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KELLY, Kevin Anthony, 1945- AN ON-AXIS FRESNEL ZONE PLATE IMAGING SYSTEM WITH A GERMANIUM GAMMA RAY CAMERA AND COMPUTER RECONSTRUCTION.

The Ohio State University, Ph.D., 1976 Engineering, nuclear

Xerox University Microfilms t Ann Arbor, Michigan 48100 AN ON-AXIS FRESNKL ZONE PLATE IMAGING SYSTEM

WITH A GERMANIUM GAMMA RAY CAMERA

AND COMPUTER RECONSTRUCTION

DISSERTATION

Presented in Partial Fulfillment of the Requirements

for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State University

By Kevin A. Kelly, B.Sc., M.Sc.

The Ohio State University

1976

Reading Committee: Approved by:

Dr, Robert F. Redmond ^ Dr. Shoichiro Nakamura si ( / Dr. Philip A. Schlosser v / & & * * * < - Advisor Department of Nuclear Engineering Dedicated to

James and Anne Kelly,

my parents ACKNOWLEDGMENTS

Initially this research was funded by the National

Institutes of Health under contract NIH-NINDS-72-2323.

I wish to thank the Mechanical and Nuclear Engineering

Departments of The Ohio State University for their support during the latter part of this research. I am indebted to Dr. Arnold H. Deutchman of the Department of Nuclear

Medicine for several invaluable discussions of the role of gamma cameras in nuclear medicine, and for reviewing the rough draft of Chapter 7. I am grateful to Dr. Mark S.

Gerber, Dr. Philip A. Schlosser, and Mr. John W. Steidley

for discussions of the electronic and mechanical limi

tations on the spatial resolution of the orthogonal strip germanium camera. My sincerest thanks go to Ms. Carol

Edger for typing this dissertation under trying circum

stances. I wish to thank my reading committee, including

Dr. S. Nakamura, who taught me almost all I know about

nuclear reactor theory, and Dr. Philip Schlosser, who

directed the overall gamma camera research effort and

took an interest in me as a person as well as a student.

Above all, I would like to thank my advisor, Dr. Robert F.

Redmond, whose interest in me brought me to Ohio State.

Dr. Redmond suggested this research area and provided

direction and scope for the research. VITA

March 29, 194S Born - Croydon, United Kingdom

1967 B.Sc., Department of Physics, Uni versity of Notre Dame, Notre Dame, Indiana

1968 M.Sc. , Department of Physics, Yale University, New Haven, Connecticut

1968-1971 Peace Corps Volunteer, Ghana

1972-1975 Research Associate, Department of Nuclear Engineering, The Ohio State University, Columbus, Ohio

1974 M.Sc. , Department of Nuclear Engi neering, The Ohio State University, Columbus, Ohio TABLE OF CONTENTS

page

ACKNOWLEDGMENTS i• • n •

VITA iv

LIST OF TABLES .... ix

LIST OF FIGURES .... x

PARTIAL LIST OF SYMBOLS x* m > *

CHAPTER

1. INTRODUCTION . . .

1.1 Purpose of the Research 1 1.2 Rationale for the Research . . . 1 1.3 Goals of the Research ...... 3 1.4 Organization of the Dissertation 4

2 . RESEARCH MOTIVATION AND RELATED RESEARCH BY OTHER INVESTIGATORS . . . 5

2.1 Motivation for the Res earch 5

2 .1.1 Imaging Methods ...... 6 2 .1.2 Coded Apertures ...... 9 2 .1.3 Fresnel Zone Pl|ate Apertures 16 2.1.4 Optical Reconstruction . . 18 2.1.5 Digital Reconstruction . . 22

2.2 Related Research by Other Investigators 23

2.2.1 Shadowgrams with Other Detectors 23 2.2.2 Digital Reconstruction of Holo grams and Shadowgrams ...... 28

v TABLE OF CONTENTS (continued)

page ’

3. RELEVANT THEORY ...... 30

3.1 Optical Theory of Shadowgrams ...... 30 3.2 Gamma Ray Imaging by Shadowgrams...... 33 3.3 Digital Signal Processing Theory ...... 34 3.4 Image Enhancement Theory ...... 40

4. OUTLINE OF SYSTEM DESIGN ...... 44

4.1 The Germanium Detector ...... 44 4.2 The Fresnel Zone Plate S y s t e m ...... 45

4.2.1 Zone Plate Paramet ers...... 45 4.2.2 Relations Among Parameters .... 50 4.2.3 Method of Choosing Parameters . . . 56 4.2.4 Parameters for the Germanium Camera 64 4.2.5 Mechanical Components ...... 68

4.3 Image Processing and Display ...... 71 4.4 The Complete S y s t e m ...... 74

5. DIGITAL RECONSTRUCTION OF SHADOWGRAMS ...... 76

5.1 The Reconstruction Model ...... 76 5.2 Reconstruction of Optical Shadowgrams . . 79

5.2.1 Experimental Apparatus and P r o c e d u r e ...... 80 5.2.2 R e s u l t s ...... 85

5.3 Reconstruction of Computer-Generated Shadowgrams ...... 98

5.3.1 Generation of the Shadowgrams . . . 98 5.3.2 Point S o u r c e s ...... 99 5.3.3 Small Extended Sources ...... 102 5.3.4 Large Extended Sources ...... 107 S. 3.5 Extended Sources with Cold Spots . 108 5.3.6 Sources of Nonuniform Intensity . . 113 5.3.7 Small Strong Sources in a Weak Source Field ...... 116 5.3.8 Sources in Different Tomographic P l a n e s ...... 117

vi TABIiE OF CONTENTS (continued) page

5.4 Summary of Results ...... 125

6. IMAGE ENHANCEMENT STUDIES ...... 127

6.1 Introduction...... 127 6.2 A Fourier Transform Technique ...... 129

7. CLTNICAL ADVANTAGES AND LIMITATIONS OF THE SYSTEM ...... 143

7.1 Limitation on Source S i z e ...... 143 7.2 Imaging Whole Organs ...... 146 7.3 Imaging Small Lesions ...... 148 7.4 Conclusions ...... 151

8. SUMMARY OF RESULTS AND CONCLUSIONS ...... 153

8.1 Summary of the R e s e a r c h ...... 153 8.2 Summary of the Results ...... 154 8.3 Extensions of the R e sea rch...... 156 8.4 Conclusion...... 157

APPENDICES

A. OPTICAL THEORY OF FRESNEL ZONE PLATES AND HOLOGRAMS - AN OVERVIEW ...... 158

A.l The Frcsnel Zone P l a t e ...... 158

A. 1.1 I n t r o d u c t i o n ...... 158 A. 1.2 Basic Zone Plate T h e o r y * . 159 A.1.3 Applications of Zone Plates .... 173 A.1.4 Manufacturing of Zone Plates . . . 173

A.2 Holography ...... 174

A. 2.1 Basic H o l o g r a p h y ...... 174 A.2.2 History of Holography ...... 181

A.3 Relationship of Fresnel Zone Plates to H o l o g r a m s ...... 187

B. THEORY OF ZONE PLATE IMAGING OF GAMMA RAYS . . 190

vii TABLE OF CONTENTS (continued)

page

C. THE GERMANIUM GAMMA CAMERA ...... 196

C.l The Orthogonal Strip Detector ...... 196 C.2 Detector A r r a y ...... 199 C.3 E l e c t r o n i c s ...... 201 C.4 Mechanical Components ...... 204

D. THEORY OF DIGITAL RECONSTRUCTION OF SHADOWGRAMS 208

D.l The Diffraction Integ ral...... 208 D.2 The Fresnel Approximation ...... 213 D.3 The Fraunhofer Approximation ...... 216 D.4 The Reconstructed Image as a Fourier Transform...... 217 D.5 Application of Sampling Theory ...... 2X9 D.6 Digitizing the Reconstruction Integral . . 223 D.7 Reconstructing the True I m a g e ...... 226

BIBLIOGRAPHY . 229 LIST OF TABLES

Table page

4-1 Parameters of the Imaging System 50

4-2 System Parameters and Equations for Optimum System Resolution 63

4-3 System Parameters and Equations for a Specified Field of View 65

4-4 Zone Plate - Germanium Camera System Parameters 67

5-1 System Parameters for 1/8-inch Diameter Circular Source Shadowgram 86

5-2 Line Printer Code 89

5-3 System Parameters for Shadowgram of the Letter "E" 91

5-4 Relative Intensity of Weaker Source and Weaker Image 114

ix LIST OF FIGURES

Figure page 2.1 Conventional apertures 7 2.2 The Fresnel zone plate aperture 17

2.3 The optical reconstruction system 19

3.1 Recovery of a sampled waveform 38

4.1 A square detector of size Di divided into resolution elements of size Axi 47

4.2 Zone plate system field of view 48

4.3 The objcct-to-zone plate distance, si, and the zone plate-to-dctcctor distance, s2 49

4.4 The relationship between object plane resolution At and detector plane resolution AXj 53

4.5 Radiation from a point source in the object plane passing through the outermost zone 55

4.6 Relationships among the zone plate, detector and object plane parameters 57

4.7 Attenuation of gamma rays passing through the outermost zone 59

4.8 Mechanical components of the zone plate system 69

4.9 Block diagram of the complete imaging system 75

5.1 Experimental apparatus for casting optical shadowgrams 81

5.2 Image of a point source from an optical shadowgram 88

5.3 Imago of the letter "E" from an optical shadowgram 92

x LIST OF FIGURES (continued) Figure page

5.4 Alignment of "E" arms with rows of resolu tion elements 95

5.5 The source pattern for the "F." 104

5.6 The shadowgram for the "E" 105

5.7 The image of the "E** 106

5.8 Image of triangle and square focussed on plane 8.0 121

5.9 Image of triangle and square focussed on plane 8.5 122

5.10 Image of triangle and square focussed on plane 9.0 123

6.1 Results of transfer function calculation for a point source at the center of the field of view 136

6.2 Results of transfer function calculation for a point source in the corner of the field of view 137

6.3 Enhanced image of the letter "E" 140

A.l Spherical wavefront expanding from point 0 160

A. 2 Wavefront geometry 162

A.3 A Fresnel zone plate 166

A.4 Imaging an off-axis point source 167

A.5 Reference beam and signal arriving at the film . . 175

A.6 The diffracted beams 177

A.7 Hologram of a point scatterer 179

A.8 Reconstructing a point image 180

A.9 Hologram of a transparent object 185

xi LIST OF FIGURES (continued) Figure page

A. 10 Hologram of an opaque object 186

C.l Schematic of the orthogonal strip, charge splitting detector 198

C.2 Illustration of the 4 x 4 detector array 200

C.3 Schematic diagram of the electronics required to process signals emerging from one quadrant of the detector array 203

C.4 The camera assembly consisting of camera stand, camera head, and adjustable counter weights 205

C.5 Cutaway view of the camera head 206

D.l Coherent radiation incident on a shadowgram 210

D.2 The reconstruction geometry 212 PARTIAL LIST OF SYMBOLS

The symbols listed here are those which recurrently appear in the text.

Symbol Definition Equation

R„ Radius of the n-th Fresnel zone 3-1 n

Ar Width of the n-th Fresnel zone 3-2 n

SNR Signal-to-noise ratio 3-5

s^ Object-to-zone plate distance 4-1

&2 Zone plate-to-detector distance 4-1

a Magnification parameter 4-1

N, Number of resolution elements per L side 4-2

Length of square detector side 4-2

Ax-. Length of square resolution element 1 side 4-2

T Field of view 4-3

At Object plane resolution 4-3

N Number of zones in zone plate 4-4

D^j Diameter of the zone plate 4-4

tz Thickness of the zone plate 4-11

* « i li CHAPTER 1

INTRODUCTION

1.1 The Purpose of the Research

This study examines the feasibility of obtaining gamma ray camera images by means of an on-axis Fresnel zone plate aperture to produce shadowgrams and by digital reconstruction of images from the shadowgrams produced. A zone plate aperture system for a germanium gamma camera is designed and evaluated by means of computer studies.

The design is based on the characteristics of a high purity germanium camera currently being developed by the Nuclear

Engineering Department at The Ohio State University. The evaluation examines the ability of the system to provide images useful for the accurate diagnosis of disease in the nuclear medicine clinic.

1.2 Rationale for the Research

The current development of a high resolution germanium camera has prompted consideration of large area apertures for efficient collection of gamma radiation. The most useful such aperture, the Fresnel zone plate, results in 2 a pattern on the detector which resembles a hologram of the source intensity distribution. Normally the image of the source is recovered by beaming coherent laser light through a transparency of the holographic pattern. The resulting diffraction pattern is the image. A difficulty with this technique for the on-axis zone plate has been the obscuring of the diffraction pattern by the undiffractcd beam of light. This difficulty has prompted those researchers using X-ray film as the detection medium to use an off- axis zone plate in combination with a half-tone screen; however, the spatial resolution of this system is then degraded by a factor of three, and the efficiency by a factor of two. This solution is not acceptable for a ger manium gamma camera.

An alternative solution to this difficulty is con sidered here, the digital calculation of the diffraction pattern. Little previous work has been done with computer reconstruction of holograms because the number of resolu tion elements in an optical hologram is usually very large, and the calculation requires a great deal of computer time. The number of resolution elements in the germanium gamma camera considered here is relatively small, and the output of the camera is necessarily in digital form.

Hence the feasibility of digitally reconstructing camera holograms merits consideration. In addition, the 3 availability of image data in digital form permits the application of image enhancement techniques, developed for space and military purposes, to nuclear medicine. These considerations are the subject of this research.

1.3 The Goals of the Research

This dissertation describes the development and evaluation of a design for a practical coded aperture system with digital reconstruction. The system contains a high purity germanium, gamma ray imaging camera with an on-axis Fresnel zone plate aperture. The design of the coded aperture system is based on constraints dic tated by the characteristics of the camera detector and its associated electronics. These characteristics were established by research into gamma camera design currently being carried out by the Nuclear Engineering Department at The Ohio State University.

There are four primary goals of the research described in this dissertation. First, the physical system is to be designed. Second, the feasibility of computer recon struction is to be demonstrated. Third, the optimal digital image enhancement scheme is to be determined.

Fourth, the clinical capabilities and limitations of the zone plate, germanium camera system and associated data processing system are to be determined. 1.4 The Organization of the Dissertation

A chapter-by-chapter overview of the dissertation is presented here. Chapter 2 is a qualitative discussion of the advantages and disadvantages of coded aperture imaging with a germanium camera; this discussion motivates the consideration of digital reconstruction to overcome the difficulties. In this chapter also, closely related research by other investigators is summarized. Theory relevant to this work, but not developed by the author, is presented in Chapter 3. The terminology, concepts and equations of Chapter 3 will be used extensively in the following chapters. Chapter 4 presents the overall system design. The germanium camera is described, and a method ology of Fresnel zone plate system design is presented.

Chapter 5 deals with the digital reconstruction of shadow grams, beginning with an outline of the theory of digital reconstruction. The simulation of shadowgrams which would be formed by the design system is discussed, and the quality of the images obtained from these shadowgrams is evaluated.

Chapter 6 discusses the improvement in image quality obtained in the image enhancement studies. The clinical practicability of the overall system is evaluated in

Chpater 7. Chapter 8 contains a concise summary of the results and the conclusions drawn from these results. CHAPTER 2

RESEARCH MOTIVATION AND RELATED RESEARCH

BY OTHER INVESTIGATORS

2.1 The Motivation for the Research

In nuclear medicine, gamma ray cameras are used to form images of the distribution of radioisotope uptake in many bodily organs. Conventional cameras use a sodium iodide scintillation crystal as the detection medium; the scintillation camera has a spatial resolution of one to two centimeters. An aperture must be placed between

the organ and the detector to form an image of the organ on the detector face. The parameters of the aperture are determined by the spatial resolution of the detector.

Germanium detectors are expected to become available in

gamma cameras during the next few years (1.-4.) » and will

be capable of a spatial resolution of about two millimeters.

Present gamma camera apertures are unsuitable for a high

resolution camera. This research investigates the feasi

bility of using a particular aperture suitable for a high

resolution camera, collecting image data in coded form,

decoding the data and enhancing the image by digital

processing.

5 6

2.1.1 Imaging Methods

Traditional gamma camera apertures are variations

of two basic types, the pinhole aperture and the multi

hole collimator (see Figure 2.1). Consider a distribution

of radioisotopes to be made up of many small point

sources of radiation. With both types of apertures,

although a point source at A emits radiation in all

directions, only radiation emitted into a small cone

passing through a hole in the aperture is allowed to

reach the detector. The point source irradiates an

approximately circular area on the detector, and the

aperture is designed so that the diameter of this circle

equals the spatial resolution of the detector.

An improvement in detector resolution requires smaller

aperture holes, and hence more radiation. As the holes

become smaller, the cone into which gamma rays are emitted

bocomes smaller. The solid angle subtended by this cone

is proportional to the square of the detector resolution.

.For example, consider a case in which the spatial resolu

tion of a camera is improved from 2 cm to 2 mm. To

realize this factor of ten improvement requireis reducing

the aperture hole diameter by a factor of ton, but then

the number of gamma rays reaching the detector face is

reduced to 1% of the former value. This reduction to 1%

applies only to the pinhole aperture; for the multihole

collimator more holes may be added to the design, but 7

source

detector aperture

(a)

source aperture detector (b)

Figure 2.1 Conventional apertures. (a) the pinhole aperture; (b) the multihole collimating aperture. 8 limitations on the hole spacing imposed by geometric, material and gamma ray penetrability considerations still result in a reduction in the number of gamma rays reaching the detector.

In addition, for either type of aperture, a 2 cm by

2 cm square on the detector constitutes one resolution element before the improvement, but constitutes 100 resolution elements after the resolution is improved to

2 mm. Suppose that an average of n gamma rays must be detected in each resolution element to obtain a statis tically well-defined image. In the former case, n gamma events must be counted in the 2 cm square. In the latter case, n counts per 2 mm square are required; equivalently,

100 n counts are required in the 2 cm square in order to obtain an image having a statistical quality equal to the quality obtainable by the former detector. As the need for radiation increases by a factor of 100 per unit area, the amount of radiation actually received per unit area decreases (for a pinhole) by a factor of 100. It is desirable to obtain images of small organs with a germanium camera such that these images have the same statistical quality obtained previously with a scintillation camera for images of organs ten times larger. This requires that the organ emit 10,000 times as many gamma rays as before. 9

To meet this requirement for more radiation, one may' either increase the dose given to the patient by a factor of 10^ or increase the time for collecting radiation by

10^. Neither alternative is acceptable. One may not

increase patient dose from several millicuries to tens of

curies. The collection time is limited by the need for many patients to use the camera, by demands on the clini

cian's time, and by the need to minimize discomfort to the

patient. In addition, a high resolution camera actually

requires a shorter collection time than conventional

cameras because patient motion is expected to be a problem

in obtaining high resolution images. Furthermore, a high

resolution gamma camera is expected to significantly

expand the role of dynamic function studies in nuclear

medicine: if the image of a changing radioisotope

distribution can be formed every few seconds, one has in

effect a motion picture of the body's internal organ.

These dynamic function studies demand a minimal collection

time. Small collection time and reasonable patient dose

impose a limitation on the spatial resolution achievable

by a gamma camera using a traditional camera aperture.

2.1.2 Coded Apertures

The collection time can be reduced for dynamic

studies and the dose to the patient may be reduced for

static images by abandoning these traditional camera apertures which reproduce directly on the detector face the radioisotope distribution in the body. There are a variety of apertures which provide a large open area for the transmission of radiation but which do not result directly in the image (5). Instead, the aperture forms a pattern on the detector face which contains the image in coded form, and the pattern must be decoded to obtain the image. In this case, the aperture is termed a coded aperture.

The principle advantage of the coded aperture is that a larger fraction of the radiation emitted by the source reaches the detector, and this fraction is independent of detector spatial resolution. Replacing a pinhole or multihole aperture with a coded aperture is comparable to replacing an optical pinhole camera with a lens camera.

The lens in a camera has two functions: it provides a large opening for collecting light efficiently and it focuses the collected light into an image. The coded aperture has the advantage that it provides a large opening for collecting gamma rays efficiently. Its disadvantage is that obtaining the image is a two-step process. The

first step, shadowcasting, is the collection of radiation to form a coded pattern. The second step, reconstruction,

is the decoding of the pattern to retrieve the image. 11

A coded aperture system is especially suitable for a ' germanium gamma camera. The coded aperture provides the high gamma ray collection efficiency, and hence short collection time, needed for optimal utilization of the germanium camera. At the same time, this camera has the qualitites of high detection efficiency, good scatter discrimination and excellent spatial resolution needed for successful implementation of the coded aperture con cept.

A number of coded apertures having arrays of multiple pinholes has been proposed. A coded aperture consisting of an array of N pinholes collects N times as many gamma rays as cither a pinhole aperture or a multihole collimator.

A multihole collimator is made thick enough so that ideally gamma rays from a point source pass through only one hole in the collimating aperture. On the other hand, the multi hole coded aperture is made thin enough so that no collimation occurs, that is, gamma rays from a point source are able to pass through any hole in the coded aperture.

Four multihole coded apertures have been proposed: the random array, the nonredundant array, the linearly modu lated array, and the time modulated array.

The random array (6) divides the aperture into a mesh of small squares, half of which are opaque and half of which are transparent holes. The positions of these holes are randomly selected. Each point source in the object to be imaged casts a shadow of this aperture pattern onto the detector face. These overlapping aperture patterns create a coded image. Several optical reconstruction schemes have been suggested, each requiring a positive photographic transparency of the coded image. One such scheme involves using the original aperture pattern as a filter for light beams directed through the coded transparency. For each of the overlapping aperture patterns in the transparency a beam direction exists for which that pattern is aligned with the original aperture pattern.

The filtered light beams arc focussed onto separate points in the image plane with a lens. For beams not encountering aligned patterns, the light intensity is reduced by a factor of two in passing through the coded trans parency and again by a factor of two in passing through the aperture pattern, for an overall reduction by four.

However, if the transparency pattern and aperture pattern are properly aligned, the light intensity is reduced only by a factor of two. Hence there will be a brighter spot on the image plane corresponding to each point source in the original object.

The biggest disadvantage of the random array aperture is that the intensity of the image is never more than twice the intensity of the background. This low contrast may not be a problem in such applications as X-ray or gamma ray astronomy where separate point souces are to be imaged. But for nuclear medicine applications concerned with imaging continuous-tone objects, the low contrast attainable with the random array aperture has limited further consideration of its use.

The nonredundant array (7) alleviates the problem of low contrast by sacrificing considerable collection efficiency. Only about 24 of the aperture area allows radiation to pass through to the detector. Typical non redundant apertures contain 15 to 27 pinholes. These pinholes are spaced nonredundantly, that is, the vector displacement between any pair of holes is unique. Coded transparency formation and image reconstruction with this aperture is similar to the procedures outlined for the random array. As before, when the transparency and aper ture patterns are properly aligned during reconstruction, light will pass through all the holes in both arrays.

However, if the nonredundant arrays are not properly aligned, the light will pass through, on the average, only a single pair of holes. As a result, both the contrast between the image and background and the gamma ray collection efficiency increase with the number of holes in the array. 14

A possible limitation on the nonredundant array idea' is the excessive computer time required to generate non redundant array patterns when the number of holes is more than about 25. Nevertheless, the nonredundant array is a promising method of coded aperture imaging which has only recently begun to receive consideration.

The linearly modulated array (8) is a single line of pinholes for which the interhole spacing decreases from one end of the line to the other. More precisely, the spatial frequency of the array at any point is proportional to its horizontal position along the array. The motiva tion for this aperture is the simplicity of the reconstruc tion process which requires no precision optical system.

The coded image is viewed with a television camera which converts each horizontal line of the coded image to a temporal signal. When this video signal is passed through an appropriate dispersive delay line, each overlapping aperture pattern in the signal is compressed in the time domain into a single pulse which results in a bright spot on a television monitor. The resulting group of bright spots is the decoded image. Although the feasibility of this scheme has been demonstrated, the quality of the images obtained has been unsatisfactory, and no current research using this scheme is being carried out. IS

The time modulated array (9) is an attractive scheme' recently introduced which promises to provide resolution

> * which is uniform over the camera field of view. Most coded aperture schemes suffer from the fact that object points near the edge of the field of view cast shadows of only part of the aperture pattern onto the detector face. The reconstruction process, which is essentially a spatial integration over the recorded data, results in

images of these points having reduced resolution because of the restricted spatial limits of integration. With

the time modulated scheme, reconstruction is performed by integrating over the time domain in such a way that

the equal time limits of integration for all points produce

equal resolution for all points.

The aperture is a long lead strip containing a pseudo

random sequence of holes. During data collection the

strip moves along a line at right angles to the line from

source to detector, so that the portion of the strip which

is the effective aperture is a function of time. Data

is recorded digitally as number of counts per detector

resolution element per unit time. The decoding scheme

requires digital data processing. Good results have been

obtained with the time modulated array. The chief

disadvantage of using this array at present lies in the 4 16 excessive computer storage and processing time required to implement it.

2.1.3 Fresnel Zone Plate Aperture

The coded aperture which has received the most attention is the Fresnel zone plate. It is attractive because, at least in theory, reconstruction is accomplished most easily with this aperture. The Fresnel zone plate consists of concentric annular regions of equal area, alternately opaque and transparent to radiation. This pattern has the property that it acts as a diffraction lens to focus monochromatic light.

The use of a Fresnel zone plate aperture with a gamma camera is illustrated in Figure 2.2. Gamma rays passing through the aperture undergo no diffraction; each point source in the organ simply casts its own shadow of the pattern onto the detector face. The distribution and intensity of the overlapping patterns represent coded information about the distribution and concentration of the source material taken up by the organ to be imaged.

The coded pattern even contains three-dimensional infor mation about the source distribution because sources close to the aperture cast larger shadow patterns than those farther away. In addition, the whole detector face contains information about each point source, pro vided the aperture material is thin enough so no source aperture detector

Figure 2.2 The Fresnel zone plate aperture. 18 collimation occurs. If half the detector pattern were somehow lost, the remaining half would still contain information about the whole source distribution. The reader familiar with holography will recognize the simi larity between the pattern encoded on the detector and a hologram. In fact, this pattern is a hologram, formed by shadowcasting rather than by interference. The Fresnel zone plate pattern is similar to the hologram of a point source interfering with a plane wave, and the pattern on the detector face is similar to a hologram of the source distribution.

2.1.4 Optical Reconstruction

As in the case of an interference hologram, the

image encoded in the shadowcast hologram can be recon

structed using a source of monochromatic light. Figure 2.3

shows a photographic transparency of the detector pattern

illuminated with monochromatic laser light. Each zone plate pattern on the transparency acts as a lens focussing

the light to a separate point on the screen. The focal

length of each "lens" depends on the size of the zone

plate pattern. Because sources at various distances cast

shadows of various sizes, the images of these sources will

be formed at varying distances from the coded pattern

transparency. Hence by moving the screen in Figure 2.3, laser lenses coded pattern transparency

screen

Figure 2.5 The optical reconstruction system. i-* to 20 one can bring into focus various tomographic planes in the radioisotope tagged organ.

Optical reconstruction of shadowcast holograms, called shadowgrams, involves a difficulty not encountered with interference holograms. With interference holograms there is an angular separation of the dim, diffracted image beam from the intense, undiffracted beam passing straight through the hologram transparency. With shadowgrams there is no angular separation; the diffracted image is overlapped and obscured by the undiffracted beam. The obscuring effect can be virtually eliminated when forming an image of a small object, but an image of a large object can be formed only with unacceptably low contrast (10J .

Angular separation can be achieved by using only a portion of a Fresnel zone plate for shadowcasting. How ever, this portion, called an off-axis zone plate, results in an image having less detail than the image formed by a complete, on-axis zone plate. Use of an off- axis plate requires sacrificing by a factor of three (11) the high spatial resolution capability of the germanium camera - the capability which motivated the use of a coded aperture. In addition, the off-axis plate cannot adequately encode information about large objects unless used in series with a second aperture acting as a half tone screen (12). This dual aperture system reduces the 21 radiation collected by a factor of two, but more seriously it sometimes results in the appearance in the image of small nonexistent artifacts (13).

There are other serious difficulties with the optical reconstruction of shadowgrams. High contrast in the image requires a large dynamic range in the film transparency, that is, a large ratio of radiant energy required for maximum film exposure to that for minimum exposure. The limited dynamic range of film limits the contrast attain able in holographic images (L4) . Another difficulty is the painstaking film processing procedures required for the production of high quality holograms (1J5); these pro cedures would be difficult to carry out routinely in a clinical situation. In addition to processing for the purpose of maintaining the high dynamic range of the film, shadowgrams should be bleached and masked (16). Bleaching enhances the brightness and contrast of the image obtained.

Since more gamma rays are collected from sources near the center of the camera field of view than from those near the edge, the central sources appear brighter in the image; masking techniques during film processing can eliminate this bias. These masking techniques in particular require much more time and attention to detail than could reason ably be expected from a technician in a busy nuclear medicine clinic. 22

The difficulties with the optical reconstruction of shadowgrams led to consideration of the feasibility of digital reconstruction.

2.1.5 Digital Reconstruction

Difficulties encountered with optical reconstruction would not be encountered with digital computer reconstruc tion. The coded shadowgrams can be formed with an on-axis zone plate. The diffraction pattern resulting from illuminating the shadowgram with a beam of coherent radia

tion can be calculated digitally, and this diffraction pattern is the desired image. In digital reconstruction the image is not obscured by an undiffracted beam. Image uniformity can be obtained by a digital masking technique.

Computer processing can simulate any optical processing possible in a research laboratory, eliminating the need

for a clinician to operate a precision laser and optical

system and to carry out the special film processing pro

cedures required for holographic work.

In addition to avoiding the difficulties of optical

reconstruction, digital reconstruction of germanium

camera data in the nuclear medicine clinic has a number of

advantages. Data from a germanium camera is digital and

hence no loss of information is involved in digitizing the

data; in fact, digital processing can extract the maximum 23

information from the data. With computer reconstruction, no information is lost by using a CRT display to transfer

the data onto photographic film having film grain noise

and limited dynamic range, and no optical system aberra

tions are introduced during reconstruction. The "dynamic

range" of computer data memory can be chosen as large as

needed. Also, digital image enhancement can be readily

included in the data processing procedure to obtain an

image of maximum diagnostic value. Finally, quantitative

results are readily available. An integrated system of

germanium camera, minicomputer and visual display would

provide in a few seconds a high resolution image of an

internal organ for immediate diagnosis.

2.2 Related Research by Other Investigators

No research has been reported by other investigators

considering the use of the Fresnel zone plate aperture with

a high-purity germanium camera. All published research

to date has looked at the suitability of this aperture

for use with other detection media and, with one exception,

only optical reconstruction was employed.

2.2.1 Shadowgrams With Other Detectors

The first application of the Fresnel zone plate

aperture to radioisotope imaging was reported by Barrett 24 in 1972 (17). An on-axis zone plate created the shadow gram pattern by transmitting the gamma radiation received from a thyroid phantom. The shadowgram was detected by an Anger scintillation camera, and photographed from a

CRT display with a polaroid camera using transparency film.

This transparency was then illuminated by a beam of coherent light from a laser to reconstruct the image.

Although the fcasilbility of the concept was demonstrated, the poor resolution of the scintillation camera (18) severely limited the attainable image quality.

A group headed by W. L. Rogers (1£) then carried out a series of experiments testing the zone plate concept by simulating radioactive sources with optical sources and detectors with photographic film. They showed that better optical reconstructions could be obtained using an off-axis

zone plate. In addition, a significant increase in the

signal-to-noise ratio of the image occurred when a half

tone screen was used together with the off-axis zone plate.

These optical experiments indicated that a zone plate

aperture could be used successfully with a gamma camera

provided (1) the camera detects gamma rays efficiently,

and (2) the camera has good spatial resolution. Rogers'

group concluded that the use of a zone plate aperture

requires a detector resolution which is a factor of three

better than the resolution of the scintillation camera. 25

The use of a zone plate with the Michigan high reso lution image intensifier camera was investigated next by this group. This camera has a theoretical line spread function of 1.5 mm full width at half maximum for 122 keV gamma rays, but has an intrinsic efficiency of only

12% for 140 keV gamma rays. In zone plate experiments a standard thyroid phantom had to be loaded with the rela tively high activity of 5 mCi of ^ mTc to reduce the ampli fier noise to acceptable levels. In addition, the actual resolution of this camera was significantly poorer than that predicted theoretically because distortion in the intensifier tube had a defocussing effect on the shadow gram. A conclusion drawn from these experiments was that the system appears useful for small sources but not for larger sources containing voids. Also, the successful use of a zone plate aperture requires a camera with high intrinsic efficiency and a good actual spatial resolution.

Rogers concluded that "the zone plate imaging system opens a new dimension for imaging techniques in nuclear medi cine. This field still is in its infancy and considerable research is still needed, both theoretically and experi mentally" [20) .

Another detector capable of high resolution which has been used with a zone plate aperture is the multiwire proportional chamber. An on-axis zone plate was used with this detector by Macdonald and his colleagues (2JL) to image arrays of point sources. The purpose of the work was to verify a theoretical formula for the signal-to-noise ratio attainable as a function of the number of point sources in the array. Although there were no imaging studies of extended sources, the multiwire proportional chamber appears to be less suitable than a germanium detector for coded aperture applications in nuclear medicine because of its low sensitivity and poor energy resolution.

Only 301 efficiency is achieved for low energy 22 keV gamma rays, and the energy resolution is only about 15%.

The most successful use of a Fresnel zone plate aper ture to date has been with X-ray film as the detection medium. Barrett (22J has developed an experimental camera employing a half-tone screen, an off-axis zone plate, and an X-ray film cassette incorportating two fast intensivying screens. The X-ray transparency, after being photoreduced, is a shadowgram which produces a reconstructed image when inserted into an optical reconstruction system.

Experiments were carried out laying radio-opaque strip patterns of various spatial frequencies over an extended source. High resolution images were obtained for strip widths and spacings ranging from more than 1 cm down to about 5 mm. For a higher frequency spacing, the image contrast went abruptly to zero. Hence the system 27 appeared to have a step-like spatial frequency response.

Further experiments with a high resolution gold 2one plate and a thyroid phantom resulted in a system resolu tion between 2 and 3 mm.

The ability of the system to separate various tomo graphic planes was also tested. When one plane is in focus, other planes are out of focus to the extent that they are distant from the in-focus plane. Experimentally,

Barrett's group was able to clearly distinguish planes

2-1/2 cm apart. An object clearly in focus would begin to be distinctly out of focus if the system were focussed on a plane 5 mm behind that object. A difficulty with the three-dimensional capability of the system is that the out-of-focus image may not be simply blurred but can be made up of scrambled, but sharply defined components of the true image. This effect, along with the introduction into the image of nonexistent artifacts, is possibly due to the high spatial frequency content created by the half tone screen. In a clinical situation, this could lead to a false interpretation of the existence or position of a source. Clinical evaluation of this system is now being carried out (23).

The major disadvantage of this system is the method of radiation detection. Barrett reports that "although it has the virtues of high resolution and extreme 28 simplicity, X-ray film is obviously not the ultimate detector. It does not have the sensitivity for dynamic

function studies and does not have an energy discrimination capability which is important for rejecting scattered

radiation" (24).

2.2.2 Digital Reconstruction of Holograms and Shadowgrams

Although some research has been carried out in the

area of computer generated holograms for optical reconstruc

tion, very little work has been done on the digital recon

struction of holograms generated either by interference or

by shadowcasting. The number of resolution elements in

most holograms is so large that the computer time required

for digital reconstruction is excessive, and therefore

optical reconstruction is more practical and inexpensive.

In addition, the various tomographic planes of the optically

reconstructed image can be brought into view by simply

moving the screen on which the image is formed. On the

other hand, a separate calculation beginning with the basic

hologram data is required with digital reconstruction to

determine the image of each tomographic plane.

Goodman and Lawrence (25J demonstrated that a digital

image formed from an electronically scanned hologram is

comparable in quality with the optical image of the same

hologram. Aoki has compared optical and digital 29 reconstructions of both microwave (26) and sound-wave holograms (27) and shown them to be of similar quality.

The computer reconstruction of Fresnel zone plate shadow grams has received published consideration only by

Macdonald's group (28) in their work with the multiwire proportional chamber: in determining the signal-to-noise ratio attainable with a zone plate system, digital recon struction was chosen to eliminate the various noise sources introduced by optical processing. CHAPTER 3

RELEVANT THEORY

This chapter presents theoretical, material relevant to this dissertation but not developed by the author.

The basic optical theory which provides an understanding of the imaging ability of shadowgrams is covered first.

Then the theory of gamma ray imaging by shadowgrams is presented. Selected topics from the theory of digital signal processing are next discussed; these topics contribute to an understanding of the computer codes developed in the course of this investigation. Finally, relevant aspects of image enhancement theory are outlined.

3.1 Optical Theory of Shadowgrams

An overview of basic Fresnel zone plate theory and

its relationship to holography is presented here. Only

those features of the theory important in later chapters are mentioned. A more complete discussion, with refer ences, occurs in Appendix A. This appendix not only explains the theory but also gives a complete historical survey of the literature which contributed to the develop ment of the theory.

30 The Fresnel zone plate consists of concentric annular zones of equal area, alternately opaque and trans parent to an advancing wavefront, as shown in Figure A.3.

Denoting the first zone radius by R ^ , the radius of the n-th zone is given by

Rn * R1 /5P ' (3"X)

The width of the zones decreases with radius and is given by

Arn “ Rj ( /TT - / n-1 ) . (3-2)

A radiant wavefront encountering a zone plate behaves

in many ways as if it had encountered a lens, but the

effects are caused by diffraction rather than refraction.

A single zone plate can both converge and diverge incident

radiation, and multiple real and virtual focal points

exist for a single zone plate. The zone plate "lens" has

extreme chromatic aberration. The principle focal lengths,

real and virtual, of a zone plate for radiation of wave

length X are given by 32

Image formation by holography is also a diffraction process. The hologram of a point source is the pattern formed by the interference of a plane wave with a point scatterer of radiation. This pattern consists of concen tric annular light and dark zones having radii also deter mined by Equation 3-1. However, instead of zones of well-defined width, the pattern has a sinusoidal intensity variation along any radius. This pattern has the ability both to converge and diverge radiation in a lenslike manner. But because the zones do not have sharply defined edges, the pattern does not focus incident radi ation to focal points other than the two principal foci.

These focal lengths are also given by Equation 3-3.

Any extended object may be considered a distribution of point scatterers of radiation, and a hologram of such an object can be considered a superposition of overlapping

and interfering zone plate patterns. Radiation incident

on a hologram is diffracted by overlapping zone plate

patterns to form both a real image and a virtual image of

the original object.

Shadowgrams, on the other hand, are formed by a super

position of overlapping, but not interfering, zone plate

patterns. A shadowgram of a small object, consisting of

few equivalent point sources of radiation, is a pattern 33 very similar to a hologram of that object. However, for a large object, the shadowgram consists of many over lapping zone plate patterns, and the noninterference of these patterns can result in a pattern quite different from a hologram of the object.

3.2 Gamma Ray Imaging by Shadowgrams

Important features of the theory of gamma ray imaging by shadowgrams are presented here. A more complete discussion, with references, appears in Appendix B.

Most of the theory of gamma ray imaging of shadow grams has been developed by H. H. Barrett and his colleagues,

His approach to imaging theory is to use the mathematical

framework of communications theory. By finding the trans

fer function of a zone plate for the spatial frequencies contained in the source distribution, Barrett shows that the source spatial frequency imaged best by the system

is given by

so Rw f - -?■ -Jl . (3-4) S1 Rj

Barrett also relates the signal-to-noise ratio for a

zone plate of diameter DN to that for a pinhole aperture 34 of diameter Dp. For imaging a point source the relation- • ship is

CSNR>zp - WIT CSNRJp . (3-5)

Because D^ is much greater than Dp, for equal exposure times clearly the zone plate will have a much better signal-to-noise ratio. However, this is true only for imaging a point source. For imaging a small object having an area of M resolution elements, the signal-to-noise

- 1/2 ratio is reduced by a factor of M ' from its value for a single point. This is because gamma rays from all resolution elements contribute to the noise but only those

from one resolution element contribute to the signal for that resolution element.

3.3 Digital Signal Processing Theory

A computer code was developed in the course of this

study for digital reconstruction of shadowgrams. Optical

reconstruction involves converting a continuous two-

dimensional shadowgram intensity distribution into a

continuous two-dimensional image intensity distribution.

It is shown in Appendix D that the image is a two-dimensional 35

Fourier transform of the shadowgram and hence that the shadowgram may be considered a map of the spatial fre quency content of the image. Digital reconstruction requires that the shadowgram be reduced to a finite number of input data values and that the image be reduced to a finite number of output data values. The theory of digital signal processing includes a theory for carrying out this reduction without loss of information. An outline of the theory is presented here in the one-dimensional time and frequency domains for clarity. The results are easily extended to the two-dimensional spatial and spatial frequency domains.

Consider a continuous function of time, h(t) , that is sampled for purposes of digital processing. If the period between samples is T, then after sampling one retains only the information contained in the time function, hR (t), given by

00 6 (t - nT) ne-oo (3-6)

00 = h(nT) 6(t - nt) , 36 where n refers to the sample number and 6 is the delta function. Let the sampling function be given by

CO d(t) » £ 6(t - nT) . (3-7)

ne .00

Then

hn (t) » h(t)d(t) . (3-8)

Let a capital letter denote a Fourier transform to the frequency domain, f. The well-known convolution theorem then leads to the result

Hn (f) = H(f)*D(f) , (3-9)

where an asterisk denotes convolution. D(f) is given by

(29) to be

D(f) - ^ Y j {(f ■ f) (3‘10) n«-»

Hence Hn (f) is a function which repeats the function H(f) at intervals of period ^ in the frequency domain. H(f) contains complete information about the original function h(t). Therefore Hn (f) also contains this information 37 provided the repetitions of H(f) are spaced far enough apart so they do not overlap. Overlapping introduces into Hn (f) false information about the frequency content of H(f). This is called aliasing.

Figure 3-1 illustrates these concepts. Figure 3-1(a) shows h(t) and its Fourier transform which is zero for any frequency value of magnitude greater than f , the cutoff frequency. Figure 3-1 (b) shows d(t) and its

Fourier transform. In Figure 3-l(c), Hn (f) is shown to consist of periodic repetitions of H(f) provided the bandwidth of H(f) is less than the period of Hn (f) , that is, provided fc < ^r.

Now H(f) can be recovered by multiplying Hn (f) by a unit rectangular function, Q(f), given by

1 -L < f < 1 2T — — 2T

Q(f) - (3-11)

as shown in Figure 3-1(d). Then

H(f) » Hn (f)Q(f) (3-12) as shown in Figure 3-1(e). Application of the convolution theorem then yields h(t) H(f) 38

(a)

1 * i >

-3T -2T 21 3T I7T 2/T y t )

-3T -2T -T 31 Q(fJ

-l------u IT h (t)*q(t)

Figure 3.1 The recovery of a sampled waveform. Figures (a) through (e) illustrate relevant functions and their Fourier transforms. 39

jh(nT) Y, 6(t ‘ »T) IqCt) L n=-°°

n»-t»

00 sinn ( £ - n ) E Mm) - u - >-T , (3 -1 3 ) B *00 * ( r - ")

where q(t) is given by (30J. Hence the original continuous function can be fully recovered from samples spaced at interval T, provided ^ 2fc.

For a two-dimensional spatial function h(x,y) of extent T and T , the corresponding result is x y

« 00 simr(~- - n)sinit^^- - m) h(x,y) m Y Y h(nT ,mT ) ---- j— 5----- r - j — *— r— , y " (rr - ») ( f • m ) a y

(3-14) provided 2fc and ^~ il 2fc » where fc is the maximum x y spatial frequency content of h(x,y). This result is 40 applied in the theory of digital reconstruction presented in Appendix D to finding the best spatial interval between sample points in the image plane.

3.4 Image Enchancement Theory

An imaging system maps a set of input functions into a set of output functions. These functions can be real valued, dealing with intensities, or complex valued, deal

ing with field amplitudes. These are usually functions

of the two-dimensional independent variable, space.

An imaging system can be described by a mathematical

operator, M, which operates on input functions to produce

output functions. Let the system output be denoted by

v(Xi,yi) and the input by u(x2 ,y2). Then the relationship

between them is

vCxj.yi) = M {u(x2 ,y2)} . (3-15)

Almost all image enhancement studies have dealt with

systems which are linear. A system is linear if the

superposition principle holds for all input functions,

that is,

M {aUj + bu2> = aM{uj) + bM{u2) , (3-16) 41 where a and b are complex constants. The output of a

linear system can be expressed as a superposition of

outputs from weighted impulse responses. An impulse

response is the response of a system to a point source

(delta function) input. For a system described by opera

tor M, the impulse response is

(3-17)

It is easily shown (31^) that input and output are then

related by

00 uU.i-OhCx. ,y ;5,n)dedn

Most imaging systems are linear if both light amplitude

and phase are considered.

Some linear systems are also spatially invariant.

For such systems the functional form of the output does

not change as an input point source moves about in the

input field; only the position of the output changes. Then

the impulse response depends only on the displacements

(Xj - £) and (y^ - n). Hence the output may be expressed

as a convolution integral, as follows, 42

v(xl»yi) m J J u(C,n)h(x1 - Z ty1 - n)d^dn . (3-19) -00

Denoting the convolution by an asterisk, we obtain

v ** u*h . (3-20)

In terms of Fourier transforms, this relationship becomes a simple multiplication; that is,

V ■» UH •» HU (3-21)

The function, H, is called the transfer function of the system. The transfer function can be extremely useful for image enhancement. But its use is limited to linear, invariant systems.

We consider here only that use of the transfer func tion which proved of some use in this study. Suppose an imaging system produces a known output, v, from an unknown input, u, which must be determined. This can be accomplished by first determining the transfer function of the system.

A known point source, up , is imaged by the system and the corresponding output, v , is determined. The Fourier r transform of Vp, denoted by Vp, can then bo determined either optically (32) or digitally (33) . The Fourier 43 transform of un is a constant because u is a delta P P function. Then to within a multiplicative constant,

Equation 3*21 reduces to

Vp - H , (3-22)

and the transfer function of the system is determined.

The output, v, from the unknown input can be Fourier trans formed, giveing V. Then we obtain the Fourier transform of input, U, given by

U - f . (3-23) P

Finally, the unknown input, u, is found by taking the inverse Fourier transform of U.

Studies of space variant systems have recently begun to appear in the literature (3£). The approach has been to try to add another parameter to the transfer function to account for the variance in cases where the variance is known to have a symmetry, such as radial symmetry.

Although some theories have been advanced, it appears that no practical results have yet been obtained. Nonlinear systems remain intractable. CHAPTER 4

OUTLINE OF SYSTEM DESIGN

Most important in this chapter is the methodology for determining the physical parameters of a Fresnel zone plate imaging system. Because the choice of these parameters depends on the detector characteristics, the detection system of the high purity germanium gamma camera is discussed at the outset. A zone plate system suitable for this camera is then designed. Alternative data processing systems are considered next, and finally the design of the overall imaging system is outlined.

4.1 The Germanium Detector

The imaging system described in this chapter is a

Fresnel zone plate system designed for use with a high purity germanium gamma camera. The features of the camera design most relevant to the zone plate design are presented here. A detailed description of the camera is presented in Appendix C.

The detection system of the camera consists of an array of orthogonal strip detectors, each of which is

44 45 cut from a single crystal of high purity germanium (see

Figure C.l). The spatial resolution of the camera depends on the spacing of the strips and the electronics.

A strip spacing of 1 mm could be achieved at present; but the electronics, as currently designed, would not allow the resolution of individual strips. Improvements

in electronic resolution are possible if the energy resolution of the system is allowed to degrade or if the cost of the electronics is allowed to increase signifi cantly. The design considered here has an electronics resolution of 2.4 mm.

The camera is designed to have 16 individual detectors

in a four-by-four square array. The side of the square will be 15.3 cm in length (see Figure C.2).

4.2 The Fresnel Zone Plate System

Several important parameters must be determined in

setting up a zone plate imaging system. This section

specifies these parameters, determines relations among

them, outlines a method for choosing these parameters

and applies this method to choosing the parameters for

the germanium camera described in the previous section.

4,2.1 Zone Plate Parameters

The choice of all zone plate system parameters

depends on the characteristics of the detector in the 46 system. It is assumed here that the detector is a square ' of side made up of resolution elements of size Ax^ as shown in Figure 4.1. The number of resolution elements per side is N^.

After the shadowgram is received on the detector and digitally processed, an image is obtained which shows a

square area of the object plane. The side of this square

is defined here to be the field of view, T, of the system,

and the dimension of the resolution element in the object plane is At, as illustrated in Figure 4.2.

A Fresnel zone plate is placed between the object plane and the detector as shown in Figure 4.3. It is

assumed for the present that the detector and zone plate

thicknesses and the object depth of field are small

compared to s^, the distance from the object plane to the

zone plate, and S2 » the zone plate to detector distance.

A very useful parameter is the ratio of these distances,

given by

The zone plate is circular having N zones. Denoting the

radius of the n-th zone by Rn , then Rjj is the radius of

the entire zone plate, that is, of the outermost trans

parent zone. The diameter of the zone plate is denoted Figure 4.1 A square detector of size Dj divided into resolu tion elements of size Axi. There are Ni resolution elements per side. Here Nj * 5. 48

| At ---1

Figure 4.2 Zone plate system field of view. The area is a square of side T, containing Nj by Nj square resolution elements of size, At. Here » 5. 49

object zone plate detector plane plane plane

Figure 4. The object-to-zone plate distance, si, and the zone plate-to-detector distance, s7.

4 so by D^. The width of the n-th zone is Arn and hence the width of the outermost zone is Ar^. The thickness of the zone plate is t2»

The parameters of importance are those summarized in Table 4-1.

Table 4-1. Parameters of the Imaging System

Symbol Definition number of resolution elements per side Ni length of square detector side Di Ax^ length of square resolution element side

T field of view

At object plane resolution

N number of zones in zone plate

diameter of zone plate d n width of outermost zone firN object-to-zone plate distance S1 zone plate-to-detector distance s2 a ratio of S2 to s^

thickness of zone plate

4.2.2 Relations Among Parameters

Consider first the relationship of the detector parameters. Because contains a number of resolution 51 elements of size Ax^, the number of resolution elements per side is

N 1 ' 25^ <4'2>

Consider next the relationship of the object plane parameters, T and At. The detector data is an by matrix of gamma counts. According to sampling theory

(Chapter 3, section 3.3.1), the maximum image information is contained in an by matrix of image data. Then the size of the object plane resolution element is

At » jj— . (4-3)

The zone plate parameters have the following relation

ships. Combining Equations 3-1 and 3-2, and using an approximation which is valid for N greater than 4, we obtain

Dn « 4NArN . (4-4)

The thickness of the zone plate is dictated by the energy

of the incident gamma radiation. A lead zone plate used with 140 keV gamma rays should have a thickness of at

least 1 mm. 52

The object plane parameters are related to the detector parameters as follows. In order to resolve two distinct point sources, the outermost zones of their zone plate shadows must not overlap, but must be separated by at least one detector resolution element. Hence the centers of the two zone plate shadows on the detector must also be separated by Axj. The similar triangles of

Figure 4.9 illustrate that to obtain a shadow center separation of Ax1# the two object points must be separated by At, wheTe

Ax 1 At *» (4-5) a and a is defined in Equation 4-1. Then by combining equations 4-2, 4-3 and 4-5 we obtain an expression for

the field of view in terms of the detector size, that is,

(4-6)

Having related object plane parameters to detector parameters, we consider next the relation of zone plate

parameters to detector parameters. To obtain maximum

system resolution, the width of the outermost zone of

the zone plate, Ar^, should be such that a point source in

the object plane fully irradiates exactly one resolution 1 1 At Ax, T T

Figure 4. 4 The relationship between object plane resolution At and detector plane resolution Ax^.

c/i u 54 element in the detector plane, as shown in Figure 4.5.

If Ar^ is smaller, the outermost zones of the shadow will not be resolved by the detector; these unresolved zones will not contribute to image improvement, but rather will constitute noise which would degrade the image. If is chosen too large, the resolving ability of the system will not be fully realized. From the similar triangles of Figure 4.5 and Equation 4-1, we obtain

Ax, ArN ■ . (4-7)

In order for the detector to receive the complete shadow of the outermost zone cast by a source at the

center of the field of view,9 • the diameter of this shadow Dg must be less than or equal to D^. But the diameter of the shadow is equal to B^Cl + a), and hence

d n ± r r r ( 4 - 8 )

An alternate expression for DN can be found by substituting

Equation 4-7 into Equation 4-4, giving

N D1 dn * 4 RJ inr-sr • C4-®) 55

1 Axj Ar, T

object zone plate detector plane plane plane

Figure 4.5 Radiation from a point source in the object plane passing through the outer most zone. 56

Comparison of Equations 4-8 and 4-9 yields the result that

N 1 N<-^ . (4-10)

For detectors having a small value for it is desirable to choose N to be the largest integer less than or equal to Nj/4.

The relationships developed in this section arc summarized by the diagram in Figure 4.6. The diagram assumes that D^, Ax^ and a are known first; the arrows and associated equations show how the other parameters arc determined.

4.2.3 Method of Choosing Parameters

Several practical considerations affect the order of choice of the parameters discussed above. In Figure

4.6, it is assumed that a is known before the parameters

Ar^ and At are determined. In practice, constraints on these parameters can limit the choice of a.

The choice of a affects the magnification of the system. If a » 1 (that is, s^ * s2^ » s*ze °f

field of view equals the size of the detector, as in the case of the multihole collimator. However, as the zone plate is moved closer to the detector and a decreases, the < Di D1 N — 1 * a T = 4 D, T

Dn = 4NArN N < h T = NxAt ■ 4 D 1 ~ N j AX j

V ArN < - Ax. ■> At Ax. Ax. At = ArN = 1 + a

zone plate detector object plane parameters parameters parameters

Figure 4.6 Relationships among the zone plate, detector and object plane parameters. C/1 ^■4 system is able to image a field of view larger than the detector. The penalty for this ability is a loss in system resolution. In the limit that a equals zero and the zone plate is pressed against the detector face, the image of the object is reduced to a single point. Conversely as the zone plate is moved toward the object and a increases, the ability of the system to resolve detail in the object improves. But at the same time the field of view decreases. It is important to keep in mind that as the zone plate is moved between the object and detector its dimensions must be changed in accordance with Equations

4-7 and 4-8, if optimum imaging ability is to be realized at each new zone plate position. The inability to construct a zone plate of extremely small dimensions is a limitation on the resolving power of the system.

Another limitation, however, is imposed by the thick ness of the zone plate. Gamma rays passing through a zone at an angle with the axis of the system are partially attenuated because of the finite thickness of the plate.

A minimum value for Sj must be maintained to keep this attenuation from becoming severe. A consideration of this problem for the case a > 1 is presented here. When a is greater than one the attenuation is most severe for gamma rays from the center of the field of view passing through the outermost zone. Figure 4.7 illustrates (a)

(b)

Figure 4.7 Attenuation of gamma rays passing through the outermost zone, (a) gamma rays from a point source at the center of the field of view are emitted toward the outermost zone; (b) the angle 0 is such that half the gamma rays enter ing the zone emerge from the other side. 60 the problem and assumes for simplicity that the plate material is completely opaque to radiation. In Figure

4.7a radiation from a point source on the axis of the system travels toward the outermost zone in a direction that makes an angle with the axis having a tangent equal to D^/Zs^. Figure 4.7b shows this radiation passing through the outermost zone. If we impose the constraint that at least half of the radiation incident at the entrance to the zone must leave the zone without passing through opaque material, then the tangent of the maximum allowable angle of incidence is Ar^/2tz. Therefore, we require that

Dm Ar„

! C 4 ' U ) and hence a minimum value of s^ is established. Substi tuting Equation 4-4 leads to the result that

s. • - 4Nt_ . (4-12) l,min z '

When a is less than one, sources at the edge of the field of view are attenuated most severely; by an argument similar to the one above, it is readily shown that

4Nt_ s a _____ 2 l.min a (4-13) Since the zone plate system is capable of magnifica

tion, it is worth inquiring about the best resolution obtainable in the center of the object field of view by a camera with given parameters. According to Equation

4-5, the object field resolution improves with increasing a. Then if s^ is fixed at Sj min, by increasing s2

resolution can be improved. But increasing s2 reduces

the gamma ray collection efficiency and increases the

importance of small angle scattering in the zone plate.

More important is that an increase in s2 requires a

decrease in zone plate dimensions in accordance with Equa

tion 4-8. The mechanical inability to construct a zone

plate of small precise dimensions while maintaining a

minimum thickness imposes the basic limitation on system

resolution. By substituting Equation 4-7 into Equation

4-S and rearranging, we obtain

At-(l*i)ArN , (4-14)

which shows that as a increases to infinity the best system

resolution obtainable is equal to the width of the outer

most zone. However, for a system in which the camera

resolution AXj is greater than the minimum attainable

value of Ar^, substituting Equation 4-7 into Equation 4-5

and eliminating o results in a more severe restriction on

system resolution, namely, 62

If, for example, Ax^ is 2 mm and the minimum value of

ArN is 1 mm, then the best object resolution is 2 mm.

In choosing the parameters of a zone plate system, one must decide whether to design for the best object resolution or to enlarge the field of view in order to form a complete image of a large object. In most cases of interest in nuclear medicine in which a germanium gamma camera would be used, the decision would be to optimize system resolution. Table 4-2 presents the system parame ters, their order of choice and equations for determining these parameters for the case where system resolution is to be optimized. The table assumes that D^, Ax^, Ar^ and t are given.

The table also assumes that s^ should be set equal to its minimum value in order to achieve maximum gamma ray collection efficiency, emflX. Expressed in terms of

the given parameters of Table 4-2, the efficiency is

approximately

em a x ” 7 ( * t ^ ) (4-16)

If the source activity in the field of view is q gamma

emissions per second, the camera must have a count rate

capability C.R., at least as great as 63

For Optimum System Resolution

Given: D ^ t AXj * ArN» tz

Parameter Equation Equation No.

N1 = D i 4-2 N i AXj

Ax, a a « 4-7 1

Ax, i At At » a 4-5

T T = N j A t 4-3

N i N N 4 x 4-10

Dn Dn “ 4NArN 4-4

Sj Sj ■ 4Nt2 4-12

s2 s2 - 0.5J 4-1 64

C.R. = qemax (4-17)

If the count rate capability of the camera is'exceeded, then either q can be reduced or can be chosen greater than the minimum allowable value. Although the second alternative reduces the efficiency, the attenuation of gamma rays passing at an angle through the zone plate is also reduced and system imaging capability is enhanced.

If the system parameters are chosen so as to obtain a specified field of view, the order of choosing the parameters is that given in Table 4-3. When T is specified,

At will not necessarily be optimum and Ar^ may be greater than the minimum width one is capable of constructing.

4.2.4 Parameters for the Germanium Camera

Let us next consider the choice of system parameters for the germanium camera described in Section 4.1. In this case is 15.3 cm, the FWHM electronic spatial resolution is 2.4 mm, and so N1 is 64 resolution elements per side. The radioisotope most often used in nuclear medicine is technctium-99m with a gamma ray energy of

140 keV; if the system is optimized for use with this

isotope and if the zone plate thickness is chosen to attenuate 95% of the incident radiation, then t£ is 1 mm.

It is extremely difficult to machine zones in lead which 65

Table 4-3. System Parameters and Equations

for a Specified Field of View

Given: Dj, AxJt t2, T

Parameter Equation Equation No.

D1 N1 N1 ■ -SJ- 4 ' 2

At t “ 4-3

X1 4-5

ArN 4rH ■ 4-7

N1 N N * 4-10

Dn Dn » 4NArN 4-4

Sj Sj ■ 4Ntz 4-12

s2 s2 “ aSj 4-1 66 are spaced closer than 1 mm; and in addition, if the width of the opaque zones is less than the zone plate thickness, gamma rays making an angle with the system axis will penetrate these supposedly opaque zones. Hence a minimum practical value for Ar^ is also 1 mm.

Table 4-4 gives the parameters of a zone plate-germanium camera system for three cases. In case 1 the parameters are chosen to optimize system resolution according to the method of Table 4-2. The minimum practical value of

Ar^ is chosen, the magnification parameter a is maximized, and a spatial resolution of 1.7 mm for the object plane is predicted. Notice that the system is able to resolve objects smaller than the spatial resolution of the camera alone. However, the field of view is only 11 cm, while the camera detector is a square of side 15 cm.

In cases 2 and 3 the parameters are chosen so as to achieve a specified field of view, according to the method of Table 4-2. In case 2, the field of view is chosen equal to the detector dimensions; a is unity, and the area imaged is the same as would result from using a multihole collimator. In this case the system resolution is equal to the detector spatial resolution. In case 3 the field of view is chosen larger than the detector dimensions.

More specifically, the area of the square field of view is chosen equal to the circular usable area of the standard 67

Table 4-4. Zone Plate-Germanium Camera

System Parameters

Parameter Case 1 Case 2 Case 3

Dj (mm) 153 153 153

AXj (mm) 2.4 2.4 2.4

Ni 64 64 64 tz (mm) 1 1 1

At (mm) 1.7 2.4 3.7

T (mm) 110 153 236

ArN (mm) 1 1.2 1.5

N 16 16 16

Dn (mm) 64 77 93 a 1.4 1 0.65

Sj (mm) 64 64 100

S2 (mm) 90 64 65 68 scintillation camera in Figure C.2. The parameters were chosen as in case 2 except that s^ was determined according to equation 4-13 instead of equation 4-12. Although a

larger area is now imaged, the system resolution is degraded to 3.7 mm. Nevertheless, it is considerably better than the resolution obtainable with a scintillation

camera using a multiholc collimator.

4.2.5 Mechanical Components

A supporting structure is needed to position the

zone plate with respect to the camera head, as shown in

Figure 4.8. The center of the zone plate pattern must

be aligned with the center of the camera detector.

Hoivever, small deviations from perfect alignment will

not affect image quality. Grooves in the supporting

structure allow precise placement and replacement of

removable zone plates. The figure shows three such

grooves to provide for three zone plates, one designed

for each of the three cases of Table 4-4.

Bach plate consists of 1 mm thickness of lead

bonded to 1 mm thickness of aluminum to provide structural

support. The transparent zones are formed by machining

away lead, leaving the aluminum which is relatively

transparent to gamma radiation. camera supporting zone camera head frame structure plate window

Figure 4.8 Mechanical components of the zone plate system. Not drawn to scale. The zone plate is in the second of three grooves in the supporting structure. Figure 4.8 also shows a lead frame aligned with the' zone plate and the detector. The frame is to be placed against the patient's body and simply consists of a lead sheet with a square window equal in size to the field of view. Three frames must be provided to match the three fields of view obtained with each of the three zone plates. The frame is important for coded aperture imaging and extremely so when computer reconstruction is used with a germanium gamma camera. The small value of this camera necessitates choosing N as large as possible, which requires that be chosen as large as possible.

Then the shadow of a centrally positioned point source fills the detector face. Under these circumstances a point source at the edge of the field of view casts a circular shadow centered on the detector edge, so that half the shadow falls on the detector and half off. For sources outside the field of view, less than a half shadow falls on the detector. Since these sources will not appear in the reconstructed image, the counts they contribute to the detector constitute only noise. Hence it is essential that these sources should be screened off from the detector by a frame. 71

4.3 Image Processing and Display

Systems considered for digital data processing include a large-scale integrated (LSI) semiconductor chip, a minicomputer and a large computer system. The present design calls for processing by a large computer system.

In the future, LSI chips may be used extensively for data processing. The most important algorithm in the data processing routine to be discussed in Chapter 5 is the fast Fourier transform (FFT) algorithm. FFT chips are currently under development (35J which can process data faster than a large minicomputer but which cost much less: a chip measuring 0.27 in. by 0.35 in. can perform a 12-bit, 1024-point FFT in only 2.5 milliseconds. These chips are not available at the present time, however, and no chips are available to perform the image enhancement algorithms. Although such chips are not likely to be available in the near future, designers of an imaging system such as outlined here should keep aware of advances in LSI technology.

Minicomputers purchased solely for use with a gamma camera have several disadvantages compared to large com puter systems used in the time-sharing mode. A sophisti cated minicomputer system could add as much as $200,000 to the cost of the complete imaging system. Apart from 72

the cost of the central processing unit, a minimum of

10,000 eight-bit bytes of memory is needed to process the

data without image enhancement; this memory alone would

cost about $10,000. Considerable extra core space and memory would be required for image enhancement processing.

Also, extra personnel would be required in the nuclear

medicine laboratory for programming, operation, and

maintenance of the minicomputer system.

A large computer system, on the other hand, is

normally present in hospitals large enough to have a

nuclear medicine laboratory and is available on a time

sharing basis. The data processing to be described in

Chapters S and 6 can be carried out on the IBM 370 at The

Ohio State University Hospital for about $10 per patient.

Also, the availability of a large computer system permits

quantitative analysis of the image as described by

Deutchman (36).

A data acquisiton system for putting gamma camera

raw data onto magnetic tape for off-line digital processing

has been designed and built in our laboratories (37).

The raw output of a gamma camera is x and y positional

voltage pulses which can be applied to the deflection

plates of an oscilloscope to cause a positionally correct

light flash; the integration of these light flashes by 73

Polaroid camera film produces the final image or the coded image, depending on the choice of aperture. The data acquisition system digitizes the x and y pulses and records this positional information on computer com patible tape. The tape serves both as the data storage medium and the computer input medium. It can be hand carried to the hospital computer center, read in, and analyzed by software programs kept on tape at the computer center. The positional data is recorded in real time in the same order as the information is received. Hence dynamic function studies are easily performed: for example, if data is collected for 30 seconds, each

5-second segment of the tape can be processed independently, providing six sequential time-elapsed images.

Use of this system with a scintillation camera at

University Hospital has shown that results are available

in about two hours. The system hardware can be built for under $11,000. Although the present system accepts only

20,000 counts per second, one accepting 100,000 counts per second can be built for the same cost (38).

Display of the final image can be on film if the

hospital data processing system has an adequate computer-

interfaced CRT display. Our own University Hospital has not, but image printout on a standard line printer has

proved satisfactory for diagnostic purposes. 74

4.4 The Complete System

A block diagram of the complete imaging system is presented in Figure 4.9. The radioisotope distribution to be imaged is framed in the field of view. Gamma radiation passing through the on-axis Fresnel zone plate creates a coded shadowgram pattern on the camera detector.

The germanium camera translates this pattern into a

series of voltage pulses. The information contained

in these pulses is recorded on magnetic tape by the data

acquisition system. The tape is processed at a computer center. The computer output is an image of the radio

isotope distribution. On-Axis Germanium Radioisotope Fresnel Gamma Distribution Zone Plate Camera

Image of Computer Data Radioisotope Acquisition Processing Distribution System

Figure 4.9 Block diagram of the complete imaging system. CHAPTER 5

DIGITAL RECONSTRUCTION OF SHADOWGRAMS

The Fresnel zone plate, germanium, gamma camera imaging system described in the previous chapter promises a significant potential improvement in diagnostic ability in nuclear medicine. The present chapter assesses the imaging capability of this system by digital simulation.

The physical model on which the digital reconstruction algorithm is based is outlined first. This model was used as a basis for developing a computer code for digital reconstruction. The effectiveness of the code was established using digitized data derived from optical shadowgrams. For the majority of the studies, however, computer generated shadowgrams proved more useful. The essential content of this chapter is the discussion of the results obtained from the simulation of shadowgrams and their digital reconstruction.

S.l The Reconstruction Model

A description of the optical model on which the theory of digital reconstruction is based is presented

76 77 here. The detailed derivation of the theory is given

in Appendix D.

The shadowgram pattern which falls on the detector

face is a continuous intensity distribution which is

sampled by the camera. A camera with by resolution

elements results in an by matrix of sampled values,

each value being the number of gamma ray counts per

resolution element. When the system parameters are chosen

according to the principles of Chapter 4, the highest

spatial frequency content in the continuous intensity

distribution is exactly equal to half the sampling fre

quency. Therefore, the requirements of sampling theory

for acquiring complete information about the continuous

shadowgram intensity distribution are satisfied.

The reconstruction model assumes that a shadowgram

transparency is illuminated with a planar beam of coherent

radiation. The shadowgram transparency is considered

to transmit radiation only through an Nj by Nj array

of equally spaced points in which a point is associated

with each camera resolution element. In effect, the

transparency becomes an array of point sources of coherent

radiation. The intensity of each point source is propor

tional to the number of counts received in the associated

resolution element. Immediately following the array of

point sources is a converging lens which focusses onto a screen the diffraction pattern resulting from the mutual interference of the many coherent point sources. The diffraction pattern consists of a radiation intensity distribution, determined by adding vectorially the radia tion amplitudes received from each of the point sources and taking the square modulus of the resultant amplitude.

The continuous radiation intensity distribution on the

screen is the true (as compared to the conjugate) image.

The image plane in focus is changed in a real

optical system by either moving the screen or moving the

lens. Mathematically each of these alternatives is

equivalent to changing the focal length of the lens. The

focal length parameter is variable in the digital recon

struction algorithm.

In a digital reconstruction algorithm one must deter

mine how to sample the continuous radiation distribution

so as to obtain the complete image with maximum resolution.

More specifically, the size of the area in the image plane

to be sampled and the spacing of the samples must be

determined. When the sample spacing is chosen optimally,

a continuous image distribution can be generated by inter

polating between sampled points. The continuous image is

then complete in the sense that the maximum information

has been extracted from the shadowgram data. Samples 79 spaced too far apart result in loss of information; samples spaced too closely result in excessive computer time, yet provide no additional information beyond that obtainable by interpolating between optimally sampled points.

A set of dimensionless variables for characterizing the optimal spacings in the shadowgram and image planes has been defined, relations between these variables have been derived, and a set of reconstruction equations incorporating these variables has been developed. These results are presented in Appendix D, and summarized here.

Maximum information from an by matrix of shadowgram data is contained- in an by matrix of image data.

The size of the image is inversely proportional to the size of the shadowgram resolution elements. The size of the image resolution element is inversely proportional to the size of the shadowgram. In both cases, the propor tionality constant depends only on the.shadowgram plane to image plane distance.

5.2 Reconstruction of Optical Shadowgrams

Initial testing of the reconstruction algorithm was carried out by reconstructing images of point sources.

Shadowgram data was generated by accurately drawing overlapping zone plate patterns on fine mesh graph paper, ' dividing the graph paper into a 32 by 32 grid pattern, and counting the number of times each small square within a grid element was within a darkened zone. The resulting

32 by 32 matrix of data was used as input for the recon struction computer code. The ability of the code to reconstruct images of several points in various parts of the field of view and in different tomographic planes was established. Once the code was tested in this way, the ability of the code to reconstruct images of extended sources had to be established. To accomplish this, an optical apparatus was constructed to record on film optical shadowgrams of extended sources.

5.2.1 Experimental Apparatus and Procedure

Optical shadowgrams were formed using the apparatus

shown in Figure 5.1. An optical bench was constructed

from a section of an aluminum duct. A 75-watt floodlamp was mounted behind a ground glass plate. Light passing

through this plate fell on a second ground glass plate

masked by heavy black construction paper. The source

distribution pattern was cut from the construction paper.

The purpose of the second ground glass plate is to diffuse

the light emerging from the source pattern. Ideally,

light from any point in the source pattern should emerge optical large 75-watt Fresnel zone bench box floodlamp plate

ground sheet glass source power small film plate pattern screen cord box holder

timer

Figure 5.1 Experimental apparatus for casting optical shadowgrams. 82

with isotropic intensity. In practice, the intensity

is greatest for light emerging along the axis of the

optical bench. Use of the first ground glass plate close

to the floodlamp was found to reduce the nonisotropy.

The floodlamp and glass plates were entirely enclosed in

a small box with a window having an area equal to the

intended field of view of the source.

To use the apparatus, light from the source pattern

was allowed to pass through a Fresnel zone plate pattern.

This pattern was a photoreduccd transparency of a photo

graph of a large zone plate pattern, constructed with

black India ink on transparent mylar and mounted on

white cardboard. The transparency was a 32-zoned plate

with the central zone opaque. Light passing through the

zone plate was allowed to fall on sheet film loaded into

a sheet film holder. A screen with a window was placed

immediately after the zone plate to insure that only the

light passing through the zone plate fell on the film.

.A movable mount was constructed for each item on the

optical bench. The entire apparatus was contained in a

large box painted black on the inside to avoid having

light from the source reflect from nearby metallic fixtures

in the laboratory onto the film.

The apparatus was used to simulate the shadowgram

intensity pattern which would be formed on a detector by 83 a source of gamma radiation and a lead zone plate; however, there were difficulties with the simulation. The noniso tropy of the source has already been mentioned. In addi tion, some light from the source was reflected by the large box to the zone plate. Some light passing through the zone plate was reflected by the box to the film, and some was also reflected by the black surface covering the opti cal bench between the zone plate and the film. Also, the transparent zones of the zone plate transparency caused some dispersion of the light passing through.

To expose a shadowgram, sheets of 4-inch by S-inch

Kodak film No. 6127 were loaded into lightproof film holders in a photographic darkroom, carried to the labora tory and mounted on the optical bench. A timer controlling the illumination time of the flodlamp was set. After the laboratory lights were turned off, the film holder cover was raised, and the timer was turned on, causing the floodlamp to expose the film. After the timer shut off the floodlamp, the film holder cover was lowered and the laboratory lights were turned on. The film was then ready for development in the darkroom. For each experi mental setup, four shadowgrams having four different exposure times were made.

In order to maintain uniform results from one experi ment to the next, a standard film developing procedure was 84 devised. The film was processed in a sheet film tank at

68°F. Processing began with Kodak developer HC-110,

dilution D, for 2 minutes 42 seconds, followed by 8 minutes

in a fixer. After washing for 30 minutes in water and

rinsing for 30 seconds in a nonwetting agent, the films

were dried for one hour. Of the four exposures, the

one having the greatest contrast was selected for scanning.

A microdensitometcr was used to scan the processed

film and provide numerical data on film exposure. A 64 mm

by 64 mm square area of each shadowgram was scanned; this

required 64 scanning runs. On each run a square 1 mm by

1 mm beam of light passed through the film as the film

traveled across the beam. The output of each run was a

graph of the film optical density versus distance. The

graph density axis was calibrated in terms of a standard

optical density scale; the distance axis was divided into

64 segments, each corresponding to one millimeter of film

traversed. The 64 scanning runs were parallel and spaced

1 mm apart. The resulting 64 graphs provided density data

for a 64 by 64 matrix of resolution elements. Film

density is proportional to the logarithm of film exposure,

and hence it is proportional to the logarithm of the number

of photons incident at each resolution element of the

film. Therefore, ignoring a proportionality constant, one can determine an input data matrix giving the number of photon counts per resolution element. In this way

the number of gamma counts per resolution element

recorded by a germanium gamma camera can be simulated.

Even allowing for the simulation difficulties mentioned

above, the results obtained by processing such data can

reasonably be expected to simulate results obtainable

from the system outlined in Chapter 4.

5.2.2 Results

Because it was known that the reconstruction code was capable of reconstructing the shadowgram of a point

source, the apparatus and procedure for forming optical

shadowgrams were tested by forming a shadowgram from

a small circular source and using the code for recon

struction. The source was a 1/8-inch diameter circle

at the center of a S-inch field of view. The first

trial involved a 32 by 32 scan, and successful

reconstruction of this data lead to a 64 by 64

scan of the same shadowgram. The parameters of the

system for the latter case are given in Table 5-1. The

system parameters were not chosen according to the optimum

method discussed in Chapter 4 because at the time of this

experiment this method had not been developed. In fact,

it was consideration of how to define the field of view

and how to select the area of the film to be scanned, for 86

Table 5-1. System Parameters for 1/8-Inch Diameter

Circular Source Shadowgram

Parameter Value

64 mm Di

AXj 1 mm

ArN 0.4 mm

64 mm N 1

a 0.5

At 2 mm

T 128 mm

N 32

48 mm d n

S1 406 mm

203 mm S2

4. Definitions and explanations of these parameters are given in Table 4-1 and the accompanying text. 87 larger extended sources and hence wide area shadowgrams, which prompted development of the method. The distances s^ and s2 were chosen large enough so that the effect of source nonisotropy was minimal. The thickness of the zone plate can be considered zero. The reconstructed image for this small circular source is shown in Figure 5.2.

Before the image itself is discussed, a brief explanation of the method of image display is required.

The line printer code used for image display is given in Table 5-2. It must be emphasized that, although an attempt was made to correlate roughly the appearance of the line printer image with the calculated image intensity, the data presented in this form can only be properly evaluated by using the code given in the table. The calculated image is a matrix containing 4,096 values.

In the studies to be discussed, these actual values were used to evaluate the success of the reconstruction, whereas the line printer display of the kind shown in Figure 5.2 was used primarily to provide an overview of the image data and direct attention to interesting regions of the matrix.

In Figure 5.2 the reconstructed small circle is present at the center of the image. The diameter of the source circle was 1/8 inch, or 3.2 mm, in a source field 88

•**•*# 0 "M* *

• • • • • • •

•• 9 9 •• 9 * t t • 9 . * !• 9 m 9 9 9 • • • • • 9* • • • • ♦ t • • • • •• • • *1 • • • 1* • • • 9 • • • • ■ • • . • « • • • • • • I* :• • ♦ • « - • • • • • • • • • • • |* •| m • •• • • • m • • • • •

Figure 5.2 Image of a point source from an optical shadowgram. 89

Table 5-2. Line Printer Code in Which A is the

Greatest Intensity Value in the Image

Intensity Value, I Line Printer Code Symbol

0.9 A < I < A # overlaid with * f

0.8 A < I < 0.9 A # overlaid with + #

0.7 A < I < 0.8 A # overlaid with : H

0.6 A < I < 0.7 A H overlaid with . H

0.5 A < I < 0.6 A H V

0.4 A < I <0.5 A A A

0.3 A < I < 0.4 A + +

• » 0.2 A < I < 0.3 A • •

0.1 A < I < 0.2 A ••

I <0.1 A blank for which parameters were chosen so that the resolution

At was 2 mm. In the image the circle appears as a single brightest point surrounded to a distance of one resolution

element by somewhat less bright points. These results

are consistent with the analysis presented in Chapter 4.

Some noise is present in the image. At a few isolated

points the noise intensity is over 30% of peak image inten

sity; however, at most points the noise is less than 10%

of this value. In view of the simulation difficulties

discussed above, this reconstruction was considered

acceptable.

The main conclusion drawn from this experiment is

that the apparatus and procedures for forming optical

shadowgrams are workable. Hence use of the same apparatus

and procedures for an extended source should result in

a practical shadowgram which can be used to test the

ability of the reconstruction code to reconstruct the image

of an extended source.

Following this experiment, an extended source having

the shape of the letter "B" was used to form an optical

shadowgram by the method described above. The parameters

of the imaging system are given in Table 5-3, and the

reconstructed image is presented in Figure 5.3. Of the

three horizontal arms of the "EM the lower two are close Table 5-3. System Parameters for Shadowgram

of the Letter "E"

+ Parameter Value

Di 64 mm

ixi 1 mm

0.7 mm ArN

N1 64

a 0.38

At 2.6 mm

T 168 mm

N 15

43.5 mm

443 mm S1

s2 164 mm

^Definitions and explanations of these parameters are given in Table 4-1 and the accompanying text. 92

#•«»* *####• • : : : : : : t: • •:♦♦♦♦♦♦::!. • ft t + . ft# » • <++♦«•++:: ft *:♦♦♦♦♦* • • • * JJ4+4S: :: :: •ft ft* * *ft ft • ft • • • • *ft ft* *f ft • ft •» • ft ft ft 1 • ft ft ft

• ft ft# ft# ft ft ft ft ft »t ♦♦$$*>♦ • :• :* . ♦: •:: • : :i * : : :. 1♦*#♦«>♦+::..

• ■»•••» •»

• »•••••» • • >s t• * • • •

Figure 5.3 Image of the letter "E" from an optical shadowgram. 93 to maximum intensity but the upper arm has only about half maximum intensity. Also, the vertical part of the letter is better defined along the lower half than along the upper half. Background intensity between the horizontal arms varies from 204 to 504 of maximum intensity. Back ground intensity in the immediate vicinity of the ME" is in the range 104 to 304. Background intensity drops to less than 104 away from the letter but increases again toward the edge of the reconstruction. This background increase is not uniform in all directions; in one corner the background intensity is close to 504 of maximum intensity.

Two important results can be drawn from this experi ment. First, the reconstruction code is capable of reconstructing an image from digitized data derived from a shadowgram of an extended source. Although the recon structed letter is not of uniform intensity and the edges of the letter are not sharply defined, a clearly recog nizable "EM appears in the reconstruction. Therefore, the physical model for digital reconstruction presented in Section 5.1 and the theory of digital reconstruction in Appendix D are demonstrated to be serviceable.

A second result is the verification of the parametric relations developed in Chapter 4 for determining the resolution obtainable by a zone plate imaging system. The' optical source was a letter "E" of overall height 30.8 mm and overall breadth 17.6 mm. The horizