rays is equal to the mean:

Sq2CKCK , 2 , (2.49)

After the stochastic spreading stage, NPS transfer follows Eq.(2.25):

'2 S2CKCKCKKCK , u,, v  S 2 , u v  q 2 , T  q 2 ,

 q2,CK (2.50)

 q01 g fK

The K x-rays are then reabsorbed to produce optical photons. The energy carried in each K x-ray is equal to the binding energy, EK, regardless of the energy of the incident photon. The mean conversion gain of Branch C is therefore energy-independent:

g W E  g 2C K esc (2.51) Wg  E  esc

with   E/EK as defined previously.

The mean signal (optical photons) of Branch C is therefore

q2CCKCKC q 2 , g 2 q 0 g 1 f g 2 (2.52)

The derivation of NPS transfer follows directly from Eq. (2.20):

S u,, v g22 S u v q 2C  2C 2 C , K  g 2 C 2 C , K (2.53) q0 g 1 g 2C g 2 C 1   g 2 C 

(d) Combining all three branches

The total optical photons emitted from stage 2 is simply the sum of all three branches:

q2 q 2ABC  q 2  q 2 (2.54)  q0 g 11  g 2ABKC   g 2   f g 2

An effective gain of stage 2 can be defined following the above equation as:

g21  g 2ABKC   g 2   f g 2 . (2.55)

The combined NPS of all three branches is given by the sum of individual NPS and their cross terms:

SSSSSSSSSS2 2A  2 B  2 C  2, AB  2, BA  2, AC  2, CA  2, BC  2, CB (2.56)

Incident quanta are assumed to be independent, i.e., a photon that produces a K x-ray is independent from one that does not. Therefore, Branch A is independent from both Branches B

and C, which means SSSS2,AB 2, BA 0, 2, AC  2, CA  0 . In comparison, the same quantum

28 undergoes both Branch B and C, so the cross terms S2,BC ad S2,CB are non-zero. Yao and

Cunningham75 derived the sum of the two cross term as:

S2,BC S 2, CB2, q 0 g 1 g 2 B f K g 2 C T K  u v . (2.57)

Adding all terms together:

SSSSSS2 2A  2 B  2 C  2, BC   1 gg222AAgA     gg 222 BBgB     fgg KCCgC 222    2  gfgTuv 22 BKCK  ,  q g g 1 0 1 2 g 2

q0 g 1 g 2  PK  u,1 v

(2.58)

where PK  u, v is defined for convenience to describe the increase of NPS as a result of K-

fluorescence. The frequency dependence of results from TK  u, v which has a low- pass characteristics. The effects of K-fluorescence are therefore most significant at low frequencies.

2.2.2.4. Stage 3: Spreading of Secondary Quanta in the X-Ray Converter This stage describes the stochastic spreading of secondary quanta (e.g., optical photons) in the converter (e.g., scintillator) according to the converter MTF, denoted T3. This process results in degradation of the overall MTF of the system, and is responsible for the well known gain- resolution tradeoff considered in detector design – i.e., a thicker scintillator gives higher gain

( gg12) but more blur (reduced T3). To mitigate this effect, a scintillator such as CsI:Tl is grown in a micro-columnar structure to reduce the lateral spread of optical photons by partial internal reflection at columnar interfaces. Several authors have modeled such effects using Monte Carlo

79 calculations of photon transport. In the work reported below, the converter MTF, T3, was determined from an empirical fit to the measured MTF. The MTF associated with stage 3 was obtained by dividing out the MTF contributions at stage 2 (K-fluorescence)78 and stage 5

(detector pixel apertures) and fitting the result to a combined Gaussian and Lorentzian empirical form:

ff22  uv 1 a T f, f a e a2 1 (2.59) 31 uv 22.  ffuv  1 H 2

The fit parameter a1 and a2 are 0.2725 and 0.9099, respectively. The parameter H was related to the surface density, s, of CsI:Tl by yet another empirical fit: He 0.739 0.00223s , derived by

Siewerdsen et al.47

For a spreading stage, the mean fluence is unchanged from the previous stage:

q q g g (2.60) 3 0 1 2

For a stochastic spreading stage, the noise transfer is:

2 Sff3 u,,, v  Sff 2 u v  qTff 2 3 u v   q 2 qggPff,  1  qggT2 ff ,  qgg (2.61) 0 1 2K u v 0 1 2 3 u v  0 1 2 2 q0 g 1 g 21 PK f u , f v T 3  f u , f v 

As is typical with stochastic spreading stages, only correlated noise is modulated by T3.

2.2.2.5. Stage 4: Coupling of Secondary Quanta to Detector Apertures Stage 4 is a binomial selection stage describing the coupling of secondary quanta (e.g., optical photons) to the detector pixel apertures (e.g., a-Si:H photodiodes). The conversion of optical photons to electrons is assumed deterministically one-to-one.

The signal and noise transfer associated with stage 4 is given by:

q q g g g (2.62) 4 0 1 2 4

S f,, f g2 S 2 f f 2 q 4 u v 4 3 u v g4 3 (2.63) and 2 qggg0 1 2 41 gPffT 4K u , v 3  ff u , v 

This stage is in fact a combination of consecutive binary selection stages that include the transmission of optical photons from the scintillator, through various overlying passivation and semiconductor layers of the photodiode, conversion to electron-hole pairs, and collection of charge from the photodiode. It is straightforward to show that such a consecutive series of binary

selection stages can be treated in conglomeration, with a mean gain given by g4 , variance by

gg44(1 ) , and noise transfer according to Eq. (2.63).

2.2.2.6. Stage 5: Integration of secondary quanta by detector pixel aperture Integration of optical photons by the photodiode is described by a deterministic spreading stage governed by the spatial extent of the sensitive aperture. Mathematically, this corresponds to multiplying the signal (fluence of optical photons) by a 2D rect function spanning the sensitive area of one photodiode aperture and integrating to a single value (conversion from fluence to a scalar number of quanta). This is in turn equivalent to convolution with a rect function of unity

uv height, denoted as , , in the spatial domain. The terms a and a represents  pd, u pd, v aapd,, u pd v the pixel aperture in the u and v directions, respectively. Note that the photodiode aperture is not necessarily equal to the pixel size (sampling distance), related instead by the “fill factor”

(denoted ff) describing the fraction of pixel area that is sensitive to optical photons:

aa ff  pd,, u pd v . (2.64) aauv

In the Fourier domain, this operation corresponds to multiplication with a scaled sinc related to the photodiode aperture:

aaTffaapdupdv, , 5( u , v ) pdupdv , , sinc( af pduu , )sinc( af pdvv , ) (2.65)

For simplicity, the aperture is assumed to be square, i.e., apd a pd,, u a pd v . The signal and noise transfer is straight forward:

q q a2 g g g , (2.66) 5 0pd 1 2 4 noting that the signal is now dimensionless (converted from fluence to a dimensionless number of quanta at stage 5), and

Sff, qaggg4 1 gPffTff , 2 , Tff 2 , . (2.67) 5 uv 0 pd 1 2 4 4 Kuv  3 uv 5  uv 

The effect of aperture size (relative to the sampling distance) is non-trivial, noting that ff does not simply enter the cascade as a binary selection stage. Rather, the effects of sub-unity ff are tied intrinsically to mean signal, spatial resolution, noise aliasing. A larger fill factor (large active

32 area) corresponds to an increase in mean signal [Eq. (2.66)] but a reduction in MTF [Eq. (2.65)].

As shown in the subsequent stage, the tradeoff is compounded by effects of aliasing.

2.2.2.7. Stage 6: Sampling of detector signal Stage 6 corresponds to sampling of the detector signal at discrete pixel locations. The mean signal is unchanged:

q q a2 g g g (2.68) 6 0pd 1 2 4

The noise transfer characteristic is described by convolution with the Fourier transform of the impulse train:

4 2 2 (2.69) Sff6 uv,  qaggg 0 pd 1 2 4 1  gPffTff 4 Kuv , 3 uv , Tff 5 uv ,  IIIff 6  uv ,  where ** again denotes a 2D convolution. The sampling grid is spaced at intervals of the detector pixel pitch, (au,av),which is distinct from the aperture size (apd.u,apd,v) in Stage 5 and related by the fill factor as in Eq. (2.64). The effect of sampling on the NPS at stage 6 amounts to an addition (aliasing) of noise from spatial frequencies above the Nyquist frequency. For band- limited systems (e.g., those experiencing a strong degree of blur at stage 3, as with indirect-

detection FPDs) such that S3 is low at high-spatial frequencies, the aliasing effect is fairly small.

For higher-resolution systems, however, (e.g., direct-detection FPDs, for which the MTF at stage

3 is near unity), the aliasing effect is considerable; in fact, the reason that an ideal system (i.e., one with no blur aside from the size of the pixel aperture) exhibits an NPS at Stage 6 that is

2 uncorrelated (i.e., “white”, as opposed to ~T5 ) is entirely due to aliasing. Moreover, a reduction in ff amounts to an increase in aliased noise.

2.2.2.8. Stage 7: Readout with Additive Electronic Noise Stage 7 is modeled as a deterministic gain stage with the gain factor equal to unity. This stage accounts for noise introduced in the process of readout. There are numerous potential sources of electronic noise, including photodiode dark current, TFT thermal noise, and analog-to-digital conversion / quantization noise, but the dominant source tends to be that of the integrating amplifier. As shown in Table 2.1, a typical value of electronic noise measured (in the absence of

x-ray irradiation) and/or specified by the FPD manufacturer is  7 ~1000 – 5000 e.

The electronic noise is assumed to be uncorrelated white noise and independent of the signal.

The mean fluence is therefore unaffected:

2 q7 q 0 apd g 1 g 2 g 4 (2.70)

The NPS is simply an addition of the “additive” electronic noise, denoted Sadd:

Sff ,  qaggg4 1  gPffTff , 2 , Tff 2  ,  IIIff ,   S 7uv 0 pd 1 2 4 4 Kuv 3 uv 5 uv 6 uvadd

(2.71)

Note that NPS at stage 7 exhibits two components – the first dependent on the incident x-ray fluence, and the second independent of such. At high exposure, the first term therefore tends to dominate (sometimes referred to as “input-quantum-limited”), while at low exposure, the NPS is dominated by the electronic noise.

2.2.2.9. Stage 8: Post-readout Binning / Sampling The following stages correspond to mathematical processes of filtered-backprojection (FBP) reconstruction. Many are deterministic filtering steps that do not affect the mean signal, so signal

34 propagation is understood to be taken to be the same as q7 unless otherwise stated. Each stage does, however, affect the spatial frequency content of the signal and noise, so the transfer functions of each mathematical step, with a final sampling stage constituting an irreversible process that aliases signal and noise in a manner governed by these transfer functions.

Post-readout binning and sampling (as well as post-reconstruction binning/sampling, as will be shown later) is optional, and may be applied at various stages in the imaging chain and result in nontrivial effects on the NPS, MTF, and NEQ. A detailed treatment of the effect of binning/sampling at various stages in the imaging chain was given by Tward et al.71

Binning at stage 8 corresponds to averaging of signals within a 2D rectangular aperture, with

aperture interval denoted by Au and Av (e.g., AAuv2 corresponds to 2x2 binning). Note the difference in signal transfer between the binning stage and the aperture integration step in Stage

5 (averaging vs. integration), i.e., no aperture width is multiplied in the binning stage, and there is no change in the units of signal. Subsequently, the signal is sampled at a desired interval, denoted by Bu and Bv, which may be the same or different from Au and Av. The two steps are described by a transfer function and a sampling impulse train, respectively, as follows:

T8  fu , fv   sincfu au Au sincfv av Av  (2.72)

III8  fu , fv   au Bu IIIav Bv fu av Bv IIIav Bv fv  (2.73)

The resulting signal and NPS are therefore:

2 q8 q 0 apd g 1 g 2 g 4 (2.74)

2 S8  fu , fv   S7  fu , fv T8 u,v**III8  fu , fv  (2.75)

2.2.2.10. Stage 9: Log-Normalization This stage describes the effect of flood-field correction, and the subsequent logarithmic transform of the signal. As shown by Barrett,80 the logarithm essentially normalizes the signal by its mean. This approximation holds when the signal is small (i.e., within a small range of exposures consistent with assumptions of linearity and locality described above). This stage can thus be treated as a gain stage, with the gain factor equal to

11  2 . (2.76) q8 q 0 apd g 1 g 2 g 4

The NPS is therefore:

1 (2.77) S98 fu,, f v  2 S f u f v  2 q0 apd g 1 g 2 g 4 

At this stage, the signal changes from a count (number of quanta) to a fraction (attenuation) – both dimensionless, but related by the mean signal scale factor in Eq. (2.76). Note also the apparent change in the dependence of NPS on exposure: up through stage 8, the NPS is proportional to incident fluence, but at stage 9 and beyond, the NPS is inversely proportional to incident fluence (i.e., increasing the exposure reduces the NPS) due to the normalization by mean signal squared.

2.2.2.11. Stage 10: Ramp Filter Application of a ramp filter to the projection data constitutes a deterministic spreading stage with transfer function:

T10 fu , f v  fu (2.78)

Intuitively, the ramp filter compensates for the radial sampling density in the Fourier domain according to the Central Slice Theorem. The filter value at zero frequency is usually set to a non- zero value in practice to preserve the correct mean image value. In describing the propagation of noise, however, we require no such “fix” at zero frequency, and the transfer function is treated as a pure ramp. The NPS is given by:

2 S10 fu , fv   S9  fu , fv T10 fu , fv  (2.79)

Stage 10 therefore imparts a major change in spatial frequency content of the NPS, changing from a typically band-limited, low-frequency NPS up through stage 9 to a markedly band-pass or high-pass NPS at stage 10. This corresponds to the amplification of high-frequency noise by the ramp filter. The signal gains unit of mm-1, while the NPS at this stage is unit-less.

2.2.2.12. Stage 11: Apodization filter The amplification of high frequency noise by the ramp filter is optionally mitigated by the application of an apodization filter. A variety of smoothing filters can be applied at this stage – for example, a cosine window given by:

T11 fu , fv   hwin  1 hwin cos2fuau Bu  (2.80)

where hwin can be adjusted to provide different degrees of smoothing and usually ranges from

0.5-1.0, with 1.0 corresponding to an all-pass filter with no smoothing applied. Decreasing hwin below 0.5 will cause undesirable side-lobes at high frequencies, observed as ringing in the reconstructed image. The NPS is transferred deterministically as

2 S11 fu , fv   S10 fu , fv T11 fu , fv  (2.81)

The resulting NPS is typically mid-pass, equal to zero at zero frequency, increasing linearly at low-to-mid frequency according to the ramp filter, and then descending at mid-to-high frequencies according to the apodization filter.

2.2.2.13. Stage 12: Interpolation of Filtered Projection Data The interpolation step corresponds to voxel-driven reconstruction at arbitrary voxel locations. A commonly used interpolation scheme involves bilinear interpolation, with transfer function:

2 2 T12 fu , fv   sinc fu au Bu sinc fv av Bv  (2.82)

The NPS is given by:

2 S12 fu,,, f v  S 11 f u f v T 12  f u f v  (2.83)

Here again, the NPS is modulated at high frequencies by the interpolation filter. The apodization and interpolation filters at stages 11 and 12 are essential in mitigating high frequency noise aliasing incurred by both 2D (stages 6 and 8) and 3D (stages 14 and 15, shown below) sampling stages.

2.2.2.14. Stage 13: 3D Backprojection The backprojection process can be divided into three substages:

(a) Magnification:

First, the signal needs to be scaled from the projection domain to the reconstruction domain, which differs by the magnification factor, M. This corresponds to a change of axes in the spatial

38 domain from uv,  to M x22 y, Mz . The Fourier transform therefore follows the scaling theorem, yielding

S f/,/ M f M  S f, f   12 rz (2.84) 13M r z M 2

22 where fr f x f y .

(b) Backprojection of Individual Projections

Backprojection corresponds to smearing of the signal along the projection angle across the field of view (FOV). If the FOV in the reconstruction is infinitely long, the transfer function of this step corresponds to a  function along the radial spoke in the Fourier domain that is sampled by

th that projection. Consider the i projection with a coordinate system  fxi,, f yi f z  rotated by the

angle of the projection, the transfer function is a radial spoke along the f yi axis written as

 ()fxi . For real systems with finite FOV with source-to-detector distance (SDD) as the upper bound, the smeared projection is truncated (or multiplied) by a rect function of width FOV and unity height in the spatial domain, which corresponds to convolution with a sinc function in the

Fourier domain. The sinc function convolved with the delta function yields the function itself; therefore, the transfer function of backprojection of a single projection is given by:

T13i f xi, f yi , f z   f xi  FOV sinc FOV  f xi  FOV sinc FOV  f xi  . (2.85)

Noise transfer is deterministic and scaled by the factor 1/FOV. The NPS of a single projection is therefore:

2 1  SfffSff13i xi,,,,, yi z  13 M xi z Tfff 13 i xi yi z  (2.86) FOV 

which is a scaled version of S13M modulated by a 3D sinc function. When the FOV approaches

infinity, T13i is equal to a delta function along xi, i.e.,

limT13i f xi , f yi , f zi   f xi  . (2.87) FOV   

The backprojection operation does not change the units of the signal (mm-1). The signal spectrum, however, now spreads across 3D instead of 2D. The NPS therefore has units of signal2mm3, which for the case of signal with dimensions of attenuation coefficient (mm-1, as in CT) can be simplified to mm with the unit of signal being mm-1. In the work reported below and in subsequent chapters, the temptation to reduce the units of the NPS to mm-1 in this way is avoided, and it will be written typically as units [2mm3] where  is understood to connote units of signal

(e.g., attenuation coefficient, mm-1 or Hounsfield Units, HU).

Each projection can be assumed to be independent, ignoring effects like image lag. The signal transfer follows a summation over the total number of projections, m, yielding a transfer function

of summed spokes denoted T13i :

M m T13f x , f y , f z  T13i f xi , f yi , f z  (2.88) m i1

1 For a large number of projections, T13  fx,, f y f z   M , which exactly cancels out the ramp fr filter up to Nyquist frequency.

The noise of each projection is added similarly:

2  M m S13 fx,,,, f y f z   S 13 i f xi f yi f z  m i1 (2.89) 2  M m 1 2  S f,,, f T f f f 12M yi z  13 i xi yi z  mdi1

 M 2 1 which can be reduced to S13 fx,,, f y f z  S 12 M f r f z  in the limited of a large number of mf projections.

Although the backprojection process is represented as a deterministic spreading stage, the noise is not simply blurred by the transfer function. We can, however, define a noise "transfer function" for the backprojection projection process as:

M 2 1 2  f , f   . (2.90) 13 z m f

When projections are sparse, the above simplification to a 1/f relationship no longer holds, and the effect of individual spokes must be more carefully considered. The MTF of the system and

NPS of the image then appear as radial spokes in the Fourier domain, corresponding to streaking in the reconstruction, an artifact commonly observed with undersampled data.

2.2.2.15. Stage 14: 3D sampling The final step in forming the 3D image (subject to optional binning, below) is to sample the 3D reconstruction at discrete voxel locations, with voxel size denoted by b. The voxel dimensions in the axial domain (bx and by) are typically equal, and that in the longitudinal domain ( ) is also referred to as the slice thickness. The corresponding Fourier domain impulse train is a 3D array of delta functions:

III14f x , f y , f z  bx IIIbx f x by IIIby f y bz IIIbz f z  (2.91a)

The NPS is therefore:

S14f x , f y , f z  S13f x , f y , f z *** III14f x , f y , f z  (2.91b) where *** denotes 3D convolution. Analogous to the 2D sampling stage (stage 6), 3D sampling would incur aliasing effects and the choice of reconstruction voxel size should be carefully considered, e.g., upsampling at this stage provides anti-aliasing, but at the expense of increasing computation time and data size.71

2.2.2.16. Stage 15: Post-Reconstruction Binning and Sampling Binning and resampling of the 3D reconstruction is optional – a common example being to average slices. This stage is similar to Stage 8, with the exception that operations are performed in the 3D reconstruction domain. The same notation of binning and sampling interval A and B are used here, with changes in the subscript to denote association with (x,y,z) axes. The transfer functions for this stage are:

T15f x , f y , f z  sincf xbx Ax sincf yby Ay sincf zbz Az  (2.92)

III15f x , f y , f z  bx Bx IIIbx Bx f x by By IIIby By f y bz Bz IIIbz Bz f z  (2.93)

The NPS of the 3D image reconstruction is therefore:

2 S15f x , f y , f z  S14f x , f y , f z T15f x , f y , f z *** III15f x , f y , f z  (2.94)

Note again the possibility for yet more aliasing introduced by a poor choice (mismatch) between the binning and sampling intervals.

2.3. The Resulting 3D Fourier Domain Performance Metrics The cascade of stages constituting 2D image formation and 3D image reconstruction are illustrated in Fig. 2.3

Figure 2.3. Illustration of the cascaded systems model for 2D image formation in a flat-panel detector followed by 3D filtered backprojection for CBCT. (Published with permission from Medical Physics)

Considering the propagation of mean signal in the cascade of Fig. 2.3, we see that the system

MTF is governed by that of the projection system (FPD) in combination with the apodization and interpolation filters applied in 3D image reconstruction:

Tfff xyz,,,,,,,,,,  TfffTfffTfffTfff3 xyz 5 xyz 11 xyz 12  xyz  (2.95a)

where conversion of axes from (fu,fv) to (fx,fy,fz) for T3 and T5 follows the simple relationship

22 fu f x f y / M and fvz f/ M . The MTF is symmetric in the fx-fy plane but may be

43 asymmetric in fz, since the apodization filter (T10, T11) is typically applied in the u direction only.

Moreover, note that the ramp filter does not enter into the system MTF.

Expanding all terms in Eq. (2.94), the 3D image NPS given by:

qaggg41 gPT 2 T 2  III  S T 2 ** III  0pd 1 2 4 4 K 3 5 6 add  8 8 1 S T2 T 2 T 2 2****** III T 2 III 2 10 11 122 13 14 15 15 q a2 g g g M  0pd 1 2 4  S T2 ** III  7 8 8 2 2 21 2 2 2 T10 T 11 T 122 13****** III 14 T 15 III 15 2 q a g g g M  0pd 1 2 4  2 tot 1 2 2 2f 2   2 S7 T 8******** III 8 T 11 T 12 III 15 T 15 III 15 22 q0 f mM apd g1 g 2 g 4  (2.95b)

2 2 where the ramp filter, T10 , and the backprojection operator, 13 , are written out explicitly in the third line of the equation. The function argument (fu,fv) or (fx,fy,fz) for each term is dropped for brevity. The 3D NPS exhibits a band-pass characteristics in the fx-fy plane because the ramp filter

(stage 10) and backprojection (stage 13) were applied only in that plane. The NPS along fz, however, is low-pass due to the detector MTF and interpolation filter. The units of the 3D signal is that of attenuation coefficient (mm-1), giving the NPS units of (mm-1)2mm3, which might be simplified to mm, but as described above, will be more typically written as [2mm3].

Substituting Eqs. (2.95) and (2.96) into Eq. (2.17), the 3D NEQ is given by:

T 2 NEQ f22 f . (2.95c) xyS

The units for 3D NEQ are mm-12mm3)-1 which can be reduced to mm-2, the usual units of fluence for NEQ. One could in principle define a 3D DQE simply by normalization of the NEQ by the incident fluence as in Eq. (2.15).

It is sometimes useful to separate the effects of quantum and electronic noise. Assuming the effects of aliasing are small, the quantum noise power spectrum, SQ, is given by:

qaggg41 gPT 2 T 2  IIIT 2 ** III  0pd 1 2 4 4 K 3 5 6 8 8 1 S T2 T 2 T 2 2****** III T 2 III Q 2 10 11 122 13 14 15 15 q a2 g g g M  0pd 1 2 4  2 tot 1 4 2 2 2 2 2 f 2  2 agggpd1241 gPTT 435 K  ** IIIT 68 ** IIITT 81112 *** III 15 T15***III 15 22   q0 f mM apd g1 g 2 g 4  (2.96a)

The electronic noise power spectrum, SE, is given by:

~    1  f 2   tot  2 2 2 2 S E  SaddT8 **III8T11T12 *** III15T15 *** III15 . (2.96b)  q 2  2 2 2 f  0  mM a pd g1 g 2 g 4   

An analytical expression of NPS in this form elucidates the dependence of image noise on various factors in the imaging chain, providing a valuable tool in optimizing imaging performance.

Extension of the basic forms for the 3D MTF and NPS are given in the Appendix of this chapter, including effects associated with focal spot blur, x-ray scatter, and cone-beam artifact.

2.4. Task-Based Image Quality Metric - Detectability Index As introduced in Chapter 1, image quality should be defined with respect to the imaging task.

Task-based performance can be reported in terms of detectability index, denoted as d', as a combination of NEQ, the imaging task, and a particular observer model:

2  T fx,,,, f y f z W Task f x f y f z  d'2  df df df (2.97) 3D x y z SQ f x,,,, f y f z  S E f x f y f z 

where T, SQ and SE are the system MTF, quantum and electronic noise-power spectrum as defined in previous sections. The subscript "3D" indicates a fully volumetric detectability index, which is distinct from the 2D "slice" detectability index to be derived in Chapter 4. Such a 3D detectability index assumes an observer who can somehow grasp the full dimensionality and correlations within volumetric data, rather than, for example, scrolling slices on a 2D display.

The task function, WTask , essentially conveys the frequencies of interest, with various specific forms relating to “detection” and “discrimination” provided below. Intuitively, detectability can be interpreted as a description of how well the frequencies of interest are transmitted through the imaging system.

The tasks considered in this work fall under the broad category of classification tasks within the framework of statistical decision theory on binary hypothesis testing. The task function is defined as the difference of the Fourier transforms of the spatial domain representations (i.e., spatial domain object functions, denoted O) of two hypotheses. Two sub-categories of such tasks include detection and discrimination. For a detection task, a "signal-present" hypothesis is represented by a volume-of-interest (VOI) or region of interest (ROI) containing the signal, and a

"signal-absent" hypothesis is represented by a background only VOI or ROI. For detection on a

uniform background, WTask can be written as:

WTask    FT[] O (2.98) where  represents the contrast between signal and background.

For a discrimination task, the two hypotheses correspond to discrimination between "signal A" and "signal B". The task function is accordingly written as:

WTask A  FT[][] O A  B  FT O B . (2.99)

Generalizations to detection and discrimination tasks on cluttered background are derived in

Chapter 4. Formulating classification tasks according to binary hypotheses is perhaps the simplest means of describing the spatial frequencies pertinent to a given task. Extensions of classification tasks to multiple hypotheses, and modeling of another broad category of imaging tasks – estimation - are subjects of ongoing research.

The detectability index in Eq. (2.97) corresponds to a prewhitening (PW) matched filter observer which is assumed to be able to completely decorrelate image noise. The PW model is so called, because in classic models of image quality (i.e., not including non-idealities, such as Poisson excess, stochastic spreading of optical photons in the converter, electronics noise, and aliasing), the NPS would be thought to go exactly as MTF2, such that the integrand cancels out to a constant “pre-whitened” form. The model can be extended to the non-prewhitening (NPW) matched filter observer81 which, instead of assuming complete knowledge of background noise, applies a detection template in the form of the signal:

2 ()T W2 df df df ' 2  Task x y z d3D  (2.100) ()()S S  T  W2 df df df  Q E Task x y z

These two models can be extended to include an eye filter, E(f), and internal noise, Ni, to account for response characteristics of the human visual system.82, 83 The PW model extended in this way is denoted PWEi and written as:

ETW22() d'2 Task df df df (2.101) 3D 2 x y z ESSN Q E i

Similarly, the NPW model with the eye filter is denoted NPWE and written as:

2 E2 ()T W2 df df df ' 2  Task y x z d3D  . (2.102) E4 ()(S S  T  W)2 df df df  Q E Task y x z and can be extended further to include the internal observer noise, denoted NPWEi and written as:

2 E2 ()T W2 df df df ' 2  Task x y z d3D  , (2.103) ESSTWN42()()    df df df  Q E Task i x y z

The equations above for the 3D detectability index are drawn from previous work58 and could in principle be related to task performance in 3D images. However, there are currently no established 3D eye filter models, no models shown to correspond to a human observer scrolling slices or viewing a slice montage, and little established experimental methodology for reliable human observer performance assessment in fully 3D images. The exact form of eye filters and internal noise appropriate to fully 3D data (e.g., slice scrolling, multi-slice montage presentation, or volumetric viewing) are subjects of ongoing work in the modeling of image perception, and

48 are therefore not invoked in this work. For these reasons, a 2D “slice” detectability index is derived from the fully volumetric detectability index in Chapter 4, with the appropriate eye filter and internal noise model adapted from previous research. The fully 3D detectability index is useful within the abstract context of a model observer who somehow appreciates the full dimensionality and correlation of the 3D image.

The above observer models were chosen in this work based on established research demonstrating good correspondence with human observers in 2D radiography.84 It is not the purpose of this work to advance or improve any particular observer model. More complex models are areas of active research in perception science and could possibly be incorporated into the image quality framework developed in this dissertation.

Appendix.2A. Extensions to the Basic Model 2A.1. Focal Spot Blur An extension to the basic model presented in this chapter includes the effects of x-ray focal spot, which imparts a deterministic blur characterized by Tspot. Assuming a Gaussian shape to the focal spot intensity profile, the transfer function for this effect can be modeled roughly as a

Gaussian:85-87

2  M 1 a f22  f spot u v Tespot  (2.104a)

where M is the magnification factor, and aspot is the width of the Gaussian focal spot distribution in the spatial domain. Alternatively, the focal spot could be modeled as a uniform source

24, 88 distribution in a square aperture of width aspot, yielding a sinc function Tspot as follows:

    . (2.104b) Tspot sinc M 11 a spot f u  sinc  M   a spot f v 

The effect of Tspot on MTF, NPS, and NEQ is considered in combination with scatter in the following section. The term “deterministic” blur in reference to Tspot means that while the effect does impart blur on the system (i.e., the numerator of the NEQ), it does not affect the NPS – i.e., it does not appear in the denominator of the NEQ – since a random distribution of independent quanta arising from an extended source is still a random distribution of independent, uncorrelated quanta. The system MTF may be degraded by Tspot, but the quantum noise NPS is unaffected.

2A.2. X-Ray Scatter Another effect that can be incorporated in the model is x-ray scatter resulting from Compton interactions of x-rays in the object, which results in contrast reduction and artifacts (e.g., shading) in the image. Scattered radiation can be modeled as arising from an "effective scatter point

89, 90 source" that is at a distance, yscat, away from isocenter and subjected to the same signal and noise transfer characteristics as primary quanta. A comprehensive treatment of the effect of scatter on MTF, NPS, and DQE was given by Kypiranou91 and Siewerdsen87, with the results

summarized below. Scattered ( qs ) and primary ( q p ) quanta are frequently related by the scatter- to-primary ratio, SPR, defined as:

q s SPR  (2.105) q p

The total quanta incident on the detector is simply the summation of the primary and scattered:

s p p qtot  q  q (1  SPR ) q (2.106)

The scatter MTF, Tscatter, can be determined experimentally from the difference in ESF measured with and without the scattering source (object) in the field of view. It tends to exhibit a rapid falloff to zero, corresponding to long-range artifacts such as shading and referred to as a “low- frequency drop.” The system MTF therefore carries the combined effect of scattered and primary radiation:91

qqps TTTTTTT3 5spot scatter 11 12 qp q s q p q s  (2.107) 1 SPR TTTTTT3 5spot scatter 11 12 11SPR SPR

with argument (fx,fy,fz) dropped for all terms for brevity.

The projection NPS, S7, from Eq.(2.71) now needs to account noise contributions from scattered photons:

SSSSps   7 77add qagggp4 p p p1  gPffTff p p , 2 , Tff 2 ,  IIIff ,  (2.108) pd1 2 4 4 K  u v 3 u v 5 u v 6  u v  qagggs4 s s s1 gPffTffTffT s s , 2 , 2 , 2 ff ,  IIIff ,  S pd1 2 4 4 K  u v 3 u v 5 u v scatter u v 6  u v add where the gain factors can be computed separately for the primary spectrum (indicated by superscripts p) and the (slightly) shifted scattered spectrum (indicated by superscripts s).

The total NPS follows a simple extension from Eq.(2.95):

 S T2 ** III  7 8 8 2 2 21 2 2 S2 T10 T 11 T 122 13****** III 14 T 15 III 15 (2.109) p22 p p p s s s s M q a g g g q a g g g  0pd 1 2 4 0 pd 1 2 4 

with S7 given by Eq.(2.99).

The effects of focal spot blur and scatter on 3D NEQ can be derived by direct substitution of Eq.

(2.107) and Eq. (2.109) into Eq. (2.15).

Various assumptions can be made to simplify Eq. (2.107) and (2.109). The energy shift in scattered spectrum is small such that the gain factors for primary and scattered photons can be assumed to be the same. The additional blur caused by scatter can be assumed to be small, in which case scatter simply imposes a reduction of contrast by a factor of 1/(1+SPR). This effect is captured in the system MTF as:

1 TTTTTT (2.110) 1 SPR 3 5spot 11 12

Similarly, the NPS can be reduced to:

qaggg41 gPT 2 T 2  III  S T 2 ** III  tot pd1 2 4 4 K 3 5 6 add  8 8 1 S T2 T 2 T 2 2****** III T 2 III 2 10 11 122 13 14 15 15 q a2 g g g M  tot pd 1 2 4  4 2 2 2 agggpd1 2 41 gPT 4 K 3 T 5  III 6  S add T 8 ** III 8 11    2 2 2 22 p 2 TTT10 11 122 13******III 14 T 15 III 15 1 SPR q2 M   0 a g g g  pd 1 2 4  1  S p 1 SPR

(2.111)

where qtot follows Eq.(2.106).

The resulting NEQ is then given by:

2 2 T 1 TTTTT3 5spot 11 12  NEQ  (2.112) S1 SPR S p

The above simplifying assumptions reveal the dependence of MTF, NPS, and NEQ on scatter – viz., scatter increases the quantum noise component of the projection NPS by a factor (1+SPR) but decreases the 3D reconstruction NPS by a factor of 1/(1+SPR). A contrast reduction of

1/(1+SPR) is reflected as a scale factor in the system MTF, resulting in an overall reduction in

NEQ by 1/(1+SPR).

2A.3. Cone-Beam Artifact For CBCT acquisition from a planar circular orbit, there is a well known artifact associated with violation of Tuy’s condition92 for exact reconstruction. Specifically, there exists a cone of spatial frequencies that are unsampled for any spatial locations away from the central plane.

Accordingly, the 3D MTF and NPS exhibit a null cone about the fz axis. The spatial frequencies

53 occupied by the null cone is space-variant, i.e., locations further away from the central plane of have a bigger null cone. Furthermore, the null cones for locations away from the axis of rotation exhibit increasing rotational asymmetry about the fz axis.

This effect can be simply modeled by multiplying the fully sampled 3D MTF and NPS by a binary mask, denoted Mcone, corresponding to whether the spatial frequency is sampled, i.e.,

Tconebeam TM cone (2.113a)

and Sconebeam SM cone . (2.113b)

Projection weightings associated with the reconstruction algorithm (e.g., the cosine weights in the Feldkamp algorithm93) need to be considered separately.

This approach can be similarly extended to other missing data cases, e.g., tomosynthesis, provided that the sampled frequencies are "fully sampled". Sparse projections require a separate model introduced in Chapter 3.

Chapter 3 Anatomical Background and Generalized Detectability in Tomosynthesis and Cone-Beam CT 3.1. INTRODUCTION Image “noise” may be broadly interpreted to describe variations in the image that impede the imaging task, including purely stochastic variations (such as quantum noise) as well as deterministic (or semi-deterministic) variations such as structured pattern noise, correlated electronic noise, and artifacts of image processing / reconstruction.94-97 Similarly, anatomical background (i.e., image variations arising from anatomy not associated with the structure of interest) presents a major impediment to detectability in various imaging applications (e.g., breast,95 chest,96, 98 dental,99 angiography,100 and liver101), often outweighing other noise sources, such as quantum or electronic noise.97 Thus, anatomical background noise presents an important factor to be incorporated in the description of system performance – not only in 2D projection imaging94, 98, 102-104 (in which 3D anatomy is entirely superimposed in the 2D image) but also in tomosynthesis (giving partial rejection of out-of-plane structure)105, 106 and CBCT (in which anatomical background is minimized to residual in-plane structure). As detailed below, the latter represent a continuum in the reduction of anatomical background in proportion to the extent of the source-detector angular orbit.

The characterization of anatomical background has been an area of considerable interest in 2D medical imaging applications, such as mammography and chest radiography. For example,

Burgess et al.107 described breast structure by an empirical power-law relationship in proportion to /f, with the power-law exponent measured approximately to be 2.8 in digitized film-screen mammograms.108 Heine et al.109, 110similarly showed that the statistical nature of mammograms

55 can be depicted by white noise, filtered with a parametric transfer function of the form 1/fand measured the value of  to be in the range 2.8-3.0 in digital mammograms. In 2D chest radiography, Samei96, 103 showed the predominance of anatomical background noise on detectability, and Richard et al.55 characterized such in terms of power-law noise with  in the range ~3.5–4. Such analysis was subsequently extended to dual-energy chest radiography as well,111 where reduction of anatomical noise through tissue discrimination (i.e., the soft-tissue image with bones removed) was associated with a decrease in  and significantly increased detectability index.

The power-law depiction of anatomical noise has been extended to 3D imaging applications as well. In the context of breast tomosynthesis and CBCT, for example, Glick et al.112 included an anatomical noise-power spectrum proportional to 1/f 3 in investigating the optimal scintillator thickness for a CT mammography system. Gong et. al.113 applied the same to simulate a digital breast phantom. Metheany et al.114 extended Burgess’ analysis to breast CT, confirming a difference of 1 between  in breast CT slices and mammograms (viz.,  = 3.01 in mammograms, = 1.86 in breast CT slices, and  = 2.06 in breast CT slices segmented such that adipose and glandular tissue voxels carried values of 1 and 2, respectively). Engstrom et al.115 similarly compared breast tomosynthesis slices and projection data and found reduced background in tomosynthesis slices evidenced by a lower value of  (2.87 compared to 3.06). In

3D chest tomosynthesis. Yoon et al.116 measured  and  in an anthropomorphic phantom and applied the findings to theoretical analysis of detectability as a function of  , and the extent of the source-detector orbit.

The work described below extends power-law analysis of background clutter to the full continuum of source-detector orbital extent, from a single projection (2D radiograph) to limited- arc tomosynthesis and fully 3D CBCT. The work involves experimental analysis based on a physical phantom presenting power-law noise and theoretical analysis of the corresponding detectability index for a variety of imaging tasks. While the phantom is not intended to simulate a particular anatomical site (e.g., breast or chest), the approach offers a general experimental and theoretical basis with specific implications for systems currently under development for diagnostic and image-guided procedures. The study extends 3D cascaded systems analysis of tomosynthesis and CBCT58 to quantify the tradeoffs among anatomical background, quantum, and electronic noise as a function of orbital extent, number of projection views, and dose. The generality as well as the limitations of the current study are addressed, and the implications for site-specific 3D imaging applications (e.g., the breast or chest) are described.

3.2. METHODS The theoretical and experimental methods are summarized below. First, an analytical basis drawn from fractal theory is presented for the design of phantoms presenting power-law noise in

2D projections and 3D reconstructions, along with an example phantom constructed of various diameter spheres. Second, image acquisition scenarios and measurement of signal, noise, and power spectrum are discussed. Finally, the cascaded systems model for tomosynthesis and CBCT introduced in Chapter 2 is extended to include anatomical background noise in a “generalized” description of the NEQ and detectability index.

3.2.1. Analytical Basis and Design of a “Clutter” Phantom for Power-Law Noise

Anatomical background power spectra intrinsic to the object, denoted as S Bobj are frequently characterized according to the fairly ubiquitous power law relationship [Eq. (3.1)] used to describe a wide range of stochastic processes:117

 obj S Bobj f   (3.1) (af ) obj

where obj describes the magnitude of fluctuation, obj denotes the degree of correlation of the variations, and f is the spatial frequency coordinate. The term a is a scale factor (taken as 1) carrying units inverse to frequency (e.g., a = 1 mm for f in units of mm-1), effectively acting as an aperture rendering the denominator dimensionless despite different values of obj. Such a power-law relationship has been used in the empirical description of anatomical background in breast108, 114, 115 and chest116 imaging. In this work, a physical phantom was designed to present the same form of power-law spectral density as found in such anatomical sites (i.e., similar obj).

In this way, the current work is intended as a general analysis of power-law noise in tomosynthesis and CBCT in a manner that is pertinent to various clinical applications, but not specific to any organ.

The phantom design was based on the properties of self-similar objects, a special class of which are fractals. Such objects exhibit power-law spectral density with the exponent obj related to the fractal dimension by:117

 3 D  E  obj (3.2a) 2

 obj  2(D  E) 3 (3.2b) where D is the fractal dimension, E is the Euclidean dimension in which the fractal is embedded, and obj is the power-law exponent of the power spectrum of the object. For example, a 3D object with fractal dimension D = 3 (e.g., 3D Hilbert curve with fractal dimension defined as the

108, Hausdorff dimension) gives obj = 3, which is close to that measured for the breast and chest.

114-116 In the current work, for simplicity, we consider objects with a symmetric 3D power spectrum, such as a random collection of spheres. The box-counting method (a means of measuring the fractal dimension) dictates that the number of boxes, denoted N , each of side- lm

117, 118 length lm , required to cover the fractal is related to the fractal dimension by:

1 N  (3.3) lm D lm

The number of boxes of size required to cover a sphere of diameter li > is equal to the

3 3 li lm ratio of their volume 3 . Similarly, a box of size can cover 3 spheres of size < . lm li

Therefore, the number of boxes of size required to cover a tightly packed volume of spheres of different sizes is approximately:

 l 3   l 3  N  n  n  i   n  j  lm lm  li  l 3   l j  l 3  li lm  m  l j lm  m  (3.4) where n is the number of spheres of diameter in the phantom. For to satisfy equation lm

(3.3) when D=3, it follows that:

1 n  (3.5) lm 3 lm which implies that equal volumes of differently sized spheres give a fractal dimension of D = 3 as well as a power-law exponent of obj = 3. In principle, higher or lower fractal dimension (and power-law exponent) can be achieved by varying the proportion and density of various size spheres comprising the object, giving a means of experimentally achieving any particular value of obj. The fairly close agreement in obj between a simple, self-similar fractal and that of actual

(breast or chest) anatomy is interesting: in a statistical sense, such anatomy approximates a collection of self-similar 3D clutter.

From this analytical basis for power-law noise, a “clutter” phantom was constructed consisting of equal volumes of acrylic spheres of five different diameters (15.88, 12.70, 9.52, 6.35, 3.18 mm), approximating an object with obj ~ 3. An equal volume of spheres of each diameter corresponds to a number of spheres in proportion to 1, 1.95, 4.63, 15.63 and 125, respectively. The spheres were randomly mixed to fill an acrylic box of dimension (20 x 20 x 12.5) cm3, as illustrated in

Fig. 3.1. While true self-similarity would require spheres ranging from infinitely small to infinitely large, the physical phantom satisfies self-similarity over a finite spatial frequency range over which the fractal dimension and obj value are expected to hold [Eq. (3.2b)]. The size of the box was consistent with the 3D field of view of the CBCT imaging bench [Fig. 3.1(a)].

Alternative containers were also investigated, e.g., a cylinder of diameter ~16 cm. A cylindrical phantom with a well-matched bowtie filter provided uniform quantum noise, but presented significant non-stationarity in the projection (and tomosynthesis) power spectrum associated with clutter. For all measurements below, a simple rectangular box was found to be a reasonable choice in that its uniform thickness gave the most spatially-uniform attenuation and best overall

60 stationarity in 2D projections and low-angle tomosynthesis images. That is, both the quantum noise and anatomical clutter were more spatially invariant in first- and second-order statistics for the rectangular box phantom compared to a cylindrical phantom, given ROIs sufficiently large to capture the longest-scale fluctuations of interest.

We distinguish between the power spectrum intrinsic to the object [SB-obj(f)] and that measured in

2 an image of the object [SB(f)], with SB()()() f T f S B obj f , where T(f) is the modulation transfer function (MTF) of the system. Similarly, obj and obj refer to power law characteristics in the object domain , while  an  refer to those in the image domain. As detailed below, the clutter phantom was used to measure the background anatomical power spectrum, SB(fx, fy, fz) as a function of the acquisition angle, tot. While it is generally appreciated that increasing tot

“rejects” anatomical clutter from slice reconstructions, these measurements provided quantitative analysis of the behavior of power-law characteristics,  and , as a function of tot.

Figure 3.1. Experimental setup. (a)The imaging bench showing basic components and coordinates for tomosynthesis and CBCT. (b) The clutter phantom was based on properties of fractal self-similarity, by which equal volumes of differently sized spheres were found to give a power-law exponent obj of approximately 3. (c) A polystyrene plate inserted into the clutter phantom was embedded with nylon

61 cylinders used for SDNR measurements. Clutter power spectrum measurements used only the random set of spheres (with the polystyrene plate removed). (Published with permission from Medical Physics)

3.2.2. Image Acquisition and Reconstruction Images were acquired using an experimental imaging bench designed for tomosynthesis and

CBCT from a circular arc source-detector trajectory. As described in previous work,56, 58, 119 the bench consists of an x-ray tube (Rad 94 in a Sapphire housing; Varian Medical Systems, Salt

Lake City, UT), a CsI:TI flat-panel detector (RID-1640A, Perkin Elmer Optoelectronics, Santa

Clara CA), and a motion control system (6K series translation stages, Parker Daedal, Harrison

PA, and Dynaserv rotation motor, Parker Hannifin, Rohnert Park, CA). Images were acquired nominally at 120 kV (1.53mm Al + 1.1 mm Cu added filtration) and 0.63 mAs per projection, giving in-air exposure of 1.84 mR/mAs at isocenter and 0.49 mR/projection at the FPD.73 The bench geometry is illustrated in Fig. 3.1(a). The (x, y, z) axes are fixed in the 3D object space such that (x, y) refers to the axial plane, (y, z) refers to the sagittal plane, and (x, z) refers to the coronal plane. Specifically, y is the inter-slice “depth” direction in 3D tomosynthesis reconstructions.

o In addition to 2D projections (tot = 0 ), 3D images were reconstructed using the FDK algorithm for 3D filtered backprojection for 15 settings representing the spectrum of tomosynthesis (tot =

o o o o o o o o o o o 10º, 20 , 30 , 40 , 50 , 70 , 90 , 120 , 150 , and 180 ) and fully 3D CBCT [tot = 200 (180 +fan),

o o o o 240 , 280 , 320 and 360 ]. For each setting of tot, two acquisition schemes were considered: (1) constant angular separation between projections (subsequently referred to as the “constant-Δθ” scheme, with Δθ =0.45˚), in which case the number of projections used for each reconstruction varies linearly with tot; and (2) constant number of projections (subsequently referred to as

“constant-Nproj”, with Nproj=89), in which case the angular separation between projections varies

62 linearly with angle. The experimental conditions are summarized in Table 1. The constant-Δθ scheme ensures that view sampling effects are constant for all settings of tot, although total dose increases with tot (such that quantum and electronic noise contributions vary with tot). The constant Nproj scheme, on the other hand, imparts the same total dose for all settings of tot

(although view sampling effects vary with tot), illustrating the scenario in which total dose represents a clinical constraint, and the system designer is considering the questions: how large an orbital extent (tot) and how many projections (Nproj)?

 Constant-Δθ Constant-Nproj tot Δθ Nproj Exposure Total Δθ Nproj Exposure Total (mR) Dose (mR) Dose (mGy) (mGy) 10º 0.45º 23 11.22 0.4 0.13º 89 43.44 1.5 20o “ ” 45 21.97 0.8 0.23º “ ” “ ” “ ” 30o “ ” 67 32.71 1.2 0.34º “ ” “ ” “ ” 40o “ ” 89 43.44 1.5 0.45º “ ” “ ” “ ” 50o “ ” 111 54.18 1.9 0.56º “ ” “ ” “ ” 70o “ ” 157 76.64 2.7 0.79º “ ” “ ” “ ” 90o “ ” 201 98.12 3.5 1.01º “ ” “ ” “ ” 120o “ ” 267 130.33 4.6 1.35º “ ” “ ” “ ” 150o “ ” 333 162.55 5.8 1.69º “ ” “ ” “ ” 180o “ ” 401 195.74 6.9 2.03º “ ” “ ” “ ” 200o “ ” 445 217.22 7.7 2.25º “ ” “ ” “ ” 240o “ ” 533 260.18 9.2 2.70º “ ” “ ” “ ” 280o “ ” 623 304.11 10.8 3.15º “ ” “ ” “ ” 320o “ ” 711 347.06 12.3 3.60º “ ” “ ” “ ” 360o “ ” 800 390.51 13.8 4.05º “ ” “ ” “ ”

Table 3.1. Experimental conditions under the constant-Δθ and constant-Nproj cases. The exposure values represent total in-air exposure to the detector, and total dose approximates the dose to the center of a 10 cm diameter water cylinder placed at isocenter – a rough approximation to the 10 cm clutter phantom. A scatter factor of 4.5 and f-factor of 0.9 was assumed. (Published with permission from Medical Physics)

Images were reconstructed at isotropic 0.518 mm voxel size, with a (357 x 357 x 222) voxel volume occupying the interior of the clutter phantom. The voxel size was chosen to adequately resolve the smallest spheres in the phantom while maintaining a reasonable size for all the image data. A Hann apodization filter was used throughout, and no additional filters (e.g., inter-slice tomosynthesis filter in the y direction) were applied. Modified Parker weights120, 121 were applied

o to reconstructions exceeding 180˚+fan. For tot = 360 , a uniform weighting of 0.5 was applied.

122 The usual normalization factor,  N proj , applied for a 180˚+fan acquisition was modified for

limited arc acquisition as tot N proj , such that the Fourier coefficients within the sampled wedge were correctly normalized to be independent of Nproj and the reconstructed voxel values were roughly proportional to tot.

3.2.3. Measurements of Signal, Noise, and Power Spectral Density

3.2.3.1. Signal-Difference-to-Noise Ratio (SDNR) As a simple, intuitive means of assessing imaging performance under conditions of varying background clutter, the signal-difference-to-noise ratio (SDNR) was measured using nylon cylinders in a polystyrene plate inserted at the central “coronal” plane of the clutter phantom as shown in Fig. 3.1(c). Analysis of SDNR was preliminary to SB measurements (below) and was intended to: 1) assess the validity of voxel values reconstructed as a function of tot (i.e., proper signal normalization in reconstructed volumes as discussed in the previous section); 2) qualitatively illustrate the influence of tot and background noise on the visibility of stimuli; and

3) give basic quantitation by way of a simple metric invoked in other studies and sometimes related to the Rose criterion for detectability. The coronal plate consisted of a 1 cm slab of polystyrene embedded with an assortment of nylon cylinders [1 cm height and diameter]. The

64 nylon cylinders presented “stimuli” within the coronal plane amid varying degrees of anatomical

o clutter under different acquisition angles: for tot=360 , the coronal plane presents stimuli within

o a uniform (polystyrene) background, while for tot < 180 +fan, out-of-plane clutter arising from the adjacent volumes of random spheres increases with smaller tomosynthesis arc. The SDNR was analyzed as:

   SDNR  Stimulus Background (3.6)  Background

where Stimulus and Background denote the mean voxel value within the region of the stimulus and

background ROIs, respectively, as illustrated in Fig. 3.2. The term  Background is the standard deviation in voxel values belonging to the background ROIs, which were chosen large enough that the standard deviation reflects the out-of-plane “anatomical noise”. Measurements were performed about each stimulus, with SDNR averaged over 6 stimuli and across 13 coronal slices within the polystyrene slab.

3.2.3.2. Measurement of Background Power Spectra i. 3D Background Power Spectra. The background power spectrum associated with “clutter” presented by the phantom of Fig. 3.1(b) was measured as a function of tot for both the constant-

 and constant-Nproj cases. Non-overlapping volumes of interest (VOIs), each 50 x 50 x 50 pixels (25.9 x 25.9 x 25.9 mm3), were selected within the reconstructed volume (total number of

VOIs = 144). The size of the VOIs was sufficiently large to encompass structures of interest

(different diameter of spheres) but small enough to avoid long-range shading artifacts. The background power spectrum was given by the square of the magnitude of the discrete Fourier transform of each VOI, with the ensemble mean subtracted:

2 x yz 1 SB fx , f y , fz   FT Wx, y, zVOIx, y, zVOI  (3.7a) Nx N y Nz NormW

th where i and Ni are the voxel size and extent of each VOI in the i direction, respectively.

2 3 Written as such, SB is the absolute power spectrum, and has units of  mm . As explained in

Chapter 2,  here denotes a shorthand for units of reconstructed voxel values and is written explicitly without cancellation of terms to clearly convey units of signal-squared times length-

o -1 cubed. For tot > 180 +fan (i.e., CBCT), the units of  are mm corresponding to attenuation coefficient, while for limited arcs (i.e., tomosynthesis), the voxel value is expected to depend on the orbital extent, and  is interpreted simply as the units of the voxel value “signal”.

To compare the background clutter power relative to the signal power, the absolute power spectrum was divided by the square of the mean signal to yield the normalized power spectrum:

1  x  y  z 1 2 S B f x , f y , f z  2 FT Wx, y, zVOIx, y, z VOI  (3.7b) VOI N x N y N z NormW where VOI denotes the ensemble mean voxel value. This form normalizes the effect of increasing signal power with orbital extent and facilitates the comparison of clutter power. Thus, normalized power spectra, with units of mm3, are presented below and used in power-law parameter fitting in the next section. The term W(x, y, z) is a 3D Hann tapering window applied to individual VOIs in the spatial domain to reduce spectral leakage:

1 1 cos( r / R) for r  R W (r)   2 (3.8) 0 otherwise where r  x 2  y 2  z 2 , and R is half the width of the VOI. The normalization factor:

2  Wi Norm  i (3.9) W n

restores the proper magnitude of the power spectrum following the tapering window, with Wi denoting the elements of the matrix W(x, y, z) and n the total number of elements in the matrix.

While a tapering window is often unnecessary in analysis of the quantum noise-power spectrum, we found it useful in reducing spectral leakage associated with the sharply decreasing (1/f) characteristic of the background power spectrum. The statistical error in the power spectrum

(denoted by error bars in plots below) was calculated as the standard deviation across the ensemble at each spatial frequency, divided by the square root of the number of VOIs in the measurement.

To analyze the component of the power spectrum owing to quantum noise and electronic noise, two repeated scans were acquired (with the clutter phantom in exactly the same position between scans), and the two reconstructed volumes were subtracted. The resulting 3D difference image presents only the stochastic noise components (quantum and electronic noise, amplified by 2 ) contributing independently between the two scans. Frequency characteristics of stochastic noise components were assumed to remain unchanged between consecutive scans (ignoring effects like detector lag). To reduce the influence of quantum and electronic noise on the anatomical background power spectrum measurement, the difference between the power spectrum of the 3D

“clutter” volume and that of the subtracted volume (divided by 2) was taken as the power spectrum of anatomical clutter alone, i.e., S  S  S / 2 . B I1 I1 I2 ii. 2D “Slice” Power Spectra. Two-dimensional “slice” power spectra were computed in a similar way as the 3D power spectra described above. Non-overlapping regions of interest (ROIs)

67 of 50 x 50 voxels (25.9 x 25.9 mm2) were drawn from coronal (x, z) slices of reconstructed volumes (total number of ROIs = 7632). A 2D Hann window and normalization factor was

applied, and the 2D Fourier transform was computed with a scaling factor of xz Nx Nz . The slice power spectrum determined in this way (i.e., slice “extraction”) corresponds to the integral

38 of the 3D power spectrum along the depth (fy) direction as described by Siewerdsen et. al. This point bears reiteration: the 2D power spectrum of an image slice “extracted” from a 3D image is not a slice from the 3D power spectrum; it is the integral over the 3D power spectrum in the orthogonal direction. The units of the 2D “slice” power spectrum and 3D power spectrum differ by a factor of length. The relevance of “slice” or “volumetric” power spectra pertain to the expected value of  and also to the observer model being considered, as discussed below – e.g., whether the observer is considered to have information belonging only to a slice or is able to completely interrogate the volume.

3.2.3.3. Power-Law Noise Parameters As a basic empirical characterization of the measured power spectra in terms that can be compared to previous work, the measurements were fit to the power-law description of Eq. (3.1).

The region of the power spectrum expected to follow a power-law characteristic is bounded by the reciprocal of the largest and smallest sizes of the spheres in the phantom. For the phantom of

Fig. 3.1(b), this gave a spatial frequency range between (1/15.88 = 0.063 mm-1) and (1/3.18 =

0.33 mm-1).

The dependence of the power-law parameters,  and , on the tomosynthesis angle (tot) was characterized in order to examine the behavior of out-of-plane clutter in the continuum from a projection image to a fully 3D reconstruction. Asymmetries in  and  in the x, y, and z directions were also examined. As described in Appendix 3A following this chapter, analysis of

68 the clutter power spectrum predicts distinct behavior in  and  in each regard (i.e., as a function of tot and asymmetric in x, y, z), showing also that the pure power-law form breaks down in tomosynthesis (although the power spectrum could still be fit to such an empirical form). To simplify the analysis, a 1D representation of the 3D power spectrum was obtained by extracting

1D profiles along the x, y, and z axes, giving SB(fx), SB(fy), and SB(fz). Linear regression with the least squares approach was performed on the log-log data points to obtain the power-law parameters  and  :

logSB   log   log f  (3.10)

Parameters corresponding to SB(fx), SB(fy), and SB(fz) were denoted as x, y, z and x, y, z. The negative of the slope of the regression line gives , while the intercept equals log . Note that

one could alternatively write logS Bobj  log obj   obj log f , and that one could infer SB-obj

2 from the measured SB by dividing T . At low spatial frequencies (where anatomical clutter predominates), division by T2 effects a small correction; at high frequencies, on the other hand, we found the correction to perturb the power spectra tails and yield poor fits. Thus, where explicitly mentioned below, the assumption  =obj and  = obj may be invoked, believed to be a fair approximation, since the fits were from the low-frequency regime. In fact, the main aim of the work is not a precise determination of obj and obj, but an investigation of their effect on detectability, assuming reasonable values of obj and obj and examining the sensitivity to broad variation in these power-law parameters.

Error bars on and  were computed using the regstats function in Matlab (vR2007a, The

Mathworks, Natick, MA) to reflect standard errors of the slope and intercept associated with the

69 least squares fit,123. The standard error of the intercept computed as such corresponds to that of log  instead of  . Assuming the error is small, propagation of errors gives the standard error in  as   log , where log  denotes the standard error of .

3.2.3.4. Detectability Index in Tomosynthesis and CBCT As described in Chapter 2, cascaded system analysis (CSA) can be used to model the imaging performance of tomosynthesis and CBCT.58, 71, 116 While the model describes the NEQ and detectability index in a continuum of tot, previous work has not included the effect of background anatomical noise on detectability. While the background power spectrum does not

124 represent a truly stochastic process, a simple approach proposed by Barrett et al. includes SB as an additional term in the denominator of the NEQ, referred to as a “generalized” NEQ (GNEQ).

This suggests a generalized detectability index as follows:

2 T(,,) f f f 2 d'(,,)2  x y z Wfffdfdfdf  Task x y z x y z SfffBxyz(,,)(,,)(,,) Sfff Qxyz Sfff Exyz

(3.11)

Note that SB, SE, and SQ used in detectability calculation are absolute power spectra with units

2 3  mm . This form allows analysis of the tradeoffs between SB, SQ, and SE with respect to, for example, tot, the exposure per projection, and the total number of projections. Under conditions of constant total dose and constant angular separation, increasing tot reduces SB (i.e., rejects out- of-plane clutter, as shown below) but increases SQ and the relative contribution of SE (due to a lower dose per projection). Similarly, spreading fewer, higher-dose projections over a larger angular range may reduce the effect of SE but at the cost of view-sampling effects that impede detectability. Tradeoffs among quantum noise, electronic noise, and out-of-plane clutter are

70 immediately apparent that are central to knowledgeable system design – e.g., in breast and chest tomosynthesis – and their inter-dependency as in Eq. (3.11) shows that they cannot be considered in isolation from other aspects of system performance.

The dependence of d’ on tot and other parameters was analyzed as a function of: i.) power-law characteristics of clutter (obj and obj); ii.) the number of projections; and iii.) total dose. In the first case, power-law parameters were taken from experimental measurements of  and  for the

phantom described above, noting the assumption above for conditions in which    obj and

   obj . We also allowed obj and obj to vary freely across wide ranges of values

-5 2 2 3 [obj=2~4obj = (10 ~ 10 ) × measured ( mm )] to examine sensitivity to each. In each case, the same system geometry, beam quality, etc. as described above were assumed, with attenuation by a 10 cm slab of water taken as an approximation to the phantom. A constant exposure to the detector from view to view was assumed. Detectability index was calculated based on Eq. (3.11) in a manner that accounted for the angular separation between projections (view sampling), as detailed in Appendix 3B.

Three idealized imaging tasks were modeled to illustrate the dependence of performance on acquisition parameters such as tot, Nproj, and dose: i.) a nominal 3 mm sphere detection task, which emphasizes low and middle frequencies, with the contrast of the sphere set to that measured for the nylon stimulus in the SDNR phantom (0.12 mm-1 in the 360o CBCT images); ii.) a delta-function detection task implemented as a constant at all spatial frequencies, and normalized to the same total signal power as (i); and iii.) a Gaussian with a width (standard deviation) of 3 mm, representing a low-frequency task. The latter two tasks carried equal signal

2 power [i.e., integral over H (fx, fy, fz)] as task (i) to emphasize frequency-dependent characteristics rather than signal magnitude.

3.3. RESULTS 3.3.1. Signal-Difference-to-Noise Ratio (SDNR)

Example coronal images acquired as a function of tot in both the constant- and constant-Nproj schemes are shown in Fig. 3.2(a). The images show the stimulus (nylon cylinders) and the ROIs used for characterization of SDNR. The ROIs were placed to represent the mean and standard deviation in the stimulus and background while avoiding the dark edge bands (edge enhancement effect of the ramp filter along the x and y directions) that are conspicuous at low tot. Out-of- plane clutter evident at small tomosynthesis arc is gradually rejected as the orbital extent increases. At large orbital extent (tot = 180˚+fan to 360˚), out-of-plane clutter is completely rejected, and distinct tradeoffs in quantum noise, electronic noise, and view aliasing are observed.

For the constant- case, a smooth quantum noise background is evident. For the Constant-Nproj cases, view aliasing effects dominate, since the angular separation between projections is large, evident as streaks in the axial plane and granular mottle in the coronal plane.

Figure 3.2. Coronal images of the clutter phantom including a conspicuous stimulus (nylon cylinder) for illustration of clutter rejection and measurement of SDNR. (a) Images acquired as a function of tot.

Regions of interest for calculation of stimulus and background mean and standard deviation are shown as black rectangles in the first image. (b) Signal difference, (c) background noise (Background), and (d) SDNR plotted as a function of orbital extent. (Published with permission from Medical Physics)

Signal difference, background noise, and SDNR are shown as a function of θtot for both acquisition schemes in Fig. 3.2(b-d). For both cases, the signal difference increases linearly as a function of tot up to 180˚+fan, beyond which it is a constant given by the true difference in attenuation coefficients (nylon - polystyrene). This behavior is consistent with expectations for

73 properly normalized projections backprojected across a limited arc – i.e, voxel value proportional to the tomosynthesis arc and independent of Nproj. The constant voxel value above tot =

180˚+fan confirms proper Parker weighting.121, 125

For the constant-Δθ case, background noise decreases with θtot as out-of-plane clutter is gradually rejected, giving a monotonic increase in SDNR with angle. For the constant-Nproj case, background noise decreases with tot up to a point (determined by the system geometry and choice of Nproj) beyond which view aliasing begins to dominate. For the conditions described above (Nproj = 89, SAD = 93.5 cm, SDD = 144.4 cm), background noise increases sharply due to view aliasing beyond tot ~120˚. This effect is also reflected in the SDNR measurements for the constant Nproj case, where SDNR improves up to tot ~120˚, beyond which SDNR declines – even after the object is "fully sampled" (180o+fan), reflecting increasing severity of view- aliasing artifacts as increases.

3.3.2. Background Power Spectra Coronal (x-z) and axial (x-y) slices of the clutter phantom reconstructed under the constant- and constant-Nproj schemes are shown in Fig. 3.3. Window and level of each image were set to its mean ± 3 standard deviations for display and intercomparison. In the coronal plane, the overall impression of contrast and noise as a function of tot is consistent with that expected from the

SDNR measurements (Fig. 3.2), with background clutter dominating at low tot (for constant-) and view aliasing dominating at large tot (for constant-Nproj). As expected in the axial plane, depth resolution along the y-direction improves with tot.

Figure 3.3. Coronal and axial slices of the clutter phantom in tomosynthesis and CBCT. Rejection of out- of-plane clutter and spatial resolution along y each improve with orbital extent. View-aliasing artifacts are evident in images reconstructed with large angular separation (e.g., the constant-Nproj scheme at large tot).

(Published with permission from Medical Physics)

Figure 3.4 shows the coronal and axial normalized power spectra corresponding to the images of

Fig. 3.3. The frequency domain plotted ranges ±fNyquist in the fx, fy, and fz directions. The logarithmic grayscale helps to visualize the broad range in spectral density, dominated by the large low-frequency component of long-range “anatomical” clutter. The overall magnitude of the normalized power spectra decreases with θtot. The coronal power spectra exhibit the “missing

75 cone” artifact about the fz axis associated with circular cone-beam CT, the extent of which is consistent with the system geometry.126 The most notable feature of the power spectra is the

“double-wedge” of spectral density about the fx-direction of the axial power spectra. The double- wedge widens with (and is equal to) tot as expected and corresponds to the sampled region of the Fourier domain according to the central slice theorem.

Figure 3.4. Coronal and axial slices of the normalized 3D power spectra of clutter phantom images. The frequency domain ranges ±fNyquist in the fx, fy, and fz directions. The large low-frequency component of

“anatomical” clutter is clearly observed, with overall magnitude decreasing with θtot. The “double wedge” of spectral density in the axial slices corresponds to the sampled region in the Fourier space. (Published with permission from Medical Physics)

Power spectra along fx and fz axes were extracted from the 3D power spectrum measurements for constant-Δθ and constant-Nproj, as summarized in Fig. 3.5. Spectra along fy are not shown for reasons of brevity and because the spectral density is extremely low (near zero, within the

o unsampled region of the double-wedge) for tot < 180 +fan. The vertical dashed lines mark the frequency range over which the power spectra are expected to obey power-law behavior (i.e., the reciprocal of the largest and smallest diameter spheres, 0.063 mm-1 and 0.31 mm-1). The log-log plots over this region demonstrate fairly good linearity (R2=0.96±0.02), consistent with a power- law form proportional to 1/f. Notably, three distinct regions were observed in the power spectra.

The frequency region bounded by the dotted lines was believed to be where the power spectra would follow power-law behavior. Below this region, the power spectrum measurements are subject to longer range fluctuations, such as shading artifacts not corrected by detrending and remained despite the subtraction of stochastic noise. Above this region, a sharp decline in the power spectra is evident, associated with correlations on scales finer than the smallest sphere in the phantom (e.g., scintillator MTF and reconstruction filter). Near the Nyquist frequency, the tails of the power spectra tend to flatten due to the presence of (white) electronic noise and aliasing.

From the power-law (linear) region of each plot, the magnitude of the background power spectrum is seen to decrease with increasing θtot, quantifying the gradual rejection of out-of- plane clutter for larger tomosynthesis angle. Comparing SB(fx) for the two reconstruction schemes, the constant Nproj case exhibits greater mid- and high-frequency noise components at large tot associated with the view sampling effects described above.

Figure 3.5. Normalized power spectra measurements along fx and fz axes extracted from the 3D power spectrum measurements for the constant Δθ [(a), (b)] and constant Nproj [(c), (d)] schemes. Power spectra are shown for seven orbital extents ranging from 10° to 360° as in the legend. The vertical dashed lines mark the frequency range over which the power spectra are expected to obey power-law behavior (i.e., the reciprocal of the largest and smallest diameter spheres, 0.063 mm-1 and 0.31 mm-1). Power spectra exhibit good linearity within this region (R2=0.96±0.02), consistent with a power-law relationship. The magnitude of the power spectra is seen to decrease with increasing θtot, quantifying the gradual rejection of out-of-plane clutter for larger tomosynthesis angle. (Published with permission from Medical Physics)

3.3.3. Power-Law Parameters

Figure 3.6 shows measurements of  and  along fx and fz (denoted respectively as x, z and

x, z) obtained from linear fits to the power spectra within the frequency range expected to follow power-law behaviour. Each plot shows and determined from the power spectrum

o of: i.) a single projection (plotted at tot = 0 ); ii.) a 3D image reconstruction from tomosynthesis or CBCT; and iii.) a coronal slice “extracted” from the 3D image (recognizing that the “slice” power spectrum involves an intrinsic integration along fy and a change in units in , as described above). As shown in Appendix 3A, a slice from tomosynthesis images of power-law clutter does

 not strictly exhibit a /f characteristic, except for special cases on the fx- and fz-axes. In light of this finding, the power-law parameters  and  shown below should be considered a purely empirical description of the 2D slice power spectra.

The results for x and z are summarized in Figs. 3.6(a-d), showing the following overall trends:

o o x increases with tot for tomosynthesis (tot <~180 ); above tot ~180 , x remains constant for the constant- case and reduces for the constant-Nproj case; z is roughly constant with tot.

Since a larger value of  corresponds to increased low-frequency correlation (qualitatively appreciated as clumpiness), the increase in  with tot may at first seem counter-intuitive; however, as evident in the images of Fig. 3.3, the effect is consistent with out-of-plane clutter rejection and better discrimination of the cross section of spheres achieved in higher angle reconstructions. As noted below, the real power of increasing tot is in the reduction of , rather than its effect upon .

Figure 3.6. Measurements of  and  obtained from linear regression of the normalized power spectra

in Fig. 3.5. The plots show  x ,  z ,  x , and  z determined under conditions of (a,b,e,f) constant-Δθ

3 and (c,d,g,h) constant-Nproj. The units of  for the 3D case are (mm ), while for projection and slice, the units are (mm2). (Published with permission from Medical Physics)

o For the constant-Δθ case [Fig. 3.6(a)], x increases with θtot, and plateaus beyond θtot ~90 for the

3D case and ~180o for the “slice” case. As expected (see Appendix 3A), the value of  is the same (within experimental error) for a single projection and a fully sampled (θtot ≥ 180˚+fan) 3D reconstruction, which is in turn equal to the  characterizing power-law behavior of the object

(obj), assuming MTF corrections at low frequencies are small as explained in Section 3.2.3.3.

For very small tomosynthesis angles,  is nearly the same for the “slice” and “3D” cases: the slice power spectrum approaches that of a filtered projection, reducing x by approximately 1

80 from obj; similarly for 3D power spectra, the effect of the ramp filter is not canceled by the backprojection operation which has a transfer characteristic of 1/f as explained in Chapter 2, thus also decreasing 3D x fromobj by ~1. This difference is quickly recovered as tot increases, although the “slice” and “3D” cases approach separate values of  at large tot, which are also

108, 114 expected to differ by 1. A difference in x of approximately 0.7 is observed in this case, the discrepancy likely owing to measurement noise and the fact that slice power spectra integrate along the fy direction over all frequencies, while the phantom only follows power-law behavior over a limited frequency range. For the constant-Nproj case [Fig. 3.6(c)], x exhibits similar behavior, except that the “slice” and “3D” curves decrease for θtot > 180˚ as view-aliasing effects impart an increasing level of “graininess” in the images.

In comparison, z is nearly constant with θtot in both cases [Fig. 3.6(b, d)] for “projections,”

“slices,” and “3D” with a notable exception above θtot =180˚+fan. According to the central slice theorem and as shown in Fig. 3.4, the 3D power spectra are empty along fy below θtot =180˚+fan; therefore, integration along the fy-direction is equivalent to “adding” the power spectra of different views, which would not affect the slope (“color”) of the spectra. Therefore, z is equal for the “projection,” “slice,” and “3D” cases before the double-wedge fills the Fourier domain.

Above θtot=180˚+fan, a precipitous drop in z occurs for the “slice” power spectrum as the

Fourier domain fills in about the fy axis, giving a decrease in z of ~0.7 from the 3D case, less than the nominal value of 1, as noted above. The values of z are comparable to that of x above

180˚+fan.

Figure 3.6 also summarizes measurements of , with a logarithmic scale better showing the strong dependence on tot. Values of  for 3D reconstructions, and 2D slices through the 3D

81 reconstruction are shown on the same graph despite the difference in units ( for 3D reconstruction carries units of mm3, whereas the units of  in a slice is mm2). The value of  for a single projection is not shown because of the difference in scale resulting from normalization

-6 2 2 -6 2 2 (in a single projection, x = 2.18x10  mm , z = 1.80x10  mm ).

o In all cases, for tot < 180 +fan, x and z decrease 2-3 orders of magnitude with tot, corresponding to stronger rejection of out-of-plane clutter for larger tomosynthesis angles. For

o the constant- case, x is constant above tot = 180 +fan (fully sampled 3D data without further rejection of out-of-plane clutter), whereas for the constant Nproj case, x actually increases with

tot due to view-aliasing noise. The value of  along both the fx and fz axes is similar for “slice” and “3D” power spectra for fully sampled CBCT images, consistent with analysis in Appendix

3A, where the two values are shown to differ by a constant of order unity.

Trends in the fz-direction [Figs. 3.6 (f, h)] are similar, although z is typically 1-2 orders of magnitude smaller than x, attributable to the “missing cone” of frequencies about the fz axis intrinsic to backprojection on a circular source-detector orbit. (See discussion in Chapter 2 regarding the cone-beam artifact “null cone” and violation of Tuy’s condition.) The z for slice

o NPS exhibits a discontinuity at tot = 180 +fan. The value of z is ~10 times smaller for the “slice” power spectrum compared to the “3D” power spectrum for θtot < 180˚+fan, owing to unsampled frequencies about the fy axis, and jumps to a value nearly the same as the “3D” case beyond 180˚.

Overall, such experimental analysis of  and  confirms numerous theoretical expectations of power-law noise, highlights non-trivial asymmetries in (fx, fy, fz), and quantifies the significant dependence of background clutter on tot. First, the measurements demonstrate the overall approach regarding an experimental phantom designed to present power-law noise in a fairly

82 general manner without reference to a specific organ or body site. Second, the measurements highlight and quantify the differences in power-law parameters ( and ) associated with a 2D projection, a 3D tomosynthesis image (0<tot<180º+fan), a 3D CBCT image (tot ≥ 180º+fan), and “slice” versus “3D” representations thereof. The value of  measured here (typically in the range 2.4 – 2.8) is similar to that reported for breast (e.g.,  = 2.7~3.0 108, 114, 115), but more importantly and as detailed below, the results pose important implications for measurement and modeling of detectability in tomosynthesis – specifically, that  (rather than the much-reported and scrutinized ) is the major factor governing detectability, and improved detectability in tomosynthesis and CBCT is primarily attributable to reduction in  (rather than ).

3.3.4. Detectability Index

Detectability index was calculated as a function of tot, obj,  obj, dose, and Nproj for various imaging tasks as summarized in Fig. 3.7. Unless otherwise mentioned, the nominal values are:

-7 2 3 obj =2.76,  obj =9.72×10 ( mm ) as measured in the clutter phantom (x and x in the constant- case); exposure per projection = 0.01 mR after attenuation by 10 cm of water, and

Nproj = 89.

Figure 3.7 illustrates the dependence of d’ on (a) the frequency content (obj) and (b) the magnitude (obj) of background clutter for four different angular extents (tot = 10º, 40º, 90º,

180º). Calculations were performed for a 3 mm sphere detection task. As expected, d’ decreases with increasing clutter magnitude (obj) and clumpiness (obj) at each tot. Consistent with

127 previous work, a smaller value of  obj gives improved detectability for all four settings of tot.

The same trend is observed with reduced clutter magnitude, but changes in obj have a greater influence on detectability as seen by the large increase in d’ over an order magnitude decrease in

obj about the nominal value. In comparison, detectability is less sensitive to  obj over the range examined, especially for low tomosynthesis angles (10º, 40º) and clinically relevant values

(~2-4). Detectability generally increases with tot around the nominal values of obj and obj, illustrating the benefit of tomographic imaging in the presence of background clutter. However,

-10 -11 -12 2 3 at very small clutter magnitude (obj = 9.72x10 , 9.72x10 , and 9.72x10  mm ), the cross- over between the 90º and 180º curves reflects the increasing contribution of quantum noise relative to anatomical clutter.

Figure 3.7(c) shows d’ versus θtot for different exposures (corresponding to different levels of quantum noise relative to anatomical clutter) ranging from 0.1 R/projection to 1 mR/projection

(exposure to the detector after attenuation by 10 cm of water). As expected, low exposure corresponds to a relatively high contribution of quantum noise, which dominates over anatomical clutter for the lowest level examined (0.1 R per projection) – evidenced by the nearly constant d’ beyond θtot ~80˚. Increasing the exposure per projection reduces the quantum noise and gives the expected monotonic increase in d’ with θtot. However, further increase in exposure does not necessarily improve d’, as the relative magnitude of quantum noise becomes small in comparison to anatomical noise, which is independent of dose. This last point illustrates the importance of the “generalized” NEQ and detectability index, whereas a conventional (quantum-noise-only) metric would suggest continuous performance improvement with dose. Through incorporation of background clutter in the “generalized” form, the point of diminishing returns can be identified,

84 and a better understanding of dose dependence (both low-dose limits and high-dose diminishing returns) can be obtained.

The effect of Nproj on detectability is shown in Fig. 3.7(d), keeping the total exposure per projection at 0.89 mR in each case. A 3 mm sphere detection task was used. The case Nproj = 1 corresponds to d’ for a (filtered and normalized) 2D projection, which is independent of tot. For a small number of projections (e.g., Nproj = 6) spreading over a large angle (θtot>40˚), view aliasing is the dominant noise source and reduces d’ significantly. Such view aliasing effects are also present for θtot < 40˚, but background clutter dominates, such that d’ still improves with increasing tot. Such effects are evident qualitatively in the images of Fig. 3.3 and in the work of

Zhao et. al,60 who observed streaks in the power spectrum for angular separation () greater than 2˚. For a greater number of projections (e.g., Nproj = 85), angular separation between views is sufficiently fine that view aliasing effects are smaller; thus d’ improves monotonically with θtot due to rejection of out-of-plane clutter. For an even greater number of projections (e.g., Nproj =

506), the exposure per projection is so low that electronic noise dominates, reducing overall detectability. Such analysis serves as a quantitative guide to a fundamental system design question: for example, for fixed total dose and angular range (θtot), what is the optimal number of projections for a given imaging task?

Figure 3.7. Detectability index (d’) as a function of (a)  obj and (b)  obj, for four orbital extents (sphere

(Gaussian detection task). (d) Plot of d’ vs. θtot for different total number of projections (Nproj), demonstrating effects of view-aliasing at low Nproj (high θtot) and electronic noise at very high Nproj

(sphere detection task). (Published with permission from Medical Physics)

3.3.5. Imaging performance optimization for specific imaging tasks The univariate analysis in Fig. 3.7 is illustrated further in the surface plots of Fig. 3.8 for three imaging tasks: (a) delta-function detection, (b) sphere detection, and (c) Gaussian detection. The delta-function task emphasizes all spatial frequencies equally and is therefore most affected by quantum and electronic noise (which dominate at high frequencies), evidenced by the rapid falloff of d’ at large θtot for cases of large Nproj (where electronic noise dominates) and low Nproj

(where quantum noise and view aliasing dominate). The “peak” of this surface identifies the optimal number of projections for a given angular range, in answer to the fundamental system design question mentioned above.

The sphere detection task emphasizes both low frequencies (associated with the sphere itself) as well as middle and high frequencies (associated with the edge of the sphere). For the low- frequency aspect of the task, pronounced improvement in d’ is observed with increasing θtot due to rejection of out-of-plane clutter, but electronic noise is seen to diminish d’ at large values of

Nproj (i.e., very low exposure per projection). In comparison, the Gaussian detection task is predominantly a low-frequency task and is largely unaffected by high-frequency noise. Such a task is almost entirely “clutter-limited”, and d’ increases monotonically with tot. The effect of view aliasing is still evident for few projections spanning a large angle. Note also the overall decrease in d’ for lower-frequency tasks, illustrating the impact of low-frequency background clutter and the importance of its inclusion in the “generalized” NEQ and detectability.

Gaussian detection task (a low-frequency task). Overall, lower-frequency tasks exhibit lower d’ due to the influence of background clutter and benefit from increasing angular extent. Higher-frequency tasks are more susceptible to quantum and electronic noise, evidenced by falloff in d’ for few projections over a large arc (small Nproj and large θtot, with correspondingly high quantum noise and view-aliasing) and large

Nproj (low dose per projection, with correspondingly high electronic noise). (Published with permission from Medical Physics)

3.4. DISCUSSION AND CONCLUSIONS This work extends previous analytical modeling of the 3D noise-power spectrum, NEQ, and task-based detectability index for CBCT and tomosynthesis to include the effect of background clutter, measured and modeled according to a power-law relationship and examined as a function of orbital extent, number of projections, and total dose.

Two simple imaging scenarios were investigated, constant-Δθ and constant-Nproj, to represent varying degrees of quantum noise, electronic noise, and view-aliasing artifacts on image quality, one noise component fixed across angles while allowing others to vary. An alternative scenario

(not studied directly in the current work) could involve constant-Δθ and constant total dose (with

88 varying Nproj, dose per projection, and total orbital extent), which would also be of considerable value to the understanding of image quality in tomosynthesis. In such a case, view-aliasing and quantum noise are unchanged despite variation of orbital extent, while electronic noise increases with angle. Other schemes have also been proposed that involve uneven angular sampling and/or uneven distribution of dose across projections128. Investigation of such cases are subjects of possible future work.

A physical phantom was designed with acrylic spheres of different sizes to simulate a background exhibiting a power-law characteristic. Other authors have used similar designs. Sain et al.129 used water filled plastic eggs submerged in radioactive water to simulate normal fatty tissue in the breast. Hestermann et al.130 used a similar design with ellipsoid polystyrene beads as random, textured background to investigate system performance. Signal-to-noise ratio measured from the bead background and that from a simulated lumpy background were found to exhibit similar behavior except for a flattening at long exposure times. Park et al.131 extended the design by using spheres of different sizes and densities. A similar computer-generated phantom was adopted by Badal et al.132 to assess geometric accuracy in Monte Carlo simulation. The work presented above provides an analytical basis that explains the power-law nature of the sphere phantom design. The physical phantom based on fractal self-similarity was found experimentally to give power-law spectral density that may be physically “tuned” to give obj and obj analogous to that of anatomical structures such as the lung and breast. The phantom did not attempt to model a particular anatomical site, investigating instead the effect of background clutter in general on task-based performance. While appealing from a general standpoint and consistent with /f descriptions of “anatomical noise,” the phantom does not, of course, present the complexity of real anatomical structure – for example, asymmetries associated with ducts and

89 vessels. The phantom design was based on fractal self-similarity, which predicts a obj of 3 in an object consisting of equal volume of differently-sized spheres. A x value of 2.74 is observed in a fully 3D reconstruction, differing from the theoretical prediction of 3, because the phantom only obeys self-similarity over finite scales. The inclusion of system MTF in the measurement may also have contributed to the discrepancy.

The “generalized” performance metrics aim to include the effect of background clutter on image quality, extending previous modeling work describing 3D quantum and electronic noise. Such is found to have significant influence on task-based detectability – especially (and to no surprise) for tasks involving low spatial frequencies (e.g., Gaussian detection). A generalized model allows investigation of tradeoffs among each noise source with selection of tot, Nproj, and dose.

While the approach includes the effect of a finite number of views over a given orbital extent

(i.e., “view aliasing”), other sources of image degradation, such as reconstruction artifacts

(shading, streaks, etc.) are not included in the model, and the results assume that such artifacts can be ameliorated by acquisition and/or post-processing techniques – e.g., a bowtie filter, scatter correction, etc.

The model for detectability index presented above [Eq. (3.11)] corresponds to the fully 3D prewhitening matched filter observer. This is, of course, an idealized situation and may or may not correspond to an observer “scrolling” slices. As introduced in Chapter 2, slightly more sophisticated observer models, such as non-prewhitening and eye filter models, can be incorporated within the same framework. The correspondence of the theoretical detectability index with real observers will be investigated in Chapter 4.

As apparent in Eq. (3.11), tradeoffs among anatomical background, quantum noise, and electronic noise are important to knowledgeable system design. Results in Section 3.3.4 investigated the univariate dependence of imaging performance on background clutter (obj, obj), exposure, and number of projections. The relevance of these tradeoffs is evident in Section 3.3.5, where the dependence on task is also demonstrated. Detection of nodules or masses (analogous to the low-frequency sphere or Gaussian detection tasks) benefit from a large angular extent for the rejection of clutter; conversely, detection / discrimination of small, high contrast structures such as microcalcifications (analogous to the delta function detection task), suggests optimal performance from a limited arc with a knowledgably selected number of projections to divide the total dose in a manner that minimizes quantum and electronic noise. Although imaging techniques and reconstruction parameters (e.g., kV, mAs, voxel size, reconstruction filters, etc.) chosen for this study are not necessarily representative of a particular clinical application (e.g., breast CT), the power-law characterization of anatomical noise still holds over the specified range of spatial frequencies, and the theoretical model can be easily adapted to specific imaging systems. Investigation of other variables that can potentially influence detectability will be subjects of future work. The CSA model accommodates different system design parameters, anatomical background, and imaging tasks, providing a general framework that can provide potential utility in a wide range of imaging applications. 

Appendix 3A. Power-Law Noise in Projections, Tomosynthesis, and CBCT The magnitude and frequency content of anatomical clutter is understood to vary between projection images, tomosynthesis, and cone-beam CT, with the first and last representing limiting cases of a continuum. The dependence of power-law noise in the limiting cases (2D projections and 3D cone-beam CT) has been described analytically by Metheany et al.,114 and experimental characterization of such has been the subject of considerable work. In this

Appendix, a brief analytical description of power-law noise is offered that agrees in the limiting cases with results of Metheany et al. and describes the power spectrum across the continuum of tomosynthesis in a manner that aids interpretation of the experimental results reported above and by other authors. The analysis also distinguishes the expected form of power-law spectra as assessed from fully 3D volumes as opposed to individual slices therein.

3A.1 The Clutter Power Spectrum in the 3D Object and the 3D CBCT Image Consider an object of extent L in the x, y, and z directions, with attenuation coefficient expressed

as a stationary random variable, obj(x, y, z) , characterizing power-law clutter and with

~ deviation from the mean denoted obj(x, y, z) . The Fourier transform of these deviations is

~ written as M obj( f x , f y , f z ) , such that the power spectral density of the object is:

~ 2 M ( f , f , f ) obj x y z  obj S Bobj( f x , f y , f z )   (3A.1) 3 obj L (af 3D )

2 2 2 ~ 2 3 3 where f3D  f x  f y  f z . The units of SB-obj and M obj are  mm and mm , respectively, with  and a consistent with definitions in Section II. For a perfect 3D reconstruction (fully

92 sampled and free of artifacts), the Fourier domain image is equal to the Fourier transform of the

~ 3D ~ object – i.e., M CBCT ( f x , f y , f z )  M obj( f x , f y , f z ) . The power spectrum of the 3D image

3D therefore equals that of the object – i.e., SCBCT ( f x , f y , f z )  S Bobj( f x , f y , f z ) .

3A.2 The Clutter Power Spectrum in a 2D Projection A projection image corresponds to integration of (x, y, z) across one dimension of the 3D spatial domain, and according to the projection slice theorem, its Fourier transform, denoted ~ ~ ~ M proj ( f x , f z ) , is a slice of that of the object, i.e., M proj ( f x , f z )  M obj( f x ,0, f z ) . The power spectrum of a projection is therefore:

~ 2 ~ 2 M ( f , f ) M ( f ,0, f ) S ( f ,0, f )L3 2D proj x z obj x z Bobj x z S proj ( f x , f z )  2  2  2 L L L (3A.2)    L obj  proj  obj  proj (af 3D ) (af 2D )

2 2 2D 2 2 where f 2D  f x  f z . The units of S proj are equal to those of proj [ mm ], and the denominator is dimensionless. Comparing to Eq. (3A.1), we see that  for a projection is equal to

that for the object (which is equal to that of the fully 3D image) – i.e.,  proj   obj, and the value

of  is scaled by the length of the object being projected, i.e.,  proj  L  obj , recognizing the change in units associated with 2D and 3D power spectra.

3A.3 The Clutter Power Spectrum in a 2D Slice of the 3D CBCT Volume

2D For a slice through the fully 3D CBCT image, the 2D power spectrum, denoted S slice ( f x , f z ) , is given by integrating in one direction (taken as fy) across the 3D Fourier domain, recognizing that

2D and 3D power spectra for the same underlying data have different units. Under the assumption of stationarity, the Fourier components are independent and add in quadrature.

Correspondingly, the power spectrum of a slice extracted from a 3D volume is given by the integral over the fy-direction as:

 2 ~ 3D M CBCT ( f x , f y , f z )df y 2D  S slice ( f x , f z )  2 L (3A.3a) ~ 3D ~ 3D  ' ' ' ' M CBCT ( f x , f y , f z )M CBCT ( f x , f y , f z )df y df y   L2

The square in the numerator is written as a double integral equal to a correlation integral, which

' ' ' by assumption of stationarity, is equal to 0 unless f x  f x , f y  f y , and f z  f z . Therefore:

 2 ~ 3D M CBCT ( f x , f y , f z ) df y 2D  S ( f , f )  (3A.3b) slice x z L2

The Fourier transform is therefore integrated in quadrature. The increment dfy’ can be taken as the inverse of the extent of the object, 1/L, just as an increment in the spatial domain (the sampling interval) defines the extent of the Fourier domain (the Nyquist region). Therefore:

2 ~ 3D  M ( f , f , f )   2D CBCT x y z  obj S slice ( f x , f z )  df y  S Bobj( f x , f y , f z )df y  df y  3   obj L (af 3D )

2 obj / 2   (af )  2 2 obj / 2 y   obj(af x )  (af z )  1   df y (3A.3c)  2 2  (af x )  (af z ) 

2 obj / 2   (af )  obj y  2 obj(af 2D ) 1   df y 0 2  (af 2D ) 

(af ) 2 df y p y Substituting 2 with , such that dp  , the slice power spectrum can be written as: (af 2D ) f 2D

 2D obj 2 obj / 2 S slice ( f x , f z )  2 obj(af 2D ) 1 p  ( f 2D dp) 0 (3A.3d) 2   / 2  obj 1 p 2  obj dp obj1 0 (af 2D )

2 The integral is equal to the hypergeometric function, 2F1(1/2,obj/2;3/2,-p ), evaluated at p  infinity. We label the value simply as C, which can be evaluated analytically only for a few values of and is more generally evaluated numerically.

2D 2C obj S slice ( f x , f z )  (af ) obj1 2D (3A.3e)   slice  slice (af 2D )

The clutter power spectrum for a slice extracted from a 3D image therefore exhibits a value of 

reduced by 1 from that of the object and the 3D image – i.e.,  slice   obj 1 and  is scaled by a

constant of order unity which depends on the value of for the object, i.e.,  slice  2C obj.

3A.4 The Clutter Power Spectrum in 3D Tomosynthesis

For reconstruction from a limited tomosynthesis arc subtending a “wedge” of angle  tot, the ~ Fourier domain image is equal to M obj within the wedge, and is zero outside:

 ~  tot  M obj( f x , f y , f z ), f y  f x tan  ~   2  M tomo ( f x , f y , f z )   (3A.4a)   0, f  f tan tot  y x     2 

The corresponding power spectra are similarly equal within the double-wedge

3D [ Stomo ( f x , f y , f z )  S Bobj( f x , f y , f z ) ] and 0 outside the wedge. Thus, extraction of a 2D slice from the 3D tomosynthesis image amounts to integration along one dimension (taken as fy) of the

tomosynthesis wedge. The upper limit of the integral in Eq. (A.3d) then depends on f x , and is

tot  given by c  f x tan  . When f  c , we have:  2  y

tot  f x tan  c  2  tot  p    tan cos( f ) , (3A.4b) 2 2 2 2  2  f x  f z f x  f z

where f is the angle in the (fx-fy) plane, measured from the fx axis. The hypergeometric function is evaluated at this value of p to obtain the factor C in Eq. (3A.3e), which here depends on , f, and obj. This implies that the clutter power spectrum in a 2D slice of a tomosynthesis image does not actually obey the commonly invoked power-law form (even if the object itself consists of power-law fluctuations); furthermore, the 2D slice tomosynthesis power spectrum is non- isotropic (even if the object is isotropic). Special cases exist on the fx and fz axes, where the

96 power spectra do obey power-law behavior. Considering power spectra along fx (when fz=0),

 tot   tot  tan cos( f ) reduces to tan  and the hypergeometric function equals a constant  2   2  independent of fx and fz. The magnitude  of the slice power spectra measured along x depends on tot and decreases with angular extent. Along fz (when fx=0), the hypergeometric function is evaluated at p=0, yielding a value of 1. Therefore, as shown in the measurements above, the power spectra on fx and fz axes can be fit to power-law curves parameterized by  and  as a descriptive, empirical characterization of the 2D power spectra.

The tomosynthesis power spectrum exhibits a continuum behavior in tot. In one extreme, as tot approaches , p tends to infinity, and the power spectrum is that of fully sampled CBCT, described by an isotropic power-law characteristic. In the other extreme, as tot approaches zero, the extent of the wedge becomes smaller than (~1/L). The Fourier components are not independent within this infinitely narrow wedge; rather, they are approximately equal. When integrating across this narrow extent, we therefore do so linearly (not in quadrature) as follows:

2 C 2 ~ 3D  tot  ~ 3D 2 ~ 3D M CBCT ( f x ,0, f z )2 f x tan  M CBCT ( f x , f y , f z )df y M ( f ,0, f )2c 2D C CBCT x z  2  lim  0 Stomo ( f x , f z )    L2 L2 L2

(3A.5a)

  Using the small angle approximation tan( tot )  tot , and     N with N 1, we 2 2 tot proj proj have:

~ 2 M 3D ( f ,0, f ) 3D 3 2D CBCT x z 2 SCBCT ( f x ,0, f z )L 2 3D 2 lim  0 Stomo ( f x , f z )  2 ( f x tot )  2 ( f x tot )  SCBCT ( f x ,0, f z )L( f x  ) L L

(3A.5b)

3D which is the power spectrum for a single filtered projection. The term SCBCT ( f x ,0, f z )L is the

2 power spectrum of a projection, and ( f xtot ) is the square of the ramp filter. In retrospect, this is not a surprising result: ramp filtered projections approximating double-wedges in Fourier space are simply the basis of filtered backprojection.133

Appendix 3B. Calculation of the Power Spectrum as a Discrete Sum Over (Few) Projections When the angular increment between projections is large, (e.g., large tomosynthesis angle and

small Nproj), the transfer function of backprojection in CBCT does not approximate a radially

58 symmetric, continuous function of ~1/f2D. Therefore, the contribution of each projection to the signal and noise transfer characteristics must be added explicitly. As illustrated in Fig. 3.9, three

frequency regions are proposed for numerical approximation, detailed below.

Figure 3.9. Three regions of the Fourier domain relevant to computing detectability for few projections covering a wide source-detector arc are identified, and the boundary frequencies are labeled. (Published with permission from Medical Physics)

Region (1) is the low-frequency regime bounded by the radius at which the sampled double- wedge becomes wider than the discrete frequency increment, f (given by the reciprocal of the extent of the ROI):

f f 1  (3B.1) 2D   2 tan tot   2 

Within region (1), projections are closely spaced such that each vane adds nearly on top of the previous, approximately equivalent to summing over Nproj projections at the same angle.

Detectability index in this region is given by:

N 2 G 2 MTF 2 ( f , f ) d '2  proj x z H( f , f ) 2 df df (3B.2) 1  2 x z x z N projS B proj ( f x , f z )  N proj[SQ ( f x , f z )  S E ( f x , f z )] where G is the gain factor associated with projection as defined in previous work58, 134, and

S B proj ( f x , f z ) is the power spectrum presented on a projection, given by

2 2  obj S B proj ( f x , f z )  G LMTF ( f x , f z ) , with L denoting the length of the anatomical obj (af 2D ) clutter projected through.

Region (2) is the mid-frequency regime bounded by Eq. (3B.1) and the angular Nyquist criterion:

f f 2  (3B.3) 2D tan  where projections are close enough to approximate a continuum similar to a fully sampled CBCT scan. The detectability index is given by:

100

2 MTF ( f , f , f ) 2 d '2  x y z H( f , f , f ) df df df 2  x y z x y z S B ( f x , f y , f z )  SQ ( f x , f y , f z )  S E ( f x , f y , f z )

(3B.4)

Finally, region (3) is the high-frequency regime, where projections are far apart and each projection contributes to detectability index independently:

N 2 MTF 2 ( f , f ) d '2  N proj x z H( f , f ) 2 df df (3B.5) 3 proj  x z x z S B proj ( f x , f z )  SQ ( f x , f z )  S E ( f x , f z )

The overall detectability index is then given by:

'2 '2 '2 '2 d  d1  d2  d3 (3B.6) where for simplicity the task function is assumed to be radially symmetric, and the NPS and

MTF are taken to be appropriate for either a filtered projection image or a CBCT image.

The major difference in contribution to detectability from the “projection-like” regions versus the

“CBCT-like” region is the factor L in the denominator. It has been previously shown that the need to optimize a 3D imaging system relies upon understanding of 3D noise aliasing (which prevents deterministic filters from cancelling out in the detectability index), and optimization of the projections alone is not sufficient.134 This analysis shows that clutter contributes differently to detectability in 3D images than in projections, and therefore presents another compelling reason that a 3D imaging system must be optimized based on the 3D signal and noise characteristics (and not simply on optimization of the projection data).

101

Chapter 4 Analysis of Fourier-Domain Task-Based Detectability Index in Tomosynthesis and Cone-Beam CT in Relation to Human Observer Performance 4.1. INTRODUCTION Tomosynthesis and CBCT using flat-panel detectors (FPDs) offer the potential for improved lesion conspicuity and localization in a wide range of diagnostic and image-guided procedures, including breast imaging,5-8 chest imaging,135, 136 and surgical interventions.32, 33 The development of new tomosynthesis and CBCT systems for such applications stands to benefit significantly from an understanding of the factors that govern image quality and a theoretical framework for the assessment and optimization of imaging performance.137

Image quality assessment commonly involves human observer-based measurements [e.g., receiver operating characteristic (ROC) or alternative forced-choice (AFC) tests] or observer- independent modeling or measurement [e.g., modulation transfer function (MTF), noise-power spectrum (NPS), detective quantum efficiency (DQE), and noise-equivalent quanta (NEQ)]. The former, although time-consuming and requiring careful attention to experimental design to minimize bias, has been applied to evaluate 3D imaging systems.113 However, due to the broad parameter space associated with system design, acquisition techniques, and reconstruction methods, human observer studies may be impractical as a rigorous approach for designing and optimizing medical imaging systems. Alternative approaches with mathematical observers have been an active area of research.138-140 The latter approach with observer-independent measurement have been commonly used in the characterization of radiographic imaging systems,46-49, 141, 142 with recent research extending to dual-energy radiography,55 tomosynthesis,60, 143-145 CBCT,57, 58, 134 and the incorporation of anatomical background noise.146

102

Despite the widespread use of both of these broad approaches, there is comparatively little rigorous understanding of the connection between the two for real imaging systems – e.g., how improvement in DQE might (or might not) relate to improvement in ROC – underscored primarily by the fact that these prevalent Fourier metrics do not in themselves account for how observers interpret the image data or, more specifically, how detector performance relates to a given imaging task.

A connection between prevalent, practical metrics like NEQ and human or model observer performance metrics like ROC offers significant value in system development – e.g., in identifying low-dose performance limits and guiding design improvements (e.g., x-ray converter efficiency, electronics noise, etc.). As generally acknowledged, imaging performance is best defined with respect to an intended task, and quantitative frameworks based on statistical decision theory have been proposed to incorporate task in the assessment of image quality.4, 102,

137, 147, 148 For example, the detectability index proposed by Green and Swets149 has been adapted in the context of medical imaging150, 151 in terms of the NEQ and a spatial-frequency-dependent template (task function) corresponding to an ideal observer. Such ideal observer models have proven useful to system optimization in some contexts.147, 152 Other observer models that aim to better describe human observer performance under various imaging conditions (e.g., statistical backgrounds) have been an active area of research.84, 96, 102, 103, 107, 108, 110, 124

The work described below extends a theoretical cascaded systems model57, 58 for the 3D NEQ of

FPD based tomosynthesis and CBCT to include: i.) spatial-frequency-dependent task descriptors to yield the detectability index for a variety of idealized imaging tasks and observer models; ii.) background power-law noise, shown to be a major factor in affecting detectability over the continuum of angular extent from low-angle tomosynthesis to CBCT; and iii.) comparison of

103 theoretical calculations of detectability with the performance measured for human observers in real image data. The first is a straightforward interpretation of task-based detectability index (d’) outlined in ICRU Report 54137 in the context of 3D imaging. The second yields “generalized”

Fourier metrics [viz., generalized NEQ (GNEQ) and generalized d’] introduced in Chapter 3, where the term “general” refers specifically to the incorporation of background noise in the NEQ, as described by Barrett et al.124 (distinct from what might be termed “system” NEQ factors such as focal spot size, x-ray scatter, etc., which may also be included87). The third directly compares imaging performance predicted from theoretical calculations using model observers (based on

GNEQ and task function) with that measured with human observers, thereby investigating the extent to which task-based detectability index provides a meaningful figure of merit for performance modeling and, ultimately, system optimization.

Note that the intent of this work is not to advance the extent to which observer models in themselves are descriptive of human observer performance; rather, the work utilizes a variety of well known observer models, and investigates the extent to which a theoretical framework for

3D NPS and NEQ yields correspondence with human observers. The significance of the work lies in the potential to predict imaging performance in a manner that corresponds to that of human observers – e.g., as a function of angular extent, number of projections, and total dose in tomosynthesis and CBCT – from first principles of signal and noise propagation in a 3D cascaded systems model of the imaging system.

4.2. METHODS 4.2.1. Generalized Detectability Index Chapter 2 detailed the cascaded systems analysis framework for deriving 3D MTF, NPS, and

NEQ in tomosynthesis and CBCT. The effect of anatomical noise was considered in Chapter 3

104 by way of the "generalized" NEQ and detectability index, where the anatomical background power spectrum, SB, was included as an additional noise source in the denominator. The generalized 3D prewhitening (PW) observer model was given in Eq. (3.11) and is reproduced here for completeness:

2 T 2 d'2  W df df df (4.1)  Task x y z SSSBQE

with the frequency arguments (,,)fx f y f z dropped for brevity. The generalization of other observer models introduced in Chapter 2 is straightforward:

Non-prewhitening (NPW) model:

2 ()T W2 df df df ' 2  Task x y z d3D  (4.2) ()()S S  S  TW  2 dfdfdf  B Q E Task x y z

PW model with eye filter and internal noise (PWEi):

ETW22() d'2 Task df df df (4.3) 3D 2 x y z ESSSN B Q  E  i

NPW model with eye filter (NPWE):

2 E2 ()T W2 df df df ' 2  Task y x z d3D  . (4.4) E4 ()(S S  S  TW  )2 dfdfdf  B Q E Task y x z

NPW model with eye filter and internal noise (NPWEi):

2 E2 ()T W2 df df df ' 2  Task x y z d3D  , (4.5) ESSSTWN42()()     df df df  B Q E Task i x y z

105

4.2.2.1. “Slice” Detectability vs. “3D” Detectability The detectability index in Eqs. (4.1)-( 4.5) is written in a form in which the observer is assumed to fully perceive the volumetric image information. Although 3D detectability can be derived from the cascaded systems analysis model, the human observer tests described below involved reading of coronal slices extracted from the 3D image for purposes of simplicity. To enable comparison with human observer performance, detectability index was derived in a form pertaining to a single 2D slice extracted from the volume. Slice extraction corresponds to

38 integration across the direction orthogonal to the slice (taken as fy). For the purpose of this work, axial images correspond to the x-y plane, with y corresponding to the depth direction.

Coronal slices (i.e., images in the x-z plane) were used in the observer study, corresponding to the usual tomosynthesis view. Therefore, 2D slice metrics can be obtained by integrating 3D metrics in the previous section over the fy direction. It is important to acknowledge that the slice detectability – chosen here simply for comparison to human observer tests – is not a complete metric for optimization of a 3D imaging system. While the fully 3D detectability index may be a suitable optimization metric, the rationale for analyzing slice detectability below was simply for purposes of measuring correspondence to human observer performance – and not as a basis for system optimization. The “slice” detectability corresponding to the 3D observer models above are therefore:

Prewhitening (PW) model:

2 ()T WTask df y d'2  df df (4.6) slice T2 S S S df x z  B Q E y

106

PW model with eye filter and internal noise (PWEi):

E22() T W df '2  Task y d df df (4.7) sliceE22 T S S  S df  N x z  B Q E y i

Non-prewhitening (NPW) model:

2 2 ()T WTask df y df x df z d '2   (4.8) slice TS22 S  Sdf () TW  df dfdf  B Q Ey  Tasky xz

NPW model with eye filter (NPWE):

2 E22() T W df df df '2   Task y x z dslice  (4.9) E4 TS 2 S  Sdf () TW  df 2 dfdf   B Q Ey  Tasky xz

NPW model with eye filter and internal noise (NPWEi):

2 E22() T W df df df '2   Task y x z dslice  (4.10) E4 TS 2 S  Sdf () TW  df 2  Ndfdf   B Q E y  Task y i x z

The eye filter employed in this study was a simple approximation of Barten’s contrast sensitivity curve of the human eye153 consistent with that used in the study by Burgess 84:

E( f ) f exp( cf ) (4.11) where f is the spatial radial frequency. The eye filter was implemented such that its maximum response occurred at 4 cycles/deg. For a typical viewing distance of 50 cm, c equals 2.2. The internal noise was implemented as uncorrelated white noise dependent on the magnitude of variation in background power spectra:119

107

2 D NSi 0.0001 eq  0,0 (4.12) 100

where D is the viewing distance estimated as 50 cm for this study, and Seq is the white NPS equivalent in total power to the image noise (sum of SB, SQ and SE). The scale factor 0.0001 was determined following variation as a free parameter as in Burgess84 to give coarse overall agreement to measurements.

The slice detectability index in Eqs. (4.6)-(4.10) formed the basis for theoretical calculations performed in comparison to human observer performance for a variety of imaging tasks described below. Note that although “slice” detectability describes imaging performance on a 2D slice, the derivation of such is only achieved via the fully volumetric analysis of the 3D NPS,

NEQ, and d’3D.

4.2.2. Imaging Tasks and Physical Imaging Phantoms The sections below describe physical experimentation conducted to allow direct comparison of theoretical d’slice with the performance measured in human observers. A variety of imaging tasks were implemented in real phantoms in a manner that imparted a range of conspicuity (from imperceptible to obvious) over a broad range of experimental conditions (number of projections, dose, and source-detector orbital extent).

4.2.2.1. Detection in Uniform Background Detection in a uniform background was investigated as the simplest case in which the NEQ was governed by quantum and electronic noise only in the absence of background clutter. The physical phantom used consists of acrylic spheres of various size embedded in a uniform polyurethane cylinder. As described below, the task corresponded to the detection of a (110 HU)

108 sphere against a uniform (90 HU) background. For tomosynthesis and CBCT, a sphere of 6.4 mm diameter was selected as the signal to present strongly varying conspicuity across a range of source-detector orbital extent (tot).

4.2.2.2. Detection / Discrimination in Background Clutter Chapter 3 described a phantom designed from principles of fractal self-similarity that contained different diameter spheres randomly mixed to give power-law spectral density. Power law parameters  and  measured previously [  =9.72e-7 (2mm3) and  =2.76] were taken as empirical parameters in calculations of the generalized detectability index. The variation in background power spectrum with tot is accounted through the product of the fully 3D SB(fx, fy, fz) with the tomosynthesis “double-wedge” corresponding to the angular range of sampled frequencies in the Fourier domain.

4.2.2.3. Imaging Tasks Imaging tasks were conceived that could be physically implemented in either the uniform or clutter phantoms and modeled according to a simple binary hypothesis (denoted as H1 and H2) testing model. Six identical objects were inserted in the phantom, giving six statistically independent trials for each imaging task. “Signal-present” images were taken from the central coronal slice through each object, and ROIs were selected such that signals were at the center

(see Fig. 4.1). On the other hand, “noise-only / signal absent” images were taken from the same or neighbouring slices without the signal. Five imaging tasks emphasizing different regions of the frequency domain (i.e., various spatial frequency content) were investigated in this study.

Task functions are plotted in Fig. 4.1 and described in the following section. i.) Sphere detection on uniform background

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As described in Chapter 2, signal detection in an otherwise uniform phantom corresponds to the case in which H1 is simply the signal represented by the object function, denoted O(x,y,z) (i.e., a sphere in the 3D image and a disk in a 2D slice), and H2 is a constant. The task function WTask is given by the product of the difference in attenuation coefficient between the signal and background (), and the Fourier transform of the signal (FT[O]). For the phantom described in

Section 4.2.2.1, the task function was computed with  taken as the measured signal difference between the acrylic sphere (110 HU) and polyurethane background (90 HU) in a full 360o CBCT reconstruction and FT[O] computed as the Fourier transform of a 6.4 mm diameter sphere in the

3D image,

 WTask   FT O   Acrylic   Polyurethane  FT O6.4 mm _ Sphere (4.13)

Signal values used in calculation of  were measured as an average from multiple ROIs at various positions on the central coronal slice of the reconstruction. ii.) Large sphere detection on cluttered background

Imaging tasks in cluttered background were modeled as signals (i.e., physical objects) embedded in the clutter phantom of Section 4.2.2.2. For tasks presented on cluttered background, the

“signal-present” images contain the signal at the center of the ROI, whereas “signal-absent” images present a background sphere of equivalent size (e.g., 12.7 mm for the large sphere) at the same location. The “signal-present” hypothesis was formulated as:

FT H1  1 FT O 1 FT B (4.14)

110 where 1 and O1 are the attenuation coefficient and object function of the signal, respectively, and B corresponds to background clutter (random collection of acrylic spheres), the power spectrum of which obeys the power-law relationship.

The “signal-absent” hypothesis is:

FT H2  2 FT O 2  FT B (4.15) 21FT O FT B

where 2 is the attenuation coefficient of the background (acrylic), and O2 (equal to O1 in this case) is the object function corresponding to an acrylic sphere of the same size in place of the signal within the background clutter. Note that the Fourier transform describes magnitude only

(disregarding phase), so FT[B] in theory may pertain to various independent realizations of the background, provided they have the same (e.g., power-law) noise characteristics. Different realizations of the background therefore cancel out in theory, such that discrimination of the signal (i.e., O1 in clutter) from a clutter-only image corresponds to:

WTask  FT H1  FT H 2 () 1   2  FT O 1     FT O 1 (4.16)

For the large sphere on a cluttered background task, the signal was a 12.7 mm diameter polypropylene sphere (-84 HU), which presented a lower contrast signal in comparison to an equivalent size acrylic sphere (110 HU) in the background clutter. The contrast, , was taken as the absolute value of the measured difference in attenuation between polypropylene and acrylic from a full 360o CBCT acquisition. The task function is thus given by:

 WTask Acrylic  Polypropylene  FT O12.7 mm _ Sphere (4.17)

111

As illustrated in Fig. 4.1, this task presented primarily low and mid-frequency components. iii.) Small sphere on cluttered background

Similar to the previous task, detection of a small Teflon sphere (710 HU) in a cluttered background of acrylic spheres (110 HU) follows Eq. (4.17), where  is the measured contrast of Teflon and acrylic, and O1 is the object function for a small (3.2 mm) diameter sphere:

 WTask Teflon  Acrylic  FT O3.2 mm _ Sphere (4.18)

As illustrated in Fig. 4.1, this task presented higher frequency components. iv.) Cube vs. sphere discrimination on cluttered background

Shape discrimination was similarly modeled as the Fourier transform of the difference of two hypotheses:

FT H1  1 FT O 1 FT B (4.19)

FT H2  1 FT O 2  FT B (4.20)

where the attenuation coefficient (1) is the same for the signal and background, but object functions of the signals to be discriminated (O1 and O2) vary. The task function is therefore:

WTask  FT H1  FT H 2  1 () FT O 1  FT O 2  (4.21)

A “cube versus sphere” discrimination task was formulated such that O1 representing an acrylic cube (of side-length 6.4 mm) is discriminated from O2 representing an acrylic sphere (of diameter 6.4 mm), i.e.:

112

    (4.22) WTask Acrylic () FT O6.4 mm _ Cube   FT  O 6.4 mm _ Sphere 

As illustrated in Fig. 4.1, this task consists of middle and high frequency components. v.) Encapsulated vs. solid sphere on cluttered background

A second shape discrimination task involved a 6.4 mm diameter acrylic sphere (110 HU) encapsulated by a 3.2 mm shell of paraffin wax (giving 12.8 mm total diameter, -50 HU) as the signal, which is discriminated from a solid acrylic sphere (110 HU) in the noise-only image. The two hypotheses can be similarly written as Eqs. (4.19) and (4.20), with O2 representing the paraffin-encapsulated sphere, and O1 representing the 12.7 mm diameter acrylic sphere. The task function can be derived as:

 WTask()() Acrylic  Wax  FT O3.2 mm _ Shell (4.23) where the difference in object functions yields the 3D encapsulating shell (an annulus on a 2D slice). As shown in Fig. 4.1, this task emphasizes higher frequencies compared to tasks in i), ii) and iv). The high-frequency content is related to the fine detail associated with the encapsulating layer.

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Figure 4.1. Fourier-domain task functions (left column) and coronal image ROIs (x-z) (images at right) for varying angular extent under the constant- and constant-Nproj cases: (a) sphere detection on uniform background; (b) large sphere detection in clutter; (c) small sphere detection in clutter; (d) cube vs. sphere

114 discrimination in clutter; and (e) encapsulated sphere vs. solid sphere discrimination in clutter. (Published with permission from Medical Physics)

4.2.3. Imaging Bench and Acquisition Parameters Images were acquired on the experimental imaging bench for tomosynthesis and CBCT described in Chapter 3 and illustrated in Fig.3.1. Acquisition techniques were held fixed at 120 kV (1.53 mm Al + 0.1 mm Cu added filtration) and 0.63 mAs per projection, imparting a constant in-air exposure per projection of 0.49 mR at the detector. Acquisition protocol closely followed those described in Chapter 3, where images were acquired for 12 orbital extents spanning a continuum of low-angle tomosynthesis to full CBCT according to the “constant-” and “constant-Nproj” schemes. The full acquisition protocol is provided in Table.3.1.

Image reconstruction was performed using the FDK algorithm for 3D filtered back-projection, with a Hann apodization filter and no additional inter-slice filter for tomosynthesis. Projections were binned at 2×2 pixels, and images were reconstructed at an isotropic voxel size of

0.518x0.518x0.518 mm3, chosen to adequately resolve the smallest sphere in the phantom.

Modified Parker weights121 were applied to redundant views for acquisitions of angle greater than 180o - fan, with the exception of a full 360o acquisition where a uniform weighting of 0.5 was multiplied to all projections.

4.2.4. Experimental Validation: Human Observer Performance

4.2.4.1. Human Observer study As a simple measure of observer-based imaging performance, multiple-alternative forced-choice

(MAFC) tests were performed in which an array of ROIs was shown to observers, with one ROI

115 containing the signal and nc-1 ROIs containing background only. A 9AFC test (nc=9, displayed as a 3×3 array of ROIs) was chosen to give optimal statistical power for the selected tasks according to the table formulated by Elliot154 which relates sensitivity of detectability and proportion correct to the number of choices. Each signal ROI was a 70×70 voxel sub-image cropped from a coronal slice of a 3D reconstruction with the signal at the center. The noise-only

ROIs were cropped from the same or neighboring slices, and care was taken to avoid out-of- plane shadow of the signal or other artifacts. Different noise realizations were used for each imaging task, but the same regions of interest were used for various tot in both the constant- and constant-Nproj schemes. In addition, both the signal and noise ROIs were randomly flipped up/down and left/right to minimize observer familiarity with the images. For each 9AFC test, the grayscale window was fixed to a range of 90% of the minimum pixel value to 110% of the maximum pixel value, and the level was set to the mean. Observers were not allowed to adjust the window/level or zoom of the images, and a constant viewing distance of ~50 cm was encouraged but not strictly enforced.

Observer studies were conducted in a darkened reading room using a monochrome diagnostic quality display (Image Systems, Richardson Electronics Ltd, Plymouth MN). Prior to each test, each observer was trained for a given task using training images acquired at the same experimental conditions as the test data (both constant- and constant-Nproj cases), typically 24 images for a given task, requiring ~10 minutes for the observer to gain familiarity with the task.

The test data for each task included images acquired at 12 levels of tot (detailed in Table.3.1) for both the constant- and constant-Nproj schemes, with five statistically independent samples acquired for each case, giving (12 angles) × (2 schemes) × (5 images) = 120 readings for each task and observer. The order of the five tasks was randomized for each observer, as was the order

116 of images presented for each task. At ~5 seconds per choice, a complete study required (5 tasks)

× (~10 min) (training) + (5 tasks) × (12 angles) × (2 schemes) × (5 images) × (~5 s) (test)  100 min for each observer.

For the fairly simple (non-clinical) imaging tasks involved in these phantom studies, medical physicists / engineers were considered suitable observers. Eight observers were used. For each data point on the graphs below, therefore (i.e., for each setting of tot, scheme, and task), a total of (8 observers × 5 independent images =) 40 responses were obtained. Assuming independence among observers, all responses were pooled, and the fraction of correct responses was computed to yield the mean proportion correct (Pcorr).

Measurements of Pcorr follow a binomial distribution with mean equal to Pcorr and standard deviation Pcorr(1- Pcorr). Statistical error in Pcorr can be estimated from the standard deviation or confidence interval. As mentioned below, Pcorr measurements were also interpreted in terms of d’slice and area under the ROC curve (Az). Based on the binomial distribution of Pcorr, the distribution of Az was derived using the relations between Az and Pcorr (below). The corresponding measurement error in Az was expressed as the 95% confidence intervals calculated from its distribution.

4.2.4.2. Comparison of Theoretical and Experimental Results

For direct comparison of theoretically derived detectability index [d’3D or d’slice] and experimentally measured Pcorr, we used the basic relationships among d’, Pcorr, and Az that follow from the simplifying assumptions of normal, equal variance distributions in the underlying decision variables and consistency in observer response over the course of the tests.137 The detectability index and Az are related by:

117

d 1 1 2 2 1   d   A   ex dx  1 erf  , (4.24) z      2  0 2   2  

d2  4inverf 2 2 A  1 (4.25)   z 2

which in turn are related to Pcorr according to:

'2 1  (xd ) n 1 P( d ', n ) exp ( x ) c dx (4.26) corr c   2 2

where nc is the number of alternatives in the AFC test (9 herein), and  is the cumulative

Gaussian distribution. Note the usual relationship: Pcorr = Az for nc = 2. A lookup table relating

Pcorr, d’, and Az was constructed using these relations. Theoretical and experimental results could therefore be directly compared in terms of any of these three performance metrics. In selecting a metric by which to display the results below, d’ (which is unbounded from 0 to ) and Pcorr

(which is less general in its interpretation due to a particular choice of nc) were felt to be somewhat less meaningful at a glance. While any of the three metrics would suffice for purposes of comparison, results below are reported in terms of Az. Compared to Pcorr, Az provides more general representation of results that can be compared to other studies (e.g., ROC performance); compared to d’, the Az metric is bounded and easily interpreted within the limits of pure guessing

(Az  0.5) and “completely obvious” (Az  1.0) which the physical phantoms and experimental conditions were constructed to span. Comparison in terms of d’ or Pcorr (not shown) exhibited the same level of agreement and does not affect the conclusions of this work.

118

4.3. RESULTS 4.3.1. Comparison of Theoretical Detectability and Human Observer Performance

Figures 4.2 and 4.3 plot Az versus total angular extent for all five imaging tasks under the constant- and constant-Nproj cases, respectively. In each graph, theoretical calculations from the cascaded systems model are plotted as solid or dashed curves, and human observer measurements are represented by individual data points with error bars corresponding to 95% confidence intervals.

Figure 4.2. Comparison of theoretical and measured performance for five imaging tasks in the constant-

acquisition scheme. Curves correspond to theoretical calculations for the five "slice" observer models

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(PW, PWEi, NPW, NPWE, and NPWEi). Reasonable correspondence between theoretical and experimental results is observed, with NPWEi showing the best agreement overall. (Published with permission from Medical Physics)

The constant- acquisition scheme (Fig. 4.2, = 0.45o for all cases) shows a monotonic increase in Az with angular extent as contrast is improved for task on uniform background (Task

1) and as out-of-plane clutter is gradually removed for tasks on cluttered background (Task 2, 3,

4, 5). Furthermore, in the constant- case, total dose increases with tot, further supporting a monotonic increase in performance. For each task, a maximum Az of 1 is reached at a certain value of tot beyond which all model and human observers identify the signal as conspicuous. Of the five tasks considered, Task 1 (sphere detection on a uniform background) and Task 4 (cube vs. sphere discrimination) were the easiest, with conspicuity predicted and realized in the region

o tot < ~30 . Conversely, Task 2 (large sphere detection in clutter), Task 3 (small sphere detection in clutter), and Task 5 (encapsulated sphere discrimination in clutter) were more challenging, as

o o shown by both theoretical calculations and observer results peaking in the range tot ~100 -150 .

Task 2 achieved better performance than Task 3, seen by a steeper increase in Az with tot. The interpretation: low-frequency tasks associated with detection of signals of larger spatial extent reside in a similar frequency range as the power-law background and thus, experience faster increase in detectability with the rejection of out-of-plane clutter (i.e., increase in tot). High- frequency tasks, on the other hand, associated with finer details exhibit frequency content in a higher range than the background and thus do not benefit as significantly from the rejection of out-of-plane clutter. The prewhitening models (PW and PWEi) tended to overestimate observer performance in all cases, while the non-prewhitening models (NPW, NPWE, and NPWEi)

120 exhibited reasonable agreement with measurement for all tasks over the entire continuum of tot.

Not surprisingly, the NPWEi model gave closer correspondence to the measured human performance than NPWE, since it includes additional terms related to inefficiency imparted by internal noise. We examined the observer efficiency implied by the ratio of measured and theoretical d’slice (squared) and found a complicated dependence on task (as might be expected

155 from Abbey et al. ) and angular extent (tot). Efficiency varied in the range ~0.1 – 0.5 but did not exhibit any clear trends in retrospective analysis as a function of task or tot.

Figure 4.3. Comparison of theoretical and measured performance for five imaging tasks in the constant-

Nprojacquisition scheme. Labeling is the same as in Fig. 4.2. Fair correspondence is observed between

121 theoretical and experimental results, including a complex non-monotonic trend in performance at large

o angles above tot~180 associated with finite sampling effects (view aliased noise). (Published with permission from Medical Physics)

Figure 4.3 summarizes theoretical and experimental results for the constant-Nproj acquisition scheme (Nproj = 89 projections for all cases). In this case, the total dose is fixed for settings of tot, giving a constant level of quantum noise amid varying levels of background clutter and view aliasing noise. Similar to the constant- case, the PW and PWEi observer models overestimated observer performance, whereas the various NPW models yielded reasonable agreement with experimental results. An exception was observed for the small sphere detection task, where

o human observer performance agreed with NPWEi model for tot < 180 , but was closer to the

o PW observers beyond tot ~180 . This result is evident also in Fig. 4.1(c), where due to the high

o contrast of the sphere, the signal was sufficiently different from the background for tot > 180 for conspicuous discrimination, and noise (including view-sampling effects) does not appear to

“masquerade” as signal.

An interesting non-monotonic trend in Az versus tot was predicted and observed: Az initially increases with tot in a manner similar to the constant- case due to reduced out-of-plane clutter; however, as the total angle increases further, performance is seen to degrade somewhat – particularly for imaging tasks involving higher spatial frequency components (Tasks 3 and 5) –

o followed by an increase towards tot = 360 . The nature of the effect is detailed more completely below, owing to the distribution of a limited number of projections over wide angular ranges.

o o While the non-monotonic trend was predicted and observed to occur in the range 180 <tot<360 ,

122 agreement between theory and measurement is far from perfect: the effect appears to be overestimated for Task 3 (small sphere detection) and underestimated for Task 5 (encapsulated sphere discrimination). In Task 1 (sphere detection on uniform background), theoretical calculations for the NPWEi observer model are consistent with the general trend of the observer measurement, but do not show exact correspondence in Az values. The other tasks (involving lower spatial frequency tasks) were in better agreement.

4.3.2. Number of Projections and Orbital Extent: Effect on Quantum Noise and View Aliasing o The initial decrease in detectability starting at tot ≈150 under the constant-Nproj scheme can be attributed to view-aliasing artifacts arising for large angular separation between projections, visible to the eye as granular mottle in the coronal images of Fig. 4.1 (evident also as familiar

o radial streaks in the correspondent axial images, not shown). As tot is further increased to 360 , detectability is seen to recover as a result of two possible scenarios of projection distribution illustrated in Fig. 4.4: (1) a range of projections overlap, forming a region of redundant sampling

(to which Parker weights are appropriately applied); or (2) projections do not overlap, but instead result in a region of finer sampling. The first scenario almost always applies for the constant-

case because the finely spaced projections approximate a continuum in which projections at

o angles above 180 -fan constitute redundant samples. In the constant-Nproj case, however, the first scenario may only occur under certain combinations of Nproj and tot, in which case the summing and averaging of redundant rays passing thought a particular voxel reduces stochastic noise components (quantum and electronics noise) associated with the projections but does not affect the magnitude of “deterministic” noise (background clutter) or the signal (task function). The number of redundant projections increases as the angular extent increases. Thus, greater

123 reduction in stochastic noise competes with deteriorating view-aliasing artifacts as angular extent increases, resulting in the non-monotonic reduction in detectability observed in Fig. 4.3. An alternative scenario may also arise due to the large  in the constant-Nproj scheme, where projections acquired above 180o-fan interleave between previously acquired projections, resulting in a region of finer sampling. In such cases, a reduction of view-aliasing artifacts also causes detectability to recover.

The distribution of projections in the Fourier domain (axial fx-fy plane) according to the central slice theorem is plotted in Fig. 4.4 for six values of tot. Note that the figure illustrates the angular sampling distribution associated with a given number of projection over a given angular range, and does not depict the actual noise power spectrum. Redundant, overlapping projections

o o o o appear as brighter spokes. The intermediate angular ranges (tot = 200 , 240 , 280 and 320 ) exhibit the behavior associated with scenario 1, in which projections overlap across a range of

o o angles above 180 . For this specific number of projections (Nproj = 89) the 360 case actually corresponds to finer sampling and gives a reduction in view-aliasing artifacts due to interleaving of projections. In relation to the “valley” observed in Fig. 4.3, detectability begins to decrease

o above tot~180 due to increased view-aliasing artifacts, followed by an increase beyond

o tot~280 due to the reduction of stochastic noise with greater number of redundant projections

(corresponding to scenario 1). At 360o, detectability further increases due to finer sampling

(corresponding to scenario 2).

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Figure 4.4. Illustration of Fourier domain projection distribution on the axial (fx-fy) plane according to the central slice theorem. A total of 89 projections are distributed across a range of tot. Under the assumption

o o o o of parallel-beam geometry, redundant projection views are sampled in the tot = 200 , 240 , 280 , and 320 cases (i.e., competing effects of increased view aliasing and reduced stochastic noise), while a 360o acquisition results in interleaved projection views and finer sampling (i.e., reduced view aliasing and an increase in detectability). The sampling distribution depends on the number of projections and angular extent. For example, evenly distributing 89 projections over 320o gives redundant, overlapping projections under the parallel beam approximation (“brighter” spokes); however, at 360o, projections do not overlap and result in finer angular sampling. (Published with permission from Medical Physics)

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4.3.3. Implications for Task-Based System Design in Tomosynthesis and CBCT Given the reasonable correspondence observed between theoretical and experimental results in

Figs. 4.2 and 4.3, we computed detectability for a variety of conditions as could apply to the design and understanding of tomosynthesis and CBCT systems. A spectrum of tasks were considered to elucidate the distinction of low- and high-frequency tasks: (a) low-frequency

Gaussian detection task (Gaussian width,  = 3.6 mm) on a uniform background; (b) low- frequency Gaussian detection ( = 3.6 mm) in a cluttered background; (c) mid-frequency discrimination of two Gaussian signals (1 = 3.1 mm versus 2 = 3.7 mm); and (d) high- frequency discrimination of two Gaussians signals (1 = 1.3 mm versus 2 = 1.8 mm). In each case, d’slice was computed as a function of Nproj and tot (with the dose per projection fixed at a level corresponding to 0.066 mR in-air exposure to the detector) to examine tradeoffs among background clutter and view aliasing. The NPWEi model was chosen, since it demonstrated the best overall agreement with human observer response (Figs. 4.2 and 4.3). For simplicity in these calculations, a parallel beam geometry was assumed, implying that projections 180o apart were considered redundant with a constant weight of 0.5 was multiplied.

Results are shown in Fig. 4.5. For any of the tasks, one may consider a “horizontal” slice of the d’slice surface as the case in which the total dose is fixed, and the angular range of the tomosynthesis / CBCT system is varied. In practical terms, this might correspond to a rotational

C-arm system in which one seeks to determine what value of tot provides a desired level of detectability. Conversely, a “vertical” slice of the d’slice surface corresponds to a fixed angular range, with the number of views and total dose varied. This might correspond to a tomosynthesis system with fixed tot (e.g., a clinical breast or chest tomosynthesis system), and one seeks to

126 determine what number of views (total dose) gives a desired level of detectability. The alternative case (not shown) in which d’slice (Nproj,tot) was computed at fixed total dose shows a complementary set of tradeoffs among quantum noise, background clutter, and view aliasing.

(d) 1.3 and 1.8 mm, each on a cluttered background. (Published with permission from Medical Physics)

As shown in Fig. 4.5, for all four tasks, when the number of projections is low (constant-Nproj with Nproj<~200), the trends observed in the experiments of Figs. 4.2-4.3 are once again observed: d’slice increases with angle, reaches a maximum, then decreases due to competing effects of view- aliasing and distribution of projections described above. As the number of projections increases

(and view-aliasing artifacts are reduced), a distinct difference is observed among the tasks in uniform and cluttered backgrounds. After reaching a maximum, d’slice decreases with tot for the uniform background task while remaining constant for the cluttered background tasks. In the absence of clutter, increasing the angular arc only increases quantum noise, therefore causing detectability to decrease; however, in a cluttered background, this effect is less pronounced due

127 to the preponderance of background noise outweighing quantum noise. Note also the overall reduction in the magnitude of d’slice in Fig. 4.5(b) compared to Fig. 4.5(a) despite equivalent task and signal power, the reduction attributable to background noise. Another difference between the uniform and cluttered background tasks can be seen from profiles at a fixed angle: for a given number of projections, the uniform background task exhibits continued improvement in detectability with Nproj as quantum noise is driven down by increasing dose, whereas the cluttered background tasks reach a background-noise-limited region beyond which increasing

Nproj or dose brings no further improvement in detectability.

Figures 4.5(b), (c), and (d) illustrate tasks with the same signal power but different frequency characteristics. The detectability index for the low-frequency task (b) is much lower than that for mid- and high-frequency tasks in (c) and (d), respectively, due to the frequency components of the task function coinciding with those of background noise. This corresponds to the observation of Myers et al.156 regarding noise “masquerading” as signal. For low-angle tomosynthesis (i.e., low tot), increasing the acquisition angle and thereby removing of out-of-plane clutter demonstrates more pronounced improvement for low-frequency tasks than for higher-frequency tasks.

On the other hand, detectability index for higher frequency tasks is more sensitive to the choice of tot and Nproj, since mid- and high-frequency tasks reside in the same frequency region as view-aliasing noise and are therefore more prone to the complex tradeoffs among noise components described above. View-aliasing artifacts within the low Nproj and high tot region degrades performance for mid-frequency (c) and high-frequency (d) tasks, but barely affects the low-frequency (a) task. For the mid-frequency task in (c), a sharp transition in the magnitude of d’slice can be observed along a diagonal with an approximately constant choice of . Similarly

128 for the high-frequency task in (d), there is a narrow selection of tot and Nproj with an approximately constant choice of  which marks optimal performance. The trend observed is highly dependent on the frequency range of the imaging tasks, choice of observer model, and relative magnitude of signal and noise. Overall, it is clear that low-frequency tasks benefit more from increasing acquisition angle and are mainly limited by background noise, while tasks involving mid- and high frequency components, require careful selection of acquisition parameters to minimize the influence of view-aliasing artifacts, quantum and electronics noise.

The framework provided by 3D cascaded systems analysis combined with idealized task functions and observer models provides a quantitative foundation from which specific trends in performance can be more rigorously investigated with respect to specific imaging systems and applications.

4.4. DISCUSSION AND CONCLUSIONS This work derived the generalized detectability index from the 3D cascaded systems model as a task-based performance metric for tomosynthesis and CBCT, and validated it in comparison to human observer response for several simple imaging tasks over a wide range of imaging conditions. A reasonable level of agreement was observed between theoretical predictions and experimental results. Interesting, non-trivial trends were revealed that suggest important design considerations for system optimization. Tradeoffs among anatomical noise, quantum and electronics noise, as well as view sampling effects result in complex behaviour (e.g., non- monotonic dependence of d’slice on tot) that suggest optimal choices of acquisition parameters specific to the imaging tasks.

Such behavior was reasonably predicted and explained by the cascaded systems model, with the level of agreement depending on the imaging task – perhaps not surprisingly, considering the

129 simplicity of Fourier hypothesis-testing task functions in comparison to the complexities of human visual perception. The case for which the model deviated most from measurement was

o the encapsulated sphere discrimination task under the constant-Nproj case for tot above 180 , where theoretical prediction did not accurately predict the large drop observed in human observer performance. This suggests room for improvement when modeling high-frequency tasks which are more prone to deterioration by stochastic noise and view sampling effects.

Overall, results demonstrate that generalized detectability index yielded reasonable correspondence with human observer performance for a variety of simple imaging tasks over a broad range of experimental conditions in both the constant- and constant-Nproj schemes, helping to bridge the gap between Fourier-based metrics (e.g., NEQ) of system performance and observer-based characterization of image quality (e.g., ROC).

Of the simple observer models considered, the non-prewhitening models, especially the NPWEi model, yielded the best overall agreement with human observer response. The fair agreement for the NPW model may seem surprising, considering that such models have been shown previously to correspond poorly with human observer performance due to a large zero-frequency (DC) response157. In the generalized detectability index calculation above, inclusion of the power-law noise in the denominator introduces a large zero-frequency noise component which diminishes

DC response – similar to the effect achieved by an eye filter in the NPWE or NPWEi models.

Therefore, the DC effect that confounded NPW model agreement in previous work158 was not a significant factor in the experiments considered above, with the exception of the uniform background task. Burgess et al84 further modified the NPW model to include an eye filter and internal noise, which improved agreement with observer response significantly, consistent with the findings above. Prewhitening observer models consistently overestimated response likely due

130 to the fact that the discrete spheres in the cluttered background were indistinguishable from the signal, and human observers were not able to completely "decorrelate" noise. That said, it bears reiteration that the purpose of this work was not to advance or improve any particular observer model; rather, this work aimed to determine the extent to which trends in human observer response may be predicted by first-principles of cascaded systems analysis (GNEQ) combined with task functions through any of these simple observer models. More sophisticated observer models through ongoing work in perception science will presumably yield further improvement, including channelized Hotelling observer models159, 160,161 and forms of eye filter and internal noise models.162, 163

This study involved simple, idealized detection and discrimination tasks as a starting point to assess imaging performance. Modeling of more complex and higher order tasks forms an important area of future work to better relate such work to clinical applications. Examples include “search” tasks analogous to an observer detecting and localizing suspicious lesions on a radiograph or “estimation” tasks in which an observer needs to approximate the size of a lesion.

Also of interest are task functions representing multiple hypotheses – e.g., discrimination of signal-absent from signal-benign and signal-malignant hypothesis. Similarly, the model can be incorporated in analysis of multiple tasks – e.g., detection of a (low-frequency) lesion followed by detection of (high-frequency) calcifications.111 In the context of volumetric imaging, it remains to be shown to what extent the fully 3D detectability index [Eqs. (4.1)-( 4.5)] corresponds to cases in which the observer “scrolls” slices, is presented with a montage of multiple slices simultaneously, or perceives a volumetric rendering all at once. Another limitation of this work is that only one reconstruction algorithm [filtered backprojection (FBP)] with one reconstruction filter (a smooth cosine Hann filter) was investigated. Cascaded systems

131 analysis is well suited to description of FBP reconstruction, and extension to other methods (e.g., iterative reconstruction) would require a substantially modified approach, as detailed in Chapter

6. Accommodating various reconstruction filters within the model is straightforward, has been examined in previous work57, but was not investigated directly here, since it had less influence on task performance than angular range and number of projections.

In summary, the generalized detectability index was compared to human observer performance for a variety of simple tasks over a broad range of experimental conditions. Reasonable

o agreement was obtained for all tasks across the tomosynthesis angular range tot < 180 .

Discrepancy was observed for high-frequency tasks (e.g., small sphere detection and encapsulated sphere vs. solid sphere discrimination) under the constant-Nproj scheme under

o conditions dominated by view-sampling artifacts (i.e., small Nproj with tot > 180 ). Such discrepancies identify areas for improvement of the model and future investigation for task performance under conditions dominated by image artifact (rather than purely stochastic noise).

Still, generalized detectability index derived from the 3D cascaded systems model demonstrates considerable promise in relating simple Fourier metrics to human observer performance and suggests utility as an objective function in the design and optimization of 3D imaging systems in

CBCT and tomosynthesis.

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Chapter 5 Cascaded Systems Analysis of Noise and Detectability in Dual-Energy Cone-Beam CT 5.1. INTRODUCTION Dual-energy computed tomography (DE-CT) and dual-energy cone-beam computed tomography

(DE-CBCT) provide numerous potential advantages over conventional single-energy CT, most importantly in providing superior contrast by selectively combining low- and high-energy image data (projections or reconstructions) in the decomposition of various materials in the 3D image.

Weighted subtraction of 3D reconstructions exploits the attenuation and scatter characteristics of materials at different energies, thereby differentiating materials that may be indistinguishable in single-energy CT - for example, in kidney stone characterization and distinguishing iodinated vessels in bone.15-17 In contrast-enhanced imaging (e.g., iodine-enhanced angiography, renal, or liver imaging), DE-CT is frequently acquired at energies straddling the K-edge of the contrast agent (e.g., 33 keV for iodine), thus allowing separation of contrast-enhanced structures from other dense materials (e.g., bone) that may confound visualization.18, 19 More complex DE decomposition algorithms can begin to transcend the traditional contrast mechanism of linear attenuation coefficients through the discrimination of tissue composition in terms of effective atomic number, electron density, basis material densities, etc.15, 164-168 In addition to improved contrast, DE-CT or DE-CBCT could potentially achieve more accurate beam hardening correction in projection-based decomposition164, 169 and reduction of radiation dose in contrast- enhanced imaging by producing a virtual non-contrast image that in some scenarios can preclude a pre-contrast scan. 170

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Dual-energy CT has shown promise in a wide spectrum of clinical applications,170-173 raising the need for imaging performance optimization to knowledgably guide applications with respect to radiation dose, image noise, and the imaging task. In addition to the parameters that govern imaging performance in single-energy CT (e.g., kV, mAs, detector performance, system geometry, etc.), DE imaging introduces additional factors that significantly affect image quality

(e.g., the selection of kV pair, application of added filtration, dose allocation between the high- and low-energy scan, and choice of noise reduction filters and decomposition algorithms).

Current methods of technique optimization rely primarily on physical experimentation and iteration across the numerous relevant parameters. Such experimentation involves acquisition of

DE images at permutations of kV pair, dose allocation, filter material and/or thickness, and analysis in terms of basic performance metrics such as contrast, noise, and contrast-to-noise ratio

(CNR).174-176 Due to the vast parameter space that needs to be taken into consideration, an accurate theoretical framework that is predictive of DE-CT image quality would greatly benefit such efforts in providing an efficient, rigorous, and more complete examination of the parameters of interest (potentially beyond the experimental limits of existing scanner technologies).

Toward this end, early work derived closed-form expressions of image noise associated with common DE-CT decomposition algorithms and used such expressions to elucidate several fundamental principles in DE-CT imaging techniques. Alvarez et al.164 described the image noise in the projection-based estimates of Compton and photoelectric contributions to linear attenuation coefficient, and in a later work, showed that linear combinations of such estimates at an optimized display energy may form a spectral shift artifact-free image that contains the same quantum noise as a single-energy image at the same total dose177 - a result proven experimentally

134 by Kalender et al.167 Kelcz et al.168 developed an equivalent decomposition method for estimating the densities of two materials and demonstrated the noise advantage of the two-kV technique (where high- and low- energy images are acquired at two different energies) compared to the two-crystal technique (where both images are acquired simultaneously with two crystal layers – the first crystal forming the low-energy image and the second the high-energy image.).

An abundance of related work also investigated the noise characteristics of particular DE decomposition algorithms.15, 165, 166, 178, 179 The work reported below presents a theoretical model for reconstruction-based DE-CBCT decomposition based on previous modeling of DE projection radiography and CBCT.46, 47, 55, 58 Analytical forms for the DE-CBCT performance metrics such as the noise-power spectrum (NPS), modulation transfer function (MTF), and noise-equivalent quanta (NEQ) are derived, allowing description of image quality with respect to a given imaging task in terms of the detectability index (d'), which has previously shown reasonable correspondence with human observer performance for a broad range of imaging conditions and simple imaging tasks. 180

As detailed below, this chapter first presents the cascaded systems model extended to reconstruction-based DE-CT or DE-CBCT, then validates the accuracy of the model by comparing theoretically calculated DE NPS and NEQ with experimental measurements. The utility of the model is demonstrated in example optimization of DE acquisition parameters (dose allocation and kV pairs) in relation to specific imaging tasks using detectability index as the objective function. The results below are restricted to analysis of the 3D NPS and NEQ near the central axial plane (ignoring effects of the Fourier "null cone" / cone-beam artifact for a circular source-detector orbit126) and assumes "local" stationarity of the first and second-order image statistics181- i.e., invariance of the NPS within a limited (central) region of the volumetric image.

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5.2. METHODS 5.2.1. Reconstruction-Based Dual-Energy Image Decomposition The most common form of reconstruction-based (also called “image-based”) DE image decomposition involves a weighted subtraction of the high- and low-energy 3D image reconstructions:

I DE  I HE  wI LE , (5.1) where IDE, IHE, and ILE denote the dual-energy, high-energy (HE), and low-energy (LE) CBCT reconstructions, respectively. The tissue cancelation factor w is a scalar broadly defined to reduce the signal value (attenuation coefficient or HU) of one material, M1, to the same intensity as another material, M2, effectively "cancelling" M1 from the DE image. Assuming perfect cancelation (i.e., ignoring effects of x-ray scatter, spatial variation of x-ray energy throughout the object, etc.), we have:

DE  DE   HE ww  LE   HE   LE , (5.2) MMMMMM1 2 1 1 2 2

yielding w as the ratio of the difference in effective linear attenuation coefficients (1 and 2) of the two materials (M1 and M2) in the high-energy image to that in the low-energy image:

HE HE MM w  12. (5.3) LE LE MM12

The effective attenuation coefficients, , are computed using linear attenuation coefficients provided in ICRU Report No. 33182 and a polychromatic beam model generated from Spektr73.

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Variants of the simple weighted subtraction algorithm have been reported in the literature, e.g., applying weighting factors to both the high- and low-energy images.183 A bit of arithmetic reveals that the signal and noise are scaled equivalently in such forms, and the detectability index

(below) is unaffected by the form of (linear) weighting factors applied if the same material is being cancelled. Alternatively, w may be freely varied for manual adjustment of the contrast between structures of interest and the background [e.g., to account for effects of scatter, beam hardening, or other reconstruction artifacts, or varied in this study to demonstrate the effect of material cancelation on image quality (Fig. 5.5)], exposing w as a parameter that may be adjusted as simply as the window and level. Results below pertain to the single-parameter, weighted subtraction DE decomposition algorithm given in Eq. (5.1).

5.2.2. Cascaded Systems Analysis of DE-CBCT The basic cascaded systems analysis model for CBCT was described in Chapter 2, with generalizations including anatomical noise presented in Chapter 3. The separate models for DE decomposition and 3D CBCT cascaded systems analysis naturally combine in a general form for reconstruction-based DE-CBCT. As illustrated in Fig. 5.1, the DE decomposition step is incorporated following the reconstruction stages. Analogous to the form derived in Richard et al. for dual-energy radiography,55 the DE NPS, SDE, for DE-CBCT is given by:

SDE S HE w2 S LE . (5.4) where SHE and SLE are the HE and LE energy NPS from Eq. (5.3). Frequency dependence of S and T [(fx, fy, fz)] is omitted for conciseness. Similar to the single-energy NPS, the units of the 3D

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DE NPS, SDE, is 2mm3, where  denotes the 3D DE image signal [i.e., the linear combination of single-energy linear attenuation coefficients according to Eq. (5.1)] with units mm-1.

Figure 5.1. Schematic diagram for cascaded systems analysis of reconstruction-based DE-CBCT. The DE decomposition step indicated by w combines the low- and high-energy reconstruction NPS (SLE and SHE) into the DE NPS (SDE). (Published with permission from Medical Physics)

The dual-energy NEQ can be defined based on DE NPS and MTF in a manner similar to the single-energy NEQ:

T 2 NEQDE   f (5.5) S DE where T is the system MTF, assumed to be equal for the low-, high- and dual-energy cases for simplicity.

The NEQ can be extended to account for the influence of anatomical background in diminishing task performance by including the background ("clutter") power spectrum, denoted SB, in the denominator as an additional noise term, yielding the "generalized" NEQ.124 In anatomical sites such as the breast and lung, SB is frequently characterized by the power-law relationship introduced in Chapter 3:

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 S  (5.6) B af  where the exponent  describes the degree of correlation (clumpiness), a is a unity scale factor of units mm-1, and  is the magnitude of the background power spectrum, carrying the same units as

2 3 107, 108 SB,  mm .

A similar expression can be written for the generalized NEQ of DE-CBCT as:

2 DE T NEQ  f DE DE 2 (5.7) SSTB

DE where SB is the dual-energy anatomical power spectrum (in the object domain). The weighted subtraction decomposition does not change the magnitude of  from a single-energy reconstruction, but does affect  in a manner proportional to the square of the contrast between the two materials that constitute the power-law pattern.24 For example in DE breast imaging, where the contrast of endogenous fibroglandular, adipose, and tumor tissues are modified by decomposition, the factor DE is proportional to the square of the contrast between fibroglandular and adipose tissue in the DE image. Selection of  and  is detailed below.

5.2.3. Experimental Validation of 3D Dual-Energy NPS and NEQ Imaging performance metrics (e.g., NPS, NEQ) computed from the cascaded systems analysis model were validated against measurements from an experimental imaging bench consisting of:

(1) an x-ray tube (DU693 in EA10 housing, Dunlee, Aurora, IL) with 14o anode angle and 0.8 mm focal spot size; (2) a PaxScan 4030CB flat-panel detector (Varian Imaging Products, Palo

Alto, CA) with 2048x1536 pixels at 0.194 mm pixel pitch and ~250mg/cm2 CsI:Tl scintillator;

139 and (3) a motion control system (Compumotor 6k8, Parker Hannifin, Cleveland, OH) that operates a central rotation stage and allows precise positioning of the source and detector. The system geometry was set to that approximating a dedicated extremities scanner23 with 43.3 cm source-to-axis distance and 55.4 cm source-to-detector distance. Acquisitions were performed with the FPD operated in 2x2 binning mode with a pixel size of 0.388 mm.

Low- and high-energy images were acquired over energy ranges of 60-90 kV and 90-130 kV, respectively, based on generator limitations and typical imaging techniques for DE-CBCT applications. Additional filtration was held fixed to 5 mm Al and 0.3 mm Cu for the low-energy dataset, and 0.3 mm Cu and 0.5mm Ag for the high-energy dataset. Previous work identified such added filtration as near optimal for DE projection imaging,184 and application of the theoretical model to optimize filtration specifically for 3D DE-CBCT is the subject of possible future work. In particular, the addition of 0.5 mm Ag to the high-energy beam has been shown to provide optimal separation between low- and high-energy spectra. Imaging techniques (mAs) were selected to deliver an approximately equal dose of 7.5 (±0.39) mGy at all kV pairs, with absolute dose measured using a 0.6 cm3 active volume ionization chamber (Accu-Pro 9096,

RadCal, Monrovia, CA) at the center of a 16 cm diameter acrylic CTDI phantom.

To minimize scatter and beam-hardening artifacts in the image data, NPS measurements were performed in reconstructions of air with a 22.8 mm of Al placed in the beam to simulate attenuation through a uniform object (equivalent to ~85 mm water in a 90 kV beam filtered with

5 mm Al and 0.3 mm Cu). This method is equivalent to measuring NPS from a uniform object, e.g., a water cylinder, but with improved stationarity in the first and second-order statistics.

Images were reconstructed from 360 projections acquired over 360o using 3D filtered-back projection with a Hann apodization filter at isotropic voxel size of 0.3 mm.

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Dual-energy images were decomposed from LE and HE reconstructions according to Eq. (5.1).

The tissue cancelation factor w was allowed to vary arbitrarily for the purposes of model validation. Two single-energy CBCT scans were acquired at each energy, allowing two DE images to be reconstructed at each energy pair. Subtraction of the 3D images provided detrending of background shading (e.g., heel effect, cupping artifact, etc.) prior to analysis of the

NPS. Thirty non-overlapping volumes of interest (VOIs) of size 53x53x53 voxels or

15.9x15.9x15.9 mm3 were placed at a constant distance from the center of reconstruction within each difference image. The NPS was calculated in a similar way as reported for single-energy image in Chapter 3 - as the ensemble average of the square of the Fourier transform of each realization, denoted as VOI DE :

2 DE1 bx b y b z DE NPS fx,,,, f y f z   FT VOI x y z (5.8) 2 NNNx y z

th where bi and Ni are the voxel size (0.3 mm) and extent (53 voxels) of each VOI in the i direction, respectively. The factor of 1/2 accounts for the amplification of noise resulted from the subtraction detrending process. The statistical error in the NPS estimation was reported as the standard deviation across the ensemble of VOIs at each spatial frequency, divided by the square root of the number of VOIs in the measurement.

5.2.4. Detectability Index as an Objective Function Previous work applied the task-based detectability index to the analysis of DE projection radiography, CBCT, and tomosynthesis by combining simple observer models and imaging tasks with the NEQ.58, 111, 180 Chapter 4 presented validation of such in comparison to human observer performance for a variety of idealized imaging tasks, providing a simple yet powerful framework that bridges the gap between objective Fourier-domain metrics and human observers.119, 180

141

Considering the linear weighted subtraction in DE-CBCT decomposition, a straightforward extension of the single-energy detectability index yields its DE analogue:

()TWDE  2 d' 2  Task df df df , (5.9) DE DE DE x y z SS B where TDE denotes the DE system MTF, which was assumed to be equal to the high- and low-

137 energy MTF. The term WTask is the Fourier domain task function, defined as the difference of the two hypotheses in a binary decision task, namely, "signal-present" versus "signal-absent" (or

"background-only") for a detection task, and "signal A" versus "signal B" for an A versus B discrimination task. The "signal-absent" hypothesis corresponds to the Fourier transform of the background, which may either be uniform (e.g., in air) or structured (e.g., power-law anatomical background in the breast or lung). The "signal-present" hypothesis, on the other hand, is given by the Fourier transform of the object function (e.g., a sphere) on the background and carries magnitude equal to the contrast between the signal and the background. For reconstruction-based

DE-CBCT, similar definitions apply, with the hypotheses weighted by the DE contrast (CDE) which, following Eq. (5.1), is a linear combination of the contrast in high-energy (CHE) and low- energy (CLE) images:

CDE C HE wC LE (5.10)

The detectability index shown in Eq. (5.9) corresponds to the prewhitening observer model, where the observer is assumed to be able to fully decorrelate 3D noise characteristics.

Alternatively, the 3D non-prewhitening observer filters the noise with a detection template in the form of the expected signal:

142

2 ()TDE  W2 df df df '2  Task x y z d  . (5.11) DE (()SDE SDE ) T DE W2 df df df  B Task x y z

Compared to the prewhitening observer, the non-prewhitening model has been shown to give better agreement with human observers. 84, 180

The detectability index as written in Eqs. (5.9) and (5.11) correspond to the 3D detectability index in which the observer is able to fully interrogate volumetric information. As mentioned in previous chapters, the extent to which such a simple formulation corresponds to observers dynamically scrolling slices or interpreting multiple slices at once is the subject of ongoing research in image perception. The 3D detectability index can also be defined in analogous forms pertaining to a single slice through the reconstruction as derived in Chapter 4. The results below are presented in terms of the 3D detectability index in order to capture the fully volumetric signal and noise characteristics without limiting results to a particular plane of visualization.

5.2.5. Optimization of DE-CBCT Imaging Techniques Example calculations were performed to demonstrate the utility of detectability index as an objective function for optimizing imaging techniques and decomposition parameters, including the tissue cancelation factor (w), dose allocation (A), and kV pairs. The tissue cancelation factor w was chosen to “cancel” a particular material relative to another, giving w as a function of kV pairs. The detectability index was then calculated as a function of both the dose allocation factor

(i.e., the fraction of dose imparted by the LE acquisition, ranging from 0 to 1) and the kV pair, and the peak detectability d'(A, kVLE, kVHE) identified the optimum combination of A and kV.

Calculations assumed a nominal total dose (HE plus LE scans) of 15 mGy. The object was assumed to be equivalent to 15 cm of water. The same filtration was used as that in the

143 experiments described in Section II.C (5 mm Al and 0.3 mm Cu for low-energy, and 0.3 mm Cu and 0.5 mm Ag for high-energy acquisition). Electronics noise in the simulation was assigned to be 7000 electrons based on measurements from Yang et al.185

Three idealized imaging tasks were computed based on a variety of potential DE imaging applications:

5.2.5.1. Identification of kidney stone composition

A kidney stone differentiation task was formulated assuming two minerals commonly found in

186 renal calculi, brushite (CaHPO4·2H2O) and calcium phosphate Ca3(PO4)2. The linear attenuation coefficient of each was calculated from the respective elemental composition and density (brushite: 2.33 g/cm3, calcium phosphate: 3.14 g/cm3). Dual-energy decomposition was set to cancel brushite to the same intensity as a soft tissue (ICRU-44)187 background. The resulting contrast between brushite (or equivalently soft tissue) and calcium phosphate was taken as the DE contrast, CDE. The detectability index was computed for a low-mid frequency detection task consisting of a 1.5 mm radius sphere:

W FT CDE  r2  x 2  y 2  z 2 (5.12) Task     where H is the heavyside step function and r = 1.5 mm.

5.2.5.2. Iodine-enhanced vessel versus bone

A vessel-in-bone task was computed to simulate a common application of contrast-enhanced angiography in which iodine-enhanced vessels need to be distinguished from surrounding bone.

The tissue cancelation factor was chosen to cancel cortical bone to the same intensity as soft tissue, leaving an iodine-enhanced vessel. The dual-energy contrast was calculated as iodine (10

144 mg/mL) versus bone (or equivalently soft tissue). This amounts to a detection task in the DE image. For simplicity, the directional dependence of the vessel was ignored and a symmetrical

3D mid-frequency imaging task was defined directly in the Fourier domain as the difference between two Gaussians:

f2 f 2  f 2   f 2  f 2  f 2  WCDE exp x y z  -exp   x y z  (5.13) Task 2222    12   

-1 -1 where 1 = 0.3 mm and 2 = 0.2 mm .

5.2.5.3. Soft-tissue tumor detection in anatomical background

A generic soft-tissue tumor detection task was modeled assuming a tumor embedded in a power- law cluttered anatomical background. Pertinent tissues include the tumor and the two materials constituting the power-law pattern. Dual-energy contrast was therefore modeled as the signal difference between the tumor and the background material closer in attenuation coefficient to the tumor. Results below are shown for a breast tumor detection task on a fibroglandular and adipose tissue background. Other tasks that can be modeled within a similar framework (the subject of future work) include detection of a lung nodule in a lung tissue and air background, detection of a sarcoma in a muscle and adipose background, etc.

The linear attenuation coefficient of breast tumor was taken from Johns and Yaffe for infiltrating ductal carcinoma:188

tumor  0.0343 Al  0.8411lu (5.14)

145 where  Al and lu are the linear attenuation coefficients of aluminum and lucite [Poly(methyl methacrylate) (PMMA)], respectively. The dual-energy contrast was taken as the difference in signal levels between breast tumor and fibroglandular tissue.

The anatomical background noise was characterized by the power-law relationship described in

Eq.(5.6), with  equal to 3 (consistent with power spectrum measurements from the breast by many authors 114, 115 and the expected value of 3D self-similar “fractal” structure59). The value of

 for anatomical background in the breast was estimated using the relationship

 () DE  DE 2 , i.e.,  was taken as proportional to the square of the contrast fibroglandular adipose between adipose and fibroglandular tissue. The proportionality constant  was found from previous measurements in a lung phantom116 and a sphere phantom59, both yielding similar values of (~0.0079 mm3).

The nominal imaging task for the tumor detection task consisted of a low-frequency 3 mm

Gaussian task defined in the spatial domain and imaged at a total dose of 15 mGy. To illustrate the benefit of tissue cancelation, w was allowed to vary from 0 to 1 instead of being held fixed as in the previous two tasks described in Sec. 5.2.5.a and 5.2.5.b. The total dose and the frequency

DE DE DE contents of the tasks were also varied to illustrate the impact of C , S and S B on detectability. Two additional dose levels were investigated, 1.5 and 0.15 mGy, spanning three orders of magnitude to examine the low-dose limits of detectability and illustrate tradeoffs between quantum noise and anatomical clutter as imaging conditions transition from an anatomical noise limited regime to a quantum noise limited regime. Two additional imaging tasks included: (1) a low-mid frequency task defined as a 1.5 mm radius sphere, and (2) a high-

146 frequency task modeled as the difference between two Gaussian tasks defined in the frequency

-1 -1 domain following Eq. (5.13) with 1 = 0.9 mm and 2 = 0.8 mm .

5.3. RESULTS 5.3.1. Comparison of Theoretical and Experimental NPS and NEQ Figure 5.2 shows example DE NPS [(a), (b), (d), (e)] and NEQ [(c), (f)] for the [60, 130] kV pair

DE image. The profiles in Fig. 5.2(b) and (c) are the radial average calculated from the axial plane of the 3D NPSDE and NEQDE, respectively. The sagittal profiles in Fig. 5.2(e) and (f) represent the angular average of line profiles taken 4 samples away from the origin of the fz axis,

-1 i.e., fz = 0.249 mm . The NPS and NEQ from single-energy reconstructions are also plotted as dashed lines on the corresponding graphs for comparison. Error bars depict the standard deviation of measurements from all VOIs. The tissue cancelation factor w was set to 0.295, resulting in a DE NPS with magnitude intermediate to that of the high- and low-energy. The magnitude of NPSDE could be greater than both the high- and low-energy NPS, depending on the choice of w, but will always be greater than the high-energy NPS as evident from Eq. (5.4).

147

Figure 5.2. Comparison of theoretical and measured DE NPS and NEQ from the [60, 130] kV DE image.

The top row shows results in the axial (fx, fy) plane. An axial slice through the 3D DE NPS is shown in (a).

The radial average of the 3D NPS and 3D NEQ for this axial plane are displayed in (b) and (c). The bottom row shows the sagittal (fx, fz) plane. A slice through the 3D NPS in the sagittal plane is shown in

(d). Line profiles along the fz direction of the single-energy and dual-energy NPS and NEQ are plotted in

(e) and (f). Theoretical predictions are shown as curves, whereas experimental results are shown as individual data points for (b), (c), (e), and (f). Reasonable agreement between theory and experiment was observed for both the single-energy and dual-energy results. (Published with permission from Medical

Physics)

Theoretical calculations of both single- and dual-energy NPS and NEQ are in reasonable agreement with experimental measurements. A slight discrepancy is observed in the NEQ tails at

148 spatial frequency above ~0.7 mm-1 where measurements exhibit a steeper decline than theory.

The discrepancy is attributed to two possible sources of error. First, a slight underestimate of the tails of the MTF causes a slight underestimate of the NPS tails, which is in turn amplified in the

NEQ since it enters in the denominator. Second, there are a variety of possible noise sources in the imaging / reconstruction system that may not be exactly accounted in the model, including small discrepancies in spectral model, electronic noise, and aliasing artifacts. Small errors in such factors may appear as subtle discrepancies in the NPS tails but are magnified in the high- frequency NEQ. Furthermore, a slightly larger discrepancy is observed in the sagittal NEQ along the fz axis [Fig. 5.2(f)] where samples are selected closer to the region of the cone-beam artifact

(null cone).

Figure 5.3 presents axial NPSDE for a wide range of energy combinations at: (a) fixed high- energy at 110 kV and low-energy ranging from 60-90 kV and (b) fixed low-energy at 80 kV and high-energy ranging from 90-130 kV. A constant tissue cancelation factor (w = 0.5) was used in

DE decomposition for purposes of illustration. Experimental results were plotted as individual data points for the [60, 110] and [90, 110] kV pairs on (a) and the [80, 90] and [80, 130] kV pairs on (b). Figure 5.3(c) plots the axial NPSDE for the [60, 130] kV image at varying levels of w. The measured NPS exhibit a fairly typical level of experimental error (standard deviation in measured spectral estimates), and theoretical results demonstrate fairly good correspondence and fall within the error bars of most experimental data points. Other kV combinations and w values were also examined and similar levels of agreement were observed. The general trend of NPS as a function of kV is consistent with the behavior of single-energy imaging and is characteristic of the filters applied. In Fig. 5.3(a), the NPS is seen to decrease with increasing low-energy kV. For an incident x-ray fluence imparting the same dose to the center of the object, the total fluence to

149 the detector after attenuation by the object increases with beam energy, therefore decreasing the

NPS. Conversely, in Fig. 5.3(b), where the low-energy kV is kept constant, the DE NPS generally increases with high-energy kV due to the introduction of the Ag filter which decreases the fluence to the detector beyond a certain kV (~80 kV in this case). This effect, in conjunction with decreased quantum detection efficiency [g1 in Eq. (5.3)], results in NPS increasing with kV.

Figure 5.3. Comparison of theoretical and experimental DE NPS for a range of experimental conditions.

(a) Fixed high-energy beam, and varied low-energy beam, also for a constant tissue cancelation factor w =

0.5. (b) Fixed low-energy beam, and varied high-energy beam for a constant tissue cancelation factor w =

0.5. (c) Varying levels of w at fixed low- and high-energy beams (60 and 130 kV). (Published with permission from Medical Physics)

5.3.2. Detectability Index The optimization results for the brushite vs. calcium phosphate kidney stone discrimination task are plotted in Fig. 5.4 (a)-(d) as a function of kV pairs (high-energy kV, kVHE, and low-energy kV, kVLE). As described in Section 5.2.5.1, the tissue cancelation factor w was chosen to cancel brushite to a soft-tissue background and has intrinsic energy dependence as well. Figure 5.4(a)

150 plots the optimal dose allocation factor, A*, identified as the dose allocation, A, that maximizes detectability at each kV pair. Some interesting trends were observed: for a fixed kVLE, A* decreases with higher kVHE as more dose is required to compensate for increased noise and decreased detective quantum efficiency (DQE) in the high-energy image; on the other hand, for a fixed kVHE, A* decreases with higher kVLE as the low-energy image becomes less noisy [Fig.

5.3(a)] as a result of increased transmission through the object; furthermore, as kVLE further increases, A* increases again due to a higher tissue cancelation factor w giving a higher weight on the low-energy image. For all kV pairs, A* was found to be less than 0.5, indicating that more dose should be allocated to the high-energy image compared to the low-energy image.

Figure 5.4. Optimization results for a kidney stone discrimination task and an iodine vs. bone discrimination task. (a) and (e): The optimal dose allocation factor, A*, identified as the dose allocation that maximizes d' at each kV pair. (b) and (f): Integral of the 3D NEQ (irrespective of task) demonstrates similar trends despite different values of w and A*. (c) and (g): The dual-energy contrast of brushite vs. calcium phosphate and iodine vs. bone, respectively, with (g) reflecting the K-edge of iodine at 33 keV

(i.e., ~65 kV). (d) and (h): Detectability index as a function of kV pair calculated using the optimal dose

151 allocation, A*, shown in (a) and (e), respectively. The detectability index of the kidney stone discrimination task (d) shows the combined effect of NEQ (b) and contrast (c), whereas that of the iodine vs. bone task (h) is mainly driven by contrast (g). (Published with permission from Medical Physics)

Figure 5.4(b) plots the integral of the 3D NEQ along the fx, fy and fz directions, which essentially reflects the d’ of a task that is fixed in contrast for all energies and unity at all spatial frequencies

(i.e., a delta function detection task). The trend of the integral of NEQ can be explained in a similar manner to the trend observed for A*: NEQ is optimized at low kVHE due to low NPS, and at intermediate or low kVLE depending on the tradeoff between decreasing NPS and increasing w. Figure 5.4(c) shows the DE contrast between brushite and calcium phosphate at different kV pairs which follows naturally from the monotonically decreasing attenuation characteristics of the two materials. Detectability index plotted in Fig. 5.4(d) was calculated using the energy- dependent A* shown in Fig. 5.4(a). The maximum d’ identifies the optimal kV pair at [45,105], with a corresponding A* of 0.32. The optimal kV selection is driven by both the NEQ and contrast, as seen by the "blending" of optima from Fig. 5.4(c) and (d).

Similar results are shown for the iodine vs. bone discrimination task in Fig. 5.4 (e)-(h). The optimal dose allocation factors for both tasks show similar trends as a function of kV pairs and are less than 0.5 for all energy combinations. The integral of NEQ shows similar trends as well despite the different w and A* used in the calculations. In contrast to the kidney stone discrimination task, Fig. 5.4(g) presents a low-energy optimum at 65 kV that reflects the K-edge effect of iodine, which consequently results in an optimal kV selection mainly influenced by the contrast term [Fig. 5.4(h)].

152

Figure 5.5 shows detectability index computed for the breast tumor detection task in the presence of anatomical background noise. The effects of tissue cancelation factor w and dose allocation factor (A) on d' are examined for three levels of total dose [(a)15 mGy, (b) 1.5 mGy, and (c) 0.15 mGy] and three imaging tasks [(a) low-, (e) mid-, and (f) high-frequency] as described in

Sec.5.2.5.3. The absolute value of the magnitude of the DE anatomical background power spectrum, |DE|, and the DE contrast modeled as the signal difference between breast tumor and fibroglandular tissue, |CDE|, are superimposed on Fig. 5.5(a). Figure 5.5(d) plots the integral of the 3D quantum noise NEQ. The energy pair is fixed at 60 and 120 kV, and the results for other energy pairs support similar conclusions (but are not shown for reasons of brevity).

For the low-frequency task at 15 mGy [Fig. 5.5(a)], the maximum d' is achieved at w  0.6, where the signal value of fibroglandular tissue is canceled to the same intensity as adipose tissue

(w = 0.58), resulting in a minimum in the absolute value of the magnitude of anatomical noise

[|DE|, superimposed on (a)]. As the total dose decreases, the peak at w = 0.58 becomes less pronounced. As the total dose reduces by orders of magnitude to 0.15 mGy, anatomical noise is superceded by quantum noise, and cancellation of anatomical background no longer results in improved performance. Instead, d' is quantum noise limited and exhibits the combined effects of contrast [|CDE| superimposed on Fig. 5.5(a)] and quantum noise NEQ [Fig. 5.5(d)], as evident in: i.) the valley at w 0.2, consistent with the point at which |CDE| reaches 0; and ii.) the peaks at A

0.4-0.6 observed at w > ~0.5, consistent with trends in the integral of the NEQ [Fig. 5.5(d)].

Variation of the frequency content of the imaging tasks yields similar results. The influence of anatomical noise becomes less severe as the frequency content of the imaging tasks extends beyond the low-frequency region dominated by anatomical noise. For the high-frequency task,

153 performance is quantum noise limited even at high doses. Such behavior is reminiscent of similar tradeoffs in single-energy tomosynthesis and CBCT.

Figure 5.5. Detectability index computed as a function of w and A for a low-frequency detection task at

(a) 15 mGy, (b) 1.5 mGy, and (c) 0.15 mGy, and for (e) a mid-frequency task and (f) a high-frequency task at 15 mGy. The integral of the quantum noise NEQ [Eq. (5.5)] at 15 mGy is plotted in (d).

Superimposed on (a) are plots of the absolute values of dual-energy contrast between a breast tumor and fibroglandular tissue and the dual-energy anatomical "clutter" magnitude DE. A clear optimum in d' is observed in (a) at w = ~0.6 where DE is reduced to 0 (i.e., cancellation of background clutter). As dose is lowered or the spatial frequency content of the tasks increases, canceling anatomical noise becomes less important, and the DE contrast becomes the dominant factor in driving d'. (Published with permission from Medical Physics)

154

The integral of the NEQ at 15 mGy plotted in Fig. 5.5(d) is representative of similar trends at other dose levels. As expected, the dose allocation that maximizes d' at each w is seen to increase with w, demonstrating that more dose needs to be imparted by the low-energy image as its decomposition weight increases. This trend is also reflected in the detectability index calculations in the same figure. When w = 0 (i.e., for a single high-energy image) the NEQ decreases monotonically as A increases (i.e., as the dose allocated to the high-energy image decreases). At higher values of w, the NEQ [vertical profiles in Fig. 5.5(d)] exhibits an optimum at intermediate values of A, since low and high extremes of A result in high NPS in the low- energy and high-energy images, respectively.

Results of kV optimization for the breast tumor detection task are shown in Fig. 5.6. The detectability index computed as a function of kV pair is plotted along with the integral of the

NEQ for three conditions: (1) the low-frequency task at 15 mGy [Fig. 5.6(a) and (b)]; (2) the low-frequency task at 1.5 mGy [Fig. 5.6(c) and (d)]; and (3) the mid-frequency task at 15 mGy

[Fig. 5.6(e) and (f)]. The DE contrast modeled as the signal difference between the tumor and the fibroglandular tissue is included in Fig. 5.6(g) for reference. Based on the optimization results from Fig. 5.5, w was chosen to cancel fibroglandular to adipose tissue, thereby reducing anatomical noise to zero. The optimal dose allocation factor exhibits similar trends as the previous two tasks (Fig. 5.4) and is not shown for brevity. Similar to the kidney stone discrimination task, the detectability index is affected by both the NEQ and DE contrast, which can be appreciated in the “blending” of optima from the corresponding figures. As the total dose is reduced, the integral of NEQ exhibits an optimum at a higher kVLE due to the stronger influence of quantum noise, consequently moving the optimal kV selection from [65, 90] kV at

15 mGy [Fig. 5.6(a)] to [75, 90] kV at 1.5 mGy [Fig. 5.6(c)]. As the frequency content of the

155 tasks is varied, however, the optimal kV selection remains for the most part unchanged [Fig.

5.6(e)].

Figure 5.6. Detectability index and integral of the NEQ computed for a low-frequency task at 15 mGy [(a) and (b)], a low-frequency task at 1.5 mGy [(c) and (d)], and a mid-frequency task at 15 mGy [(e) and (f)].

Dual-energy contrast between the tumor and fibroglandular tissue in the DE image is plotted in (g). The optimal kV selection demonstrates a combined effect of NEQ and DE contrast, and is affected by total dose more than the frequency content of the task. (Published with permission from Medical Physics)

5.4. DISCUSSION AND CONCLUSIONS This work reported a cascaded systems model for DE-CBCT based on 3D filtered backprojection and reconstruction-based decomposition. Fourier-domain imaging performance metrics such as the DE NPS and NEQ were derived from the model and validated against measurements from an experimental CBCT imaging bench across a broad range of imaging techniques and decomposition parameters. The dual-energy detectability index was derived from such Fourier metrics for a variety of imaging tasks, including renal stone discrimination, iodinated vessel-

156 bone discrimination, and soft tissue tumor detection in anatomical clutter. The detectability index provided an objective function in the optimization of DE imaging parameters including the tissue cancelation factor (w), dose allocation factor (A), and selection of kV pair.

For all tasks investigated, the optimal dose allocation factor was found to be below 0.5, indicating that more dose should be allocated to the high-energy image. The endogenous breast tumor detection task illustrated the benefit of dual-energy imaging in eliminating anatomical background clutter and improving detectability beyond traditional single-energy scans. Tradeoffs reminiscent of single-energy tomosynthesis in the battle between quantum and anatomical noise are also revealed - namely, that dual-energy imaging is most beneficial for low-frequency tasks and dose levels above the quantum-limited regime. The calculations also indicated that optimal kV selection is dependent on total dose levels more than the spatial frequency content of the imaging tasks at a constant tissue cancelation factor.

The kidney stone discrimination task exemplified the application of dual-energy imaging in discerning two materials with similar attenuation coefficients not possible in single-energy reconstructions. For both of these tasks, the optimal kV selection was found to follow a combined effect of both NEQ (quantum noise) and DE contrast (tissue cancellation). Therefore, in these cases, optimizing imaging techniques based on noise or contrast alone is not sufficient, and a fuller appreciation of NEQ in relation to the imaging task was important to identifying the optima. The iodine vs. bone task demonstrated the situation in which a material of interest exhibited a strong K-edge effect that resulted in a sharp contrast profile within the energy range of interest; therefore, the optimal kV selection in this case was mainly contrast-driven.

157

Assumptions implicit in this work are significant and consistent with previous application of cascaded systems analysis of Fourier metrology in DE and CBCT imaging performance. These include the assumption of linearity and shift-invariance intrinsic to linear systems analysis.

Furthermore, perfect tissue cancelation was assumed with the weighted subtraction algorithm, and the effects of x-ray scatter, spatial variation of x-ray energy as a result of attenuation through the object, etc., were ignored. In future work, a straightforward extension of the model to linear noise reduction algorithms that further improve image quality and reduce dose Moreover, the present study focuses on the weighted subtraction algorithm for material decomposition, which is limited in the extent to which it provides quantitative measure of material composition. More complex decomposition algorithms, such as basis material decomposition either in the reconstruction or projection domain, could offer quantitative information of materials and beam- hardening artifact reduction169. Modeling and optimization of such algorithms within the general framework is subject of possible future work. A straightforward extension of the model to linear noise reduction algorithms56 that further improve image quality and reduce dose is also possible within the same cascaded systems framework.

158

Chapter 6 Non-Stationary Noise and Detectability in Filtered Backprojection and Penalized Likelihood Estimation 6.1. INTRODUCTION As expounded upon in previous chapters, accurate description of image quality is integral to system design, optimization, and assessment. A wealth of literature has been devoted to studying the noise characteristics of x-ray computed tomography (CT) in terms of the pixel variance, spatial domain covariance matrix, or Fourier domain noise-power spectrum (NPS).70, 122, 189-192

The latter two are simply related by the Fourier transform when noise is wide-sense stationary.

Moreover, it is generally accepted that image quality should be defined with respect to the imaging task,2, 4 where detectability is calculated to account for both metrics of noise and spatial resolution as well as the task function and observer model. Such task-based frameworks are increasingly employed in system design, performance assessment, and optimization. 24, 112, 139

An aspect of image quality that is somewhat less frequently reported is the spatially varying – i.e., nonstationary – characteristics of noise, spatial resolution, and detectability. In x-ray CT and cone-beam CT (CBCT), nonstationarity may arise from the discrete nature of reconstructed images, detector defects and nonlinearity, angular sampling, the divergent beam, variable fluence transmitted to the detector at different view angles, and the reconstruction algorithm. Several researchers have examined non-stationarity in filtered-backprojection reconstruction (FBP). For example, Pineda et al.193 calculated the local NPS at locations throughout FBP reconstructions, showing the variation in noise and NPS associated with variable fluence transmitted to the detector (and the effect of a bowtie filter). They furthermore quantified the difference between the diagonal and off-diagonal elements of the Fourier transform of the covariance matrix as a

159 metric of stationarity. Baek and Pelc 44, 194 studied nonstationary noise in fan-beam and cone- beam FBP reconstructions as a result of varying magnification, projection weighting, and cosine weighting and quantified the NPS at different spatial locations within reconstructions of air and a water cylinder. Nonstationarity in relation to task-based performance was investigated by

Brunner et al.,191 who evaluated the location-dependent NPS and covariance matrix in FBP reconstructions and calculated Hotelling observer performance for simple tasks. Wunderlich and

Noo 195 estimated the covariance in FBP reconstruction of fan-beam CT data and examined the location-dependent noise and lesion detectability using a channelized Hotelling observer.

Bartolac et al.126 investigated the nonstationary signal and noise characteristics in CBCT arising specifically from sampling along a circular source-detector trajectory (i.e., the spatially varying null cone associated with violation of Tuy’s condition). Such work demonstrates the spatially varying noise characteristics intrinsic to CT and CBCT and motivated investigation of the corresponding effect on detectability.

The need for a framework that can describe nonstationary imaging performance is pronounced in light of growing interest in statistical reconstruction methods. Compared to conventional FBP, statistical reconstruction has demonstrated potential for dose reduction, sparse reconstruction, artifact reduction, and the incorporation of prior information.196 Such methods also carry distinct non-stationary characteristics, as well as the means to explicitly enforce stationarity – e.g., the ability to enforce uniformity in the point-spread function (PSF) throughout the image.197-199 In the past several years, clinical CT systems with the capability for statistical reconstruction have emerged in clinical settings, raising the need to understand the imaging performance associated with such algorithms for assessment, optimization, and possible dose reduction. Compared to

FBP, image quality assessment in statistical reconstruction faces the additional challenge that the

160 nonlinear algorithm itself imparts nonuniformity in the image. As with FBP, noise and resolution are intrinsically nonstationary, but nonlinear algorithms carry additional dependence on the contrast of the structure of interest. Image quality in statistical reconstruction has been evaluated empirically in terms of the variance, kurtosis, contrast-dependent edge-spread function, and associated modulation transfer function (MTF).200-203 Efforts toward task-based assessment and optimization have been advanced in emission tomography by Yendiki and Fessler,204, 205 where the effect of regularization was examined in location-known and location-unknown tasks.

This chapter investigates the nonstationary noise, resolution, and task-based performance in

CBCT reconstructed with both FBP and penalized likelihood (PL) algorithms. For the latter, the current analysis pertains to PL estimation with a quadratic penalty. The FBP model builds on the cascaded systems model in Chapter 2 to account for noise nonstationarity as a result of variable fluence transmitted to the detector at different views. Noise and resolution estimation for PL is adapted from the work by Fessler for implicitly defined function estimators.197, 206 Imaging performance is reported in terms of the local MTF, NPS, NEQ, and detectability index (d’), analogous to the approach adopted by Wilson207 in SPECT. Although imaging performance in both FBP and PL is examined below, the purpose of this work is not to rate the performance of the two algorithms. In fact, as apparent in the results below, the performance of the two algorithms is highly dependent on the imaging task, object, spatial location, and imaging conditions, and it is difficult to propose a fair, general comparison of the algorithms. Finally, we present a simple task-based method for optimal selection of the PL regularization strength parameter (), including spatially varying regularization to improve detectability.

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6.2. METHODS 6.2.1. The Spatially-Varying NPS and MTF for FBP The cascaded systems model introduced in Chapter 2 primarily reported results and validation pertaining to the center of a uniform cylinder (or averaged along circular annuli concentric with the cylinder), although calculation of nonstationary NPS as a function of spatial location can be done naturally within the same model. As introduced in Chapter 2, the backprojection stage

2 (stage 13) has distinct signal and noise transfer characteristics denoted by T13 and 13 , respectively. In the limit of a large number of projections:

1 TM , (6.1a) 13 f and

 M 2 1 2 . (6.1b) 13 mf

22 where f is the radial spatial frequency ( ffxy ), M is the system magnification, and m is the number of projections. The 1/f characteristic is associated with the radial sampling density.

The main source of nonstationarity investigated in this work is variable fluence transmitted to the detector at different view angles as illustrated in Fig. 6.1. At an arbitrary voxel location within an

th image, the line integral of the i projection ray passing through that voxel (denoted as li) is different for each view angle. For example, at location 2, the ray incident at 90o has a larger line

o integral compared to the ray at 135 , i.e., l1>l2. Meanwhile, at location 4, l2 is greater than l1 due to high attenuation through the high density region. The spectrum from the x-ray tube is therefore attenuated differently, resulting in view-dependent fluence (and, therefore, noise) at the

162 detector, which is in turn propagated through the reconstruction process according to the central slice theorem. Therefore, noise (both the magnitude and correlation) in the reconstruction is dependent on both the object and the voxel location. This nonstationarity is intrinsic to CT and is evident in linear reconstruction algorithms like FBP.193 Note that such non-stationarities would be present even with a bowtie filter (also illustrated in Fig. 6.1) for the general case of a heterogeneous, non-circular object.

163

The nonstationarity of signal and noise was characterized locally around a voxel by considering the contributions from different projection views and then analyzing such local signal and noise characteristics throughout the image reconstruction. Image quality metrics derived below carry a location dependence [either the (u,v) detector domain or (x,y,z) reconstruction domain – or their

Fourier domain counterparts]. The incident spectrum at the detector from the ith projection ray passing through a particular voxel is calculated according to Beer's Law:

i q0  E  qsource Eexp l i ( E ) (6.2)

The line integrals, l, are correspondingly:

l()() E A E , (6.3) where A is the forward projection operator, and  is the vectorized representation of attenuation coefficients in the object, written as a function of energy, E, to account for a polyenergetic beam.

The subscript i indicates a single value in l associated with a particular voxel at the ith view. The forward projection operator was implemented following a separable footprint algorithm208 elaborated in Sec. 6.2.3. The detector element (and, therefore, the line integral) associated with a given voxel was identified through a projection matrix that relates the voxel location in the object to the location of its projection on the detector. The line integrals (and, therefore the local signal and noise characteristics) can then be calculated for any voxel location.

As a result of the view-dependent incident spectrum, any parameters in the model that are dependent on the spectrum also need to be calculated separately for each view. Such parameters

include the effective (energy-averaged) gain factors g1 (quantum detection efficiency), g2

(conversion gain from x-ray to optical photons in the scintillator), and Pk (gain and spreading

164 associated with K-fluorescence). The spectral dependence arises from energy-averaging according to:

g()() E q E dE g   (6.4) q() E dE  allowing calculation of the energy-dependent gain factor, gE(), for a given spectrum. For simplicity, the spatial spreading of optical quanta (T3) and K x-rays (TK) in the scintillator are assumed to have negligible dependence on the incident energy. This assumption could be removed in refinement of the model in future work. Aperture integration (T5), the ramp filter

(T10), apodization (T11), and interpolation (T12) of the projections are independent of the incident energy.

The NPS of the ith projection is therefore as follows, with view-dependent terms indicated by the superscript i and notation otherwise consistent with Chapter 2:

Si qaggg i4 i i1 gPTT i 2 2 ** IIIS + (6.5) proj0 pd 1 2 4 4K 3 5 6 add

The NPS from each projection is then propagated through the FBP reconstruction process. Each view is backprojected according to the transfer function:

M i  T13  FOVsinc FOV fx cos i f y sin i  (6.6) m 

where i is the view angle and FOV is the reconstruction field of view. Back-projecting each view and summing over all projections, the NPS of the 3D image reconstruction is given by:

165

m S i proj 2 2 211 i 2 2 . (6.7) S 2 T10 T 11 T 12 T 13****** III 14 T 15 III 15  i2 i i 2 i1 q a g g g M FOV  0pd 1 2 4 

i If the noise contribution from each view is equal, e.g., at the center of a uniform cylinder, S proj ,

i i i q0 , g1 , and g2 can be moved out of the summation, and in the limit of a large number of projections, the transfer function for Stage 13 can be simplified to the view-independent form:

2 mm2 M Ti  dsinc d f cos f sin  13   x i y i  ii11m , (6.8)  M 2 1 tot  2 mf 13 yielding the familiar form of NPS reported in Chapter 2.

Similarly for the MTF,

m i T  TffTffTffTffTffTfff3(,)(,)(,)(,)(,)(,,)uv 5 uv 10 uv 11 uv 12 uv 13 xyz (6.9a) i1

i Since T13 is the only term dependent on view angle, the MTF reduces to:

m i TTffTffTffTffTff 3(,)(,)(,)(,)(,)(,,)uv 5 uv 10 uv 11 uv 12 uv Tfff 13 xyz (6.9b) i1

The summation of over all projections produces the same form as Eq. (6.1a), which is independent of the incident spectrum and view angle, and is therefore the same for all voxel locations.

166

As illustrated in Fig. 6.1, the presence of a bowtie filter (or equivalently, other fluence modulation device, e.g., a dynamic bow-tie209 or multileaf collimators210) can be readily incorporated into the model by accounting for attenuation through the device as part of the object model,  E . A bowtie filter was excluded in the work below, since many CBCT systems do not employ them (including many breast, C-arm, and IGRT imaging systems) and to better demonstrate the basic nonstationary noise characteristics. The response of each detector element was assumed to be the same, thereby assuming an essentially perfect correction of gain and offset variations in the detector. We also excluded the heel effect, which would imply a spatially varying spectrum incident on the object. Such sources of noise nonstationarity, e.g., detector nonlinearity, defects, etc., could potentially be incorporated in the model by accounting for the response of each detector element.

From the resulting NPS and MTF, the noise-equivalent quanta (NEQ) can also be defined locally at a point:

2 T(,,) fx f y f z NEQ(,,) fx f y f z   f (6.10) S(,,) fx f y f z where f is the radial frequency in the axial plane defined in Eq.( 6.1). Together, Eqs. (6.7), (6.9b), and (6.10) demonstrate the capacity of the cascaded systems framework (usually described in terms of performance strictly at the center of reconstruction) to describe the spatially varying signal and noise characteristics in FBP in terms of local descriptors of MTF, NPS, and NEQ.

Task-based imaging performance was quantified in terms of detectability index, d', which relates metrics of MTF, NPS, and NEQ to the imaging task and observer model. As shown in previous

167 chapters, a variety of observer models can be employed in calculation of d’, including the basic prewhitening (PW) form:

2 T(,,)(,,) fx f y f z W Task f x f y f z d'2  df df df (6.11)  x y z S(,,) fx f y f z and various ‘anthropomorphic’ observer models intending to better approximate the performance of human observers, such as the non-prewhitening matched filter observer model with eye filter and internal noise (NPWEi):

2 22 Tfff(,,)(,,)(,,)x y z W Task fff x y z Efff x y z dfdfdf x y z 2  d '  24 Sfff(,,)(,,)(,,)(,,)(,,) TfffW  fff Efff  Nfffdfdfdf  xyz xyz Taskxyz xyz ixyzxyz (6.12) where WTask is the imaging task defined as the difference in the Fourier transforms of the two hypotheses in a binary task. The eye filter was:84

E( f ) f exp( cf ) (6.13) where f is the spatial radial frequency and c equals 2.2 for a typical viewing distance of 50 cm, yielding peak response at 4 cycles/deg. The internal noise was modeled as uncorrelated white noise according to:

2  D  Ni  0.001  NPS eq 0,0 (6.14) 100 

where D is the viewing distance (50 cm), and NPSeq is the white NPS equivalent in total power to the image noise. The scale factor 0.001 was found to yield reasonable agreement with experimental measurements in Chapter 4. Since the current work only considered reconstructions

168 at the central slice (elaborated in Sec. 6.2.3), the effect of out-of-plane (z-direction) beam divergence and associated cone-beam artifacts were neglected. Just as Eqs. (6.7), (6.9b), and

(6.10) allow calculation of the NPS, MTF, and NEQ at any location within the image, Eqs. (6.11,

6.12) allow calculation the spatially varying detectability index, d’(x,y).

6.2.2. The Spatially Varying NPS and MTF for Statistical Image Reconstruction Statistical reconstruction algorithms seek a reconstruction estimate, ˆ , that maximizes an objective function, (;) y relating the measured projections, p, to the object, . This work focuses on the penalized likelihood (PL) family of algorithms where the objective function includes the likelihood term Lp(;) and a penalty function R() scaled by the regularization strength parameter, :

ˆ argmax  (  ;p )  argmax[log L (  ; p )   R (  )] (6.15)

The measured projections, p, are assumed for simplicity to follow a Poisson distribution, with the mean, p , related to  by the forward model:

l A p()  I00 e I e (6.16)

where I0 is the bare-beam fluence (x-ray quanta / pixel), and l is the line integral following Eq.

(6.3). For simplicity, the energy dependence in  was dropped. Incorporating polyenergetic forward models in the work reported below is the subject of possible future work. Moreover, the assumption of Poisson noise is an obvious simplification in light of the known complexities of the image noise distribution – for example, Poisson excess, blur, and additive noise as described

169 by the cascaded systems model in the previous section. Incorporation of such complexities into model-based statistical reconstruction is an area of ongoing and future work211.

We consider a quadratic penalty, R(), of the form:

1 R() T R  , (6.17) 2 where R is a constant matrix that defines how voxels are combined and penalized. It is also the

Hessian of R(). In the work below, we used a particular class of quadratic penalty that penalizes voxel differences in a 4-neighborhood about a voxel location, j, i.e., in a 2D axial slice,

1 2 Rw() j, k  j  k  where wjk = 1 for the four nearest neighbors and 0 otherwise. The 2 jk strength of the penalty is governed by the regularization strength parameter, . While normally treated as a constant in most PL implementations,  can also be implemented as a regularization

“map,” denoted (x,y) to impart a spatially-varying penalty in the reconstruction.197-199

Fessler206 derived the covariance estimate of PL reconstructions for transmission tomography using the first-order Taylor expansion and the Implicit Function Theorem. The covariance of

voxel j with every other voxel in the reconstruction, denoted as [cov{ˆ }] j , is given by:

[cov{ˆ }] [ADARAAADART {p (  )} +  ] 1 T cov{ p } [ T { p (  )} +  ] 1  (6.18) jj where the transpose of the forward projection operator, AT, is the back-projection operator and

D{ } represents a square diagonal matrix with the vector argument corresponding to the diagonal entries. The unit vector j effectively extracts a column from the full covariance matrix and allows the calculation of covariance at any arbitrary voxel location (x, y). The term p() is the

170 noiseless forward projection [Eq. (6.16)] of the reconstruction from noiseless data, i.e.,

argmax ( ;p ) . When true and  are not available (e.g., when performing variance estimation for real data) to compute p() , the measured projection data, p, can be used as a substitution, because both forward- and back-projection are smoothing operations that reduce the effect of noise on the resulting estimate. Equation (6.18) reveals a dependence of noise on the spatial location (through j) and on the object itself, but only through the projections (i.e., through p() and p). Therefore, knowledge of the true object is not required to estimate the variance for PL reconstruction with a quadratic penalty. The covariance can then be solved iteratively using the conjugate gradient algorithm.206

Assuming the image noise to be slowly varying, the noise can be considered locally stationary – i.e., within a small neighborhood, N, the covariance of voxel j with every other voxel in the

neighborhood, [cov{ˆ }] jN , is the same for all voxels:

[cov{ˆ }]jkNN [cov{ˆ }] , jk , N . (6.19)

The local covariance matrix is therefore circulant, i.e., each row or column is a shifted version of the previous. For such matrices, the magnitude of the DFT of a row/ column is equal to the diagonal of the DFT of the full matrix, which is in turn equal to the NPS of the neighborhood by definition. We can therefore write the local NPS in a neighborhood, N, for quadratic PL image reconstruction as:

NPS =DFT[cov{ˆ } ] diag {DFT[cov{ˆ } ]} (6.20) j,NN

171

Here, diag{ } denotes the extraction of the diagonal from the argument matrix, not to be confused with D{ }. The local NPS about voxel j was estimated using the above relationship as

the DFT of cov{ˆ }j,N .

The point spread function (PSF) for PL reconstruction was derived using similar principles as the covariance:206

[PSF{ }] [ADARADAT {pp (  )} +  ] 1 T { }  (6.21) jj

The equation reveals a dependence of spatial resolution on location and on the object itself

(again, through the projection data). This finding is in contrast to that in FBP: whereas PL carries an explicit spatial dependence in the PSF (and, therefore, the MTF), FBP carries uniform spatial resolution to the extent described by Eq. (6.9b), recognizing the potential for non-stationary spatial resolution in FBP associated with sampling as mentioned above and described in Refs

194 and 212.

Under the same assumption of local stationarity, the PSF at each voxel in a neighborhood N is the same, or equivalently, the system is locally shift-invariant. The first-order Taylor expansion used in the derivation of Eqs. (6.15) and (6.18) amounts to a local linearity approximation.197

Therefore, the system is locally linear and shift-invariant, and the MTF can be calculated as the magnitude of the DFT of the PSF normalized at zero frequency:

MTF H ( f ) / H (0) , where H(f ) = DFT[PSF{ } ] (6.22) j,N

The above relation can also be derived in a manner similar to the covariance case, where the object transfer matrix is circulant with shifted copies of the PSF as its row entries.191

172

6.2.3. Digital Phantoms and Image Simulation To validate theoretical predictions of noise and spatial resolution in FBP and PL, model calculations of NPS and MTF were compared with that measured in simulated data. Three digital phantoms of increasing complexity were investigated in this study as illustrated in Fig. 6.2: (1) a uniform Circle phantom of radius 9 cm with attenuation coefficient 0.02 mm-1; (2) an Ellipse phantom with major axis of 24 cm and minor axis of 16 cm. The background of the Ellipse was uniform with attenuation coefficient 0.0164 mm-1 (corresponding to water in a 90 kV beam filtered with 2 mm Al and 0.4 mm Cu), with two circular inserts of 7.75 cm diameter and attenuation coefficients 0.0211 mm-1 and 0.0144 mm-1 corresponding to acetal and polypropylene, respectively; (3) an anthropomorphic Thorax phantom constructed from a slice through a phantom imaged on a clinical CT scanner. For simplicity, the soft tissue (skin, muscle, fat and heart) was assigned a uniform attenuation coefficient 0.0203 mm-1, and bones (ribs and spine) were assigned attenuation 0.0432 mm-1. Attenuation values for the lung tissue were preserved from the CT reconstruction, with an approximate mean value of 0.0074 mm-1. For the monoenergetic (57 keV) simulations considered in this work, the attenuation coefficient of each component was a scalar. The phantoms can be adapted to polyenergetic simulations in future studies by using energy-dependent attenuation coefficient [(E)] for each tissue component.

173

Figure 6.2. Three digital phantoms investigated in this work. (a) A uniform circular object. (b) An elliptical object composed of three materials approximating water, bone, and fat. (c) A thorax approximating a realistic distribution of tissue heterogeneities. The numeric symbols (1-4) mark four locations at which the local MTF, NPS, NEQ, and detectability index were analyzed specifically.

(Published with permission from Medical Physics)

5 The monoenergetic x-ray source was simulated with bare-beam fluence, I0, equal to 2.1×10 photons/mm2, corresponding to a 90 kV beam at 1 mAs with an exposure of 0.70 mR to the detector. Using a monoenergetic beam implies that the energy dependence in the cascaded

i systems analysis model was dropped, and the only view dependent term was q0 . System geometry was set to SAD = 120 cm and SDD = 150 cm. For simplicity and to focus on the intrinsic non-stationarities associated with FBP and PL reconstruction methods, an ideal detector model was considered – i.e., there was no detector blur beyond that of the integrating aperture, and Poisson-distributed quantum noise constituted the only noise source. Detector pixel pitch was simulated at 0.834 mm. Poisson noise was added independently to the projection data according to Ref. 213. All simulations were performed for a single-row detector, and only the central axial slice was reconstructed.

Both FBP and PL reconstruction algorithms were implemented in Matlab with external calls to a custom C++ library to perform forward- and back-projection operations on GPU. Projection data were generated with a separable footprint projector.208 For PL, image updates were performed using a paraboloidal surrogate approach.214 Ordered-subset subiterations215 were adopted to speed convergence, where the number of subsets decreased from an initial value of 60 in increments of -5 at every 50th iterations, with an additional 200 iterations performed after the

174 number of subsets decreased to 1, giving a total of 800 iterations for each PL reconstruction. A sufficiently converged solution was confirmed by examining the image difference between reconstructions at 800 iterations from those at 2000 iterations, with a maximum image difference on the order of 10-4 mm-1. Reconstructions were performed for different values of smoothing parameters for each algorithm. For PL, the regularization parameter, spanned four orders of magnitude from 104.7 to 108.5. For FBP, variable smoothing was achieved by changing the apodization window, hwin, and the cutoff frequency, f0, of the apodization filter, T11:

 2uf h1  h cos f  f T  win win 0 11  ff0 / Nyq  (6.23) 0 ff 0

where u denotes the voxel size. Apodization parameters, hwin, was varied from 0.5 to 1.0, and the ratio of the cutoff frequency to the Nyquist frequency, f0/fNqy, from 0.1 to 1.0.

For the purpose of initial qualitative validation of the model, two low-contrast line-pair detection tasks were implemented with varying levels of conspicuity. Figure 6.3(a) shows a “vertical” line- pair detection task consisting of 5 lines of 1x10 voxels, each separated from its neighbor by 2 voxels. The contrast of the lines from the background was +0.0010 mm-1, and the voxels between the lines were assigned a contrast of -0.0005mm-1 from the background. (Note the positive and negative contrast.) The detection task function on a uniform background is simply the Fourier transform of the object function, shown in Fig. 6.3(b). The majority of the signal power is concentrated in two regions along the fx axis. The second task was a “horizontal” line-pair detection task, constructed from the same line-pair rotated 90o, with the majority of signal power therefore concentrated along the fy axis.

175

Two additional imaging tasks were defined directly in the Fourier domain to investigate the more general dependence of d’ on contrast and frequency content: (1) a radially symmetric mid- frequency task defined as the difference between 2 Gaussians of different widths:

f2 f 2   f 2 f 2  WCexp x y   exp   x y  (6.24) Task 2222    12   

-1 -1 where C=0.04 mm-1, 1 =0.05 mm and 2= 0.02 mm ; and (2) an all-frequency task (constant in the Fourier domain) corresponding to a delta function detection task of contrast 0.02mm-1.

Figure 6.3. Vertical and horizontal line-pair patterns and associated task functions. (Published with permission from Medical Physics)

6.2.4. Analysis of Noise and Resolution Conventional NPS measurements, as presented in previous chapters, involve computing the NPS over multiple regions-of-interest (ROIs) [or volumes-of-interest (VOIs)] within an image and averaging the outcome (the Bartlett method for non-overlapping ROIs or the Welch method for overlapping ROIs). This method invokes stationarity assumptions both locally within the ROI

(intrinsic to the Fourier transform) and globally over the ensemble of ROIs contributing to the

NPS estimate (by way of the averaging operation). In CBCT, for example, ROIs are often

176 selected at a fixed distance from the center of reconstruction of a uniform cylinder under the assumption that the NPS is invariant within such an annulus.

An alternative method to compute noise that does not invoke the stationarity assumption involves calculating the full covariance matrix. For an N×N ROI, the covariance matrix is

2 2 th th N ×N , with the entry on the j row and k column, [cov{ˆ }] jk , calculated as:

[cov{ˆ }] EEE ˆ  ˆ ˆ  ˆ (6.25) jk j j k k 

The diagonal of the DFT of the covariance matrix corresponds to the NPS – i.e., the NPS describes the variance of the Fourier components of the noise. As mentioned in Sec. 6.2.2, if noise is stationary, the covariance is a circulant matrix diagonalizable by the DFT. In this case, the NPS captures the full noise characteristics. If noise is nonstationary, the NPS is an incomplete descriptor of the noise characteristics in that it does not describe the off-diagonal elements in the DFT of the covariance matrix. The magnitude of the off-diagonal elements has been shown to be small (~2 orders of magnitude smaller) compared to the main diagonal at different locations within a 16 cm uniform water cylinder.193 Calculating the full covariance matrix requires a large number of noisy realizations to achieve reasonable accuracy in the noise estimate. As a rule of thumb,191, 192 the number of reconstructions required is at least 10 times the number of samples, i.e., 10×N2. This is challenging even in simulated data, although efforts have been made to improve estimation accuracy using fewer data sets with assumptions on the correlation length in linear reconstruction algorithms (e.g., FBP).216

In this work, noise measurements were performed under the assumption of local stationarity (i.e., within the ROI itself, but not necessarily throughout the image) just as in the theoretical methods

177 detailed in previous sections. Noise was characterized by the local NPS within a ROI at a fixed location - i.e., instead of marching the ROI through the image (e.g., a centered annulus), NPS was measured at the same location in a large ensemble of reconstructions and then averaged. The method for NPS measurement is otherwise consistent with previous work.38 For each phantom

(Circle, Ellipse, and Thorax), reconstruction algorithm (FBP and PL), and choice of smoothing parameter (hwin and ), 1000 reconstructions were simulated. Each image was subtracted from the next to form 999 difference images. The subtraction process eliminated anatomical structures, leaving only quantum noise in the difference image (scaling the NPS by a factor of 2 from image subtraction). In physical (non-simulated) data, subtraction of two images also helps to minimize deterministic image artifacts, e.g., shading. The NPS was measured as the ensemble average of the square of the Fourier transform of a fixed ROI in all difference images according to:

1 aaxy 2 NPS fxy,, f  FT ROI x y (6.26) 2 NNxy where N is the number of voxels along each direction of the ROI (N=49 in this case) and a is the voxel size. Note that the NPS considered here is a simple 2D form (with units denoted

[(mm2mm2]) appropriate to simulation of a 2D slice.

Spatial resolution at a given location in the image can be measured as the local impulse response,197, 217 calculated by injecting a small impulse of magnitude  into at voxel j of the true object true. No noise was added to the simulated projection following the forward model. The resulting noiseless reconstruction subtracted from the noiseless unperturbed image was divided

178 by the magnitude of the impulse to yield the local impulse response. At voxel j, the PSF is therefore given by:

ˆ ˆ  pp true   j     true  [PSF{ˆ }]  . (6.27) j 

The MTF was computed as the modulus of the Fourier transform of the PSF within a 49×49 voxel neighborhood following Eq. (6.22) with the same assumption of linearity and shift- invariance within the local extent of the ROI.

To reduce the MTF to a simple scalar metric of spatial resolution, the value denoted f50 was analyzed as the frequency at which the MTF drops to a value of 0.5. Since the MTF in PL reconstruction is often anisotropic, we analyzed the radial average f50 from 49 radial spokes through the MTF over 180o in the Fourier domain, recognizing that this is still a fairly coarse representation of the MTF.

6.2.5. Optimization of Reconstruction Parameters in PL Just as cascaded systems analysis has been applied in previous chapters to optimize various aspects of the CBCT imaging chain using FBP reconstruction, the analysis described above for the NPS and MTF for quadratic PL estimation can be used to optimize aspects of the iterative image reconstruction process. In particular, optimizing the PL reconstruction algorithm holds enormous promise due to the freedom in designing custom, spatially-varying smoothing in the image domain (by way of the  map, the penalty function, or both). The sections below show how the task-based framework can be applied in selecting the reconstruction parameter in three scenarios:

6.2.5.1. Choosing a constant  to maximize local d'

179

Optimizing detectability in a local region amounts simply to computing the local d' for a range of constant  values and picking the  that gives the maximum d'. A preliminary study (not shown) demonstrated that  values at locations far from the ROI have a negligible effect on the noise and resolution characteristics within the ROI in fully sampled reconstructions without long-range correlations. An example application scenario is one in which the location of interest in the patient is known – e.g., in image-guided interventions.

6.2.5.2. Choosing a constant  to maximize global d'

When the location of the signal is unknown, we may aim to optimize d' over the entire image. A simple model involves the global average detectability, denoted , defined as the average d' over the entire object, or over multiple ROIs (e.g., in the left and right lungs in a thorax image).

More generally, can be calculated as a weighted average of local d', where the weights could be assigned to areas of the image based on their importance or disease prevalence (e.g., d' in the air region assigned a weight of 0). Note that although this model of d' optimizes performance over the entire image, it does not describe the process associated with search.

The local detectability index was calculated throughout the image in 49×49 voxel ROIs across a rectangular grid with neighboring ROIs separated by 25 voxels. The calculation was performed for PL reconstructions using a range of constant  values, and the value that maximized was identified as the optimum.

6.2.5.3. Choosing a spatially varying  map to improve global d'

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The value of  that maximizes d’ at a given location in the image varies throughout the image due to the nonstationary noise and resolution characteristics in PL reconstruction. A simple extension of the previous method is to identify the optimal local value at each grid point and interpolate the results across the reconstruction, yielding a spatially varying  map that could potentially improve performance beyond that achievable with a constant . The interpolation assumes the local optimal  to be smoothly varying, which is reasonable to the extent that noise and resolution in the image are also smoothly varying. Interpolation was performed using radial basis functions218 to produce a smooth  map, (x, y). The performance achieved with the spatially varying  map was compared to that achieved with a constant  in terms of the global average .

6.3. RESULTS 6.3.1. Spatially Varying NPS and MTF Figure 6.4 illustrates the variance at each voxel (i.e., “variance map”) computed from 1000 FBP and PL reconstructions for each of the three phantoms. In each case, reconstruction parameters

(indicated on the respective variance maps) were chosen to match the radial average f50 at the center of the phantom, yielding a reasonably fair comparison of noise at “matched” resolution, recognizing the previously mentioned challenges in this respect. The variance maps for FBP show a greater degree of spatial variation than PL in all phantoms. The highest variance is observed at regions through which rays traverse with the longest path lengths – namely, the center of the Circle, the high density insert of the Ellipse, and the spine and heart of the Thorax.

The variance map for PL reconstructions is considerably more uniform in each case,

181 demonstrating lower noise than FBP in highly attenuating regions (e.g., the dense insert of the

Ellipse and the spine of the Thorax) and higher noise in less highly attenuating regions. The PL variance map also shows a sharp drop at the boundary of the object due to the non-negativity constraint.

Figure 6.4. Variance maps, 2(x,y), calculated from multiple realizations of FBP (top) and PL (bottom) reconstructions of the three phantoms shown in Fig. 6.2. (Published with permission from Medical

Physics)

Figures 6.5-6.7 illustrate the NPS and MTF evaluated at four locations in the three phantoms of

Fig. 6.2 reconstructed with FBP and PL. Reconstruction parameters were the same as in Fig. 6.4.

In each plot, the left half shows theoretical prediction [Eqs. (6.7), (6.9b), (6.20), and (6.22)], and the right half shows the measurement from simulated images, separated by a vertical dotted line.

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Such presentation results in no loss of information due to the radial symmetry of the Fourier transform.

The NPS for both FBP and PL is anisotropic and dependent on both the object and the location within the object. As illustrated in Fig. 6.1, such characteristics are due to variable fluence transmitted to the detector at different view angles. Such a characteristic is intrinsic to x-ray tomography and is independent of the reconstruction algorithm; therefore, the NPS at corresponding locations in FBP and PL reconstructions exhibit similar anisotropic nature. For example, at the center of the Ellipse, rays traversing the major axis (sampling along the fy axis) are attenuated more strongly and therefore carry higher noise than rays traversing the minor axis

(the fx axis). Due to the presence of the higher density insert (resulting in longer line integrals through region 1), the NPS at location 1 has a broader extent compared to that at location 3 (the low density insert). By comparison, all rays through the center of the Circle have the same pathlength and therefore carry the same noise, giving the familiar "doughnut" shaped NPS reported in previous work. In the Thorax, the NPS is highly anisotropic with noisy rays associated with attenuation through the ribs, spine (location 3), and heart (location 4).

The MTF in FBP is isotropic and shift-invariant within first-order approximation as shown for the Circle phantom in Fig. 6.5. Similar results were found for the Ellipse and Thorax but are not shown for brevity. The MTF in PL reconstructions, however, is anisotropic and exhibits dependence on both the object and spatial location. As evident from Eq. (6.21), spatial resolution in PL reconstructions is affected by the amount of noise in the projection data – even when  is held constant. When the projection data are noisy, the algorithm places greater weight on the penalty term and applies greater smoothing.206 Therefore, when projections traversing a voxel from different view angles carry different amount of noise, the radial frequencies sampled by

183 noisier projections will be smoother compared to the less noisy views. This is evident in that the shape of the MTF is roughly complementary to that of the NPS - e.g., lower MTF along ~-45o at location 3 and 4 in the Thorax, corresponding to noisy rays through the heart and spine. Note that even though greater smoothing is applied to the noisier rays, the noise carried by those rays is still higher than the less noisy rays.

Excellent agreement is observed between the measured and theoretical NPS and MTF for all three phantoms for both the FBP and PL reconstructions. The NPS for the Thorax shows a slight discrepancy likely due to violation of the local stationarity assumption (i.e., second-order statistics varying within the ROI) for an object containing heterogeneous structure (e.g., the lung parenchyma). This discrepancy highlights the subtle difference in interpretation of theoretical and measured NPS. The theoretical model computes the NPS "at a point": the cascaded systems model uses line integrals of rays through one particular voxel, and the PL theoretical noise model calculates the covariance of one particular voxel with every other. The notion of correlation, however, implicitly assumes a (local) region of support. Therefore, the theoretical calculations of

NPS should be interpreted as the NPS of a hypothetical ROI centered at a given voxel within which the noise is stationary. The measured NPS, on the other hand, explicitly assumes that noise is stationary within the extent of the ROI from which the NPS is estimated, which may begin to violate the stationarity assumption, depending on the extent of the ROI and the degree of sharp heterogeneity in the object (as in the Thorax). The difference between theory and measurement is small in all cases considered, suggesting that local stationarity is for the most part obeyed in these studies, with the most pronounced difference seen in Fig. 6.7 in regions containing structures with large differences in attenuation - e.g., in the region of the spine

(Thorax location 3) where even the fairly small ROI includes bone, muscle, and lung. Note,

184 however, that this is a limitation in the experimental measurement and does not refer to the validity of the theoretical model. Rather, the observation illustrates the challenges in quantifying nonstationary noise from experimental data, and the extent of the ROI for which the local stationarity assumption holds should be carefully chosen.

185

Figure 6.5. Measured and theoretical NPS and MTF at four locations within the uniform Circle phantom, illustrating the non-stationary, anisotropic characteristics of each for FBP and PL. Reconstruction parameters were chosen to "match" spatial resolution (radial average f50) in FBP and PL at a given location. In each plot, the left half shows theoretical prediction, and the right half shows measurement from simulated images. The anisotropic nature of the NPS is consistent with attenuation of rays traversing different path-lengths through the phantom. (Published with permission from Medical Physics)

Figure 6.6. The same as Fig. 6.5 but for the Ellipse phantom. Excellent agreement in NPS and MTF is observed between theory and experiment. The anisotropic nature of NPS is consistent with sampling of rays carrying different amount of noise - e.g., rays traversing the major axis of the Ellipse (corresponding

186 to sampling along the fy axis of the Fourier domain) carrying higher noise than rays traversing the minor axis of the Ellipse (corresponding to sampling along the fx axis of the Fourier domain). The MTF for FBP is not shown, since it is the same as in Fig. 6.5. (Published with permission from Medical Physics)

Figure 6.7. The same as Fig.6.5 but for the Thorax phantom. Slight discrepancies between theory and measurement are likely associated with violation of the local stationarity assumption in regions of strongly heterogeneous structures where noise is non-stationary within the ROI. Note that this violation pertains to the measured NPS (not the theoretical NPS, which computes the NPS at a point). As in the other phantoms, the anisotropic nature of the NPS is consistent with sampling of rays carrying different amount of noise – i.e., increased noise with stronger correlation along directions associated with more highly attenuating paths through the object. The MTF for FBP is not shown, since it is the same as in Fig.

6.5. (Published with permission from Medical Physics)

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To further compare theory and measurement, the variance at each location in each phantom was estimated from: (1) the theoretical models as the area under the theoretical NPS; (2) the area under the measured NPS; and (3) the variance of the same voxel across ~1000 reconstructions.

The variance estimated by method (3) represents the sample variance and does not rely on assumptions of local stationarity, giving a standard reference to which both theory and measurement can be compared. Averaging over all 12 locations in three phantoms, the theoretical prediction of variance (1) was within 4.9±2.3% of the sample variance for FBP, and within 5.7±4.3% of PL. Variance estimates from the measurements (2) were within 4.9±6.4% for

FBP and 4.9±3.4% for PL.

6.3.2. Optimization of Reconstruction Parameters Detectability index for the vertical line pair (top) and horizontal line pair (bottom) detection tasks in PL reconstructions is plotted in Fig. 6.8 as a function of  at three locations in the Ellipse.

Corresponding images (37x37 voxels about the line-pair pattern) are shown at example values of

. For the vertical line pairs, detectability exhibits an optimum at  ~105.4 in coarse qualitative agreement with the example images. Furthermore, detectability is highest at location 1, although the line-pair task is fairly conspicuous in each case, since the noise correlation (see NPS in Fig.

6.6) is in a region of the Fourier domain (along the fy axis) not associated with the vertical line- pair task (along the fx axis as in Fig. 6.3).

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Figure 6.8. Detectability index at location 1, 2 and 4 in the Ellipse (Fig. 6.2) as a function of regularization parameter  for the vertical line pair detection (top) and horizontal line pair detection

(bottom). Example ROIs in reconstructions at select values of  are shown on the right. Both the trends and magnitude in d' agree qualitatively with visual assessment of the images. (Published with permission from Medical Physics)

Detectability for the horizontal line-pair task is much lower than for the vertical line pair task at locations 1 and 2, since the task (Fig. 6.3) and noise correlation (Fig. 6.6) occupy the same region of the frequency domain (along the fy axis). This presents a clear example of noise

"masquerading" as signal, where the middle and high frequency noise components of the NPS at

189 location 1 and 2 coincide closely with the spatial frequencies of interest. Detectability is highest at location 4, in qualitative agreement with the example image.

Figure 6.9 plots the detectability index for the mid-frequency and all-frequency task as a function of smoothing parameters in FBP and PL reconstructions of the Ellipse. The plots show calculations at locations 1, 2, and 4 as well as the global average within the Ellipse. The magnitude of the optimal global average in FBP is comparable to that in PL for all tasks considered, demonstrating that PL with a quadratic penalty achieves approximately the same performance as FBP, as observed experimentally in other work.219

For the mid-frequency task in Fig. 6.9, an optimal choice of smoothing parameter is observed for both FBP and PL, illustrating the familiar noise-resolution tradeoff: toward the left of the curves, the images are over-smoothed, and task performance is impeded by a lack of spatial resolution; toward the right of the curves, the images are sharper, but task performance is impeded by image noise. For the all-frequency task, maximum detectability occurs at parameter settings associated with higher spatial resolution, shifting the optimum to sharper kernels for FBP (a pure ramp filter or even sharper) and weaker regularization for PL.

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Figure 6.9. Detectability index computed as a function of smoothing parameters for FBP [(a),(c)] and PL

[(b),(d)] reconstructions for a mid-frequency task [(a),(b)] and an all-frequency task [(c),(d)]. The optimal smoothing required for PL is dependent on the location within the reconstruction, while the difference is considerably smaller in FBP. (Published with permission from Medical Physics)

The detectability map, d'(x,y), for the mid-frequency task is shown in Fig. 6.10 for both FBP and

PL reconstructions of the Circle, Ellipse, and Thorax phantoms. The object boundaries (black dotted line) demark the region for computing the global average . Smoothing parameters were chosen to give the best global average within the object (maximum in Fig. 6.9) and are indicated on each plot. Overall, the magnitude of d' is comparable in FBP and PL, although the latter exhibits a somewhat greater range of spatial variability. In the Circle phantom,

191 for example, PL exhibits a slightly greater reduction in d’ at the center. Similarly for the Ellipse

– a slightly greater reduction in the vicinity of the high-density insert. The greater degree of spatial variability in d’ for PL presents an interesting counterpoint to the variance maps shown in

Fig. 6.4 (where the noise for PL was shown to be spatially uniform): specifically, whereas the

MTF of FBP is shift-invariant to a first-order approximation, the MTF of PL reconstruction not only varies spatially, but does so in a way that may be disadvantageous to the task. For example, comparing the high and low density regions in the Ellipse, the difference in d' in FBP results solely from the difference in noise; however, in PL not only is the NPS higher in the high density region, but the MTF is reduced, resulting in a stronger decrease in detectability. This effect is more pronounced for tasks involving mid- to high-frequencies due to a greater sensitivity on spatial resolution.

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Figure 6.10. Detectability maps, d'(x, y), computed for FBP and PL reconstructions in each phantom for the mid-frequency task. Reconstruction parameters were chosen to maximize the global average for the mid-frequency task. (Published with permission from Medical Physics)

6.3.3. Design of an “Optimal” Regularization Map Looking closely at the optima in Fig. 6.9 for PL reconstruction, one notes that the optimal smoothing parameter is different among the various locations, suggesting location-dependent noise-resolution tradeoffs. The effect is less pronounced for FBP, since the spatial resolution for

FBP is less spatially variant (i.e., invariant to a first order approximation) as discussed above.

The location-dependent optimal  motivates the design of a spatially varying  map that could potentially improve performance overall. As described in Sec. 6.2.4, the spatially varying  map

[denoted *(x,y)] constructed from local optimal  values is shown in Fig. 6.11: (a) the Ellipse phantom and vertical line-pair task; (b) the Ellipse phantom and mid-frequency task; and (c) the

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Thorax phantom and mid-frequency task. The resulting detectability map shown in Fig. 6.11 (d- f). The global average resulting from the spatially varying  map is compared to that for a constant value of  is shown in Fig. 6.11(g-i), and Fig. 6.11(j-l) show the ratio of d'(x,y) resulting from the spatially varying  map to that using a constant , giving a fairly modest improvement up to ~10% in detectability.

Figure 6.11. Spatially varying regularization. (a-c) Spatially varying  maps designed to maximize d’ at each point in the image. (d-f) The resulting detectability maps. (g-i) Comparison of the global average resulting from the spatially varying  map (straight horizontal line) plotted in comparison to achieved with a constant value of  (plotted as a function of  for the latter). A slight improvement in detectability is achieved with the spatially varying map. (j-l) Ratio of detectability index achieved with a spatially varying and constant , showing improvement up to ~10%, particularly in more heavily attenuating regions of the phantom. (Published with permission from Medical Physics)

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6.4. DISCUSSION AND CONCLUSIONS This work investigated the nonstationary noise and resolution characteristics of FBP and PL reconstruction and related the findings to task-based imaging performance. The work yielded at least three advances: explicit extension of cascaded systems to describe the spatially varying

(i.e., nonstationary) NPS and detectability index in FBP; extension of task-based imaging performance analysis to iterative image reconstruction; and a method for designing spatially varying regularization maps in iterative image reconstruction to increase task performance.

As illustrated in Fig. 6.1, the presence of a bowtie filter or other more complex fluence modulation device could be incorporated as part of the system model and accounted in the modeling for both FBP and PL. Such devices have been shown to produce more stationary noise and detectability. However, some level of non-stationarity will persist even for well matched bowties or other fluence modulation devices in the presence of objects with complex shapes and heterogeneous structure. The current work did not include a bowtie - consistent with many

CBCT imaging systems and to better illustrate the fundamental underlying nonstationary noise characteristics for purposes of initial investigation. Stationarity and isotropy are even harder to achieve in PL reconstructions due to the nonuniformity imparted by the algorithm itself, and image quality models that account for noise nonstationarity – as demonstrated here – could provide a valuable guide to understanding the effect on imaging performance.

The cascaded systems analysis model for FBP has been validated extensively in terms of NPS,

MTF, and d' in previous studies.58, 61, 119 For PL, the work presented above validated theoretical predictions of NPS and MTF in comparison to measurements in simulated data in a variety of phantoms of increasing complexity. Future work will validate such theoretical predictions in comparison to real data, requiring a more realistic forward model accounting for non-idealities in

195 detector response (e.g., blur and additive noise) and a polyenergetic x-ray beam.220 An additional challenge with experimental validation is anticipated to be the large number of acquisitions required for accurate noise estimation. Validation with simulated data was sufficient for initial investigation and offered the advantage of being able to isolate the dependence on the object and reconstruction algorithms apart from other complications that could also cause nonstationary noise (e.g., detector nonuniformities).

Although image quality was examined in both FBP and PL, this work did not intend to provide a strict comparison between the two algorithms. As evident in Figs. 6.5-6.7, comparison of noise and resolution in FBP and PL is challenging due to the highly anisotropic nature, challenging a fair “matching” of spatial resolution and/or noise. Direct comparison of the task-based (d’) performance in FBP versus PL should similarly be considered carefully. For the imaging conditions, imaging tasks, and observer models considered in this work, the results suggest comparable task performance between FBP and quadratic PL, which is consistent with experimental observations in other work.219 The advantages of iterative reconstruction are more apparent in non-quadratic penalties, and extension of the model to include such algorithms is the subject of future work.

The assumption of “locality” encompasses a multitude of implications. First, the use of Fourier metrics assumes “spatial locality” within a small neighborhood within which the noise is wide- sense stationary, and the system can be assumed to be linear and shift-invariant. Note, however, that the assumption of locality with respect to the NPS primarily affects the validity of measurements (which require an ROI within which the first and second-order statistics are constant); the theoretical analysis – both for FBP and PL – compute the NPS “at a point” without such explicit reliance on a stationary region of support. Second, with respect to detectability

196 index, the locality assumption furthermore implies that the perturbation associated with the signal (stimulus) is small and does not change the local noise and resolution characteristics – analogous to the classic “small signal difference” common to image quality and perception analysis. Third, in the derivation of PSF and covariance in PL reconstruction, the system is locally linearizable.

Optimizing the regularization parameter employed in statistical reconstruction has been investigated in emission tomography. Fessler proposed a spatially varying  map to enforce more uniform spatial resolution throughout the image.206 Qi et al.221 proposed a method for selecting regularization to optimize the contrast-to-noise ratio. Yendiki and Fessler204, 205 further examined the effect of  on detectability in both location-known and location-unknown tasks. The work shown above proposed a simple method for selecting both a uniform (constant)  and a spatially- varying  map to improve task-based performance. A task-based approach in the optimization of

PL reconstruction holds particular promise. First, compared to hardware-based fluence modulation, only one set of projections needs to be acquired, and the regularization parameters can then be changed to produce different reconstructions suitable for different tasks. Second, regularization in PL is applied in the spatial domain (i.e., in the domain of the image reconstruction), which imparts greater freedom in differential smoothing within different parts of the image that is not achievable with conventional FBP (which applies smoothing filters in the domain of the projection data). One can extend this further to a scenario in which different parts of the image could be optimized to accommodate different tasks. For example, in imaging the thorax, one could design a regularization map that optimizes for low-frequency, low-contrast soft-tissue tumor detection in the mediastinum, and optimizes for high-frequency, high-contrast detection in the lung parenchyma. By further example, in imaging the head, one could design a

197 regularization map that optimizes for low-frequency, low-contrast detection of blood in the brain, while optimizing for high-frequency, high-contrast detection of fracture in the surrounding cranium. Finally, within the general context of “ROI imaging,” one can design a regularization map that optimizes task-based detectability within a specific ROI without regard for the consequences in surrounding regions. For efficient implementation of  map design in practical situations, noise and resolution of PL reconstructions can be estimated using Fourier approximations instead of iterative solutions.222

Calculation of task-based detectability in this work used the anthropomorphic NPWEi model, although many observer models could be considered within the analytical framework. Figure

6.12, for example, illustrates the dependence of d' on  for the vertical and horizontal line pair detection tasks at location 1 of the Ellipse using the PW observer model [Eq. (6.11)] in comparison to the NPWEi model. The PW model suggests significantly higher d' than the

NPWEi model, and shows minimal dependence on , consistent with findings by Yendiki. 204, 205

The PW model represents the ideal observer for a signal-known exactly, background-known- exactly task, therefore yielding d' considerably higher than what one might expect from visual inspection of the images in Fig.6.9. Such observer models may be useful for optimization when the images are to be analyzed by a computer program which can potentially make use of image information not appreciable by humans. Another popular observer model, the channelized

Hotelling observer, has also been shown to exhibit optima dependent on different implementations of channels in emission tomography. 204, 205, 223 The question of which observer model to use for optimization is the subject of ongoing research. In the current work, the NPWEi model demonstrated qualitative agreement with observations in terms of the optimal ,

198 recognizing that more rigorous observer studies are required in future work to validate the model observer with human performance.

The PW model suggests a much higher level of performance and a lack of optimum, whereas the NPWEi model suggests a stronger dependence of detectability on , in qualitative agreement with Fig. 6.8.

(Published with permission from Medical Physics)

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Chapter 7 Summary and Future Work 7.1. SUMMARY This dissertation extended a task-based image quality framework to address several challenging and timely issues pertinent to the development and optimization of advanced x-ray tomographic imaging methods. Following the introduction and theoretical background provided in Chapters 1 and 2, respectively, Chapter 3 addressed the point of anatomical background as a noise component in “generalized” descriptions of NEQ and detectability. A physical phantom was created based on fractal theory to present power-law noise characteristics comparable to anatomical sites such as the breast or lung, and the effect of the background clutter power spectrum was investigated in tomosynthesis and CBCT. In Chapter 4, the theoretical framework was validated in comparison to human observer performance over a broad range of imaging conditions and imaging tasks. Chapter 5 extended task-based modeling to reconstruction-based

DE-CBCT, providing an efficient and rigorous framework for the optimization of DE imaging applications. Finally, Chapter 6 extended the analytical framework to describe not only the nonstationary characteristics of noise and detectability in FBP but also to statistical reconstruction methods, where the task-based approach was applied to PL iterative reconstruction and used to optimize image regularization .

Anatomical noise has been recognized to be a major contributing noise component in radiographic and tomographic imaging that impedes the detection of soft tissue tumors.95, 97-100

Findings in Chapter 3 confirmed such and demonstrated the importance of incorporating anatomical noise in the description of imaging performance. Specifically, when performance is anatomical noise-limited, increasing dose brings no further benefit to detectability. Results also

200 reveal that the effect of the magnitude of the anatomical noise, , has a much greater effect on detectability than the much scrutinized . Implications for system design with respect to the orbital extent of tomosynthesis and number of projections were investigated and, shown to depend on the imaging task: low frequency tasks, such as soft tissue tumor detection, require a larger tomosynthesis orbital extent to reject anatomical noise; on the other hand, high frequency tasks such as microcalcification detection, were more susceptible to performance degradation from quantum and electronic noise, allowing detectability with shorter orbital extents and a smaller number of projections.

The basic cascaded systems analysis framework introduced in Chapter 2 has been extensively validated with physical measurements of NPS, MTF, and NEQ for a broad range of imaging conditions.46, 58 The study presented in Chapter 4 provided further validation that imaging performance computed from first principles of cascaded systems analysis yields good correspondence with human observers. Of the five simple observer models investigated, the

NPWEi model was found to best agree with the performance of human observers, a finding consistent with Burgess84 and a later study by Richard et al.224 This study provided the first known validation of 3D cascaded systems analysis in comparison to human observers and helped to bridge the gap between subjective and objective image quality metrics. It also provided confidence that cascaded systems analysis provides a reliable framework for system design and optimization.

Such a theoretical framework is especially useful for advanced imaging applications, e.g., DE

CBCT, as addressed in Chapter 5. Current methods of optimization in DE-CT and/or DE-CBCT frequently involves exhaustive physical experimentation to iterate across numerous parameters of interest, including kV pairs, dose allocation, filter material/thickness, tissue cancelation factor,

201 etc. Moreover, the outcome measures of such experimentation tend to be either overly simplistic metrics, such as SDNR, or exhaustive observer-based measurements. Cascaded systems analysis extended to DE CBCT not only provided an accurate, general, and rigorous method for the optimization of DE imaging applications across a broad range of imaging conditions and reconstruction / decomposition techniques, but it also allowed imaging performance to be quantified in terms of more rigorous, task-based metrics of imaging performance. Chapter 5 investigated DE-CBCT optimization in the context of several DE imaging applications of interest in the imaging community, including imaging of the breast, abdomen, and musculoskeletal systems. Results suggest that the DE contrast (related to the tissue cancelation factor) and NEQ

(related to factors affecting quantum noise, including kV pair and dose allocation) need to be considered in combination, again emphasizing the importance of a task-based approach in optimization. The soft tissue tumor detection task in breast further illustrated that the power of reconstruction-based DE-CBCT, in fact, lies in the ability to remove anatomical noise, echoing the importance of incorporating the anatomical power spectrum in the “generalized” performance descriptors advanced in Chapter 3.

Chapter 6 first extended the cascaded systems analysis model to account for nonstationary (i.e., spatial variation) of the NPS due to variable fluence transmitted to the detector from view to view, which is often the most significant source of nonstationarity. The extension of the framework to explicitly describe nonstationary noise allowed calculation of the local NPS, MTF,

NEQ, and detectability index at any location in the object. This extension resolved, to a large extent, a common criticism of Fourier domain metrics with regard to shift-variance in spatial resolution and nonstationary in noise. Second, the task-based framework was applied to describe the imaging performance in statistical reconstruction of 3D images – commonly referred to as

202 iterative reconstruction. Drawing from noise estimate derivation for implicitly defined biased estimators,197, 206 the analysis demonstrated the non-stationary behavior of the NPS and MTF in such model-based reconstruction and combined the approach with task-based assessment of detectability index. The utility of the approach was demonstrated in the selection and design of optimal regularization parameters for specific tasks. The regularization strength (and spatial regularization “map”) required to improve detectability was found to be task-dependent.

Modeling statistical reconstruction within the framework of task-based assessment is anticipated to further the understanding and assessment of such algorithms in their translation to clinical systems and applications.

Studies presented in this work pertain to various imaging tasks relevant to a variety of tomographic imaging modalities, including single-energy and dual-energy tomosynthesis and

CBCT. Findings common to all of the applications included the importance of a task-based approach and incorporation of anatomical noise in describing image quality. The cascaded systems analysis framework proved to be fairly general in adapting to various modalities and reconstruction methods, and extensions of the framework to still more modalities and technologies is evident in recent literature – including phase-contrast imaging64, 65 and photon counting detectors.63 Task-based modeling and optimization in statistical reconstruction similarly opens new opportunities not only in the design of optimal regularization methods but also methods of image acquisition and reconstruction driven by analysis of task-based performance.211

7.2. FUTURE WORK Possible areas of future work in image quality modeling fall into two main areas: image perception and imaging physics. As stated in previous chapters, the form of observer model that

203 most accurately predicts human visual response remains an area of active research in perception science. An observer model that has been gaining popularity is the channelized Hotelling observer, which adopts multiple frequency channels capable of accommodating different responses to model the complexities of the human visual system. Such model observers have been found to yield good agreement with human observer performance160, 225 and can be readily incorporated into the image quality framework established in this work. Imaging tasks investigated in this work corresponds to the highly idealized signal-known-exactly, background- known-exactly task, which, of course, does not present the complexities of clinical tasks where radiologists do not always know the nature of the lesion, need to search for the lesion, or estimate the size of the lesion. Observer models that account for scrolling of slices are also being investigated.226 Extension of observer models to more accurately predict human visual response and represent clinical tasks represent a promising area of ongoing and future research.

With respect to related future work in imaging physics, research underway seeks to incorporate more realistic physics models of both detector response and beam characteristics into the process of statistical reconstruction. The underlying, well-understood physics model of FPDs and

CT/CBCT with analytical or statistical reconstruction methods to yield realistic noise and resolution models is an area of considerable interest. More general statistical reconstruction algorithms (i.e., beyond the quadratic PL method shown in Chapter 6) pose new challenges for image quality modeling. The optimization algorithms used are usually nonlinear, irreversible, and can be irreproducible. The noise and resolution estimates used in Chapter 6 assumed a converged solution, but this is not always achievable in clinical settings where time is a constraint. Noise estimates as a function of iteration for the expectation-maximization algorithm has been investigated by Barrett.227 Whether the same can be done for other optimization

204 algorithms remains to be seen. In addition, statistical reconstruction possesses considerable freedom in the selection of (1) object function, with possible inclusion of prior information,228 registration,229 known component,230 etc.; and (2) penalty functions. Propagating image quality estimates through a non-quadratic objective function and/or penalty function presents a challenge yet to be addressed. Task-based description of image quality is applicable in such cases by means of experimental measurement of the noise and resolution, but a robust theoretical model of image quality in such non-quadratic model-based reconstruction methods is yet to be realized.

With the rapid emergence of new imaging tomographic modalities, spurred in large part by the development and availability of high-performance detectors and new reconstruction methods over the last decade, conventional x-ray contrast limits are transcended by means of spectral imaging. An interesting question to consider is whether spectral imaging in which materials are discriminated and re-labeled / displayed, for example, as color images suggests a wholly different form of imaging task. Detection or discrimination in this case tends more toward a classification task, the performance of which is highly related to the signal and noise characteristics of the images but may not be straightforward to describe. If the true composition of the object is known, one can evaluate the performance of the classification outcome with

ROC-type metrics of true positive, false negative, etc. By varying a threshold that affects the accuracy of classification, one can then compute the area under curve of the ROC and relate it to the detectability index. When more than two materials are present, the ROC has multiple outcomes and the analysis falls to more recent developments in multiple-choice statistical decision theory.

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The last two decades have seen exciting developments in x-ray detector technologies, image processing and reconstruction techniques, and an increasing scope of applications. The challenges and opportunities arising from such advances include the reduction of patient dose and key areas of application for technologies such as tomosynthesis, statistical reconstruction, photon counting detectors, DE and spectral imaging, and phase contrast imaging. Such technologies – combined with the distinct constraints and imaging tasks in a spectrum of potential applications – pose a vast challenge to understanding, optimization, and assessment of image quality which may at least in part be addressed by the research advanced in this dissertation.

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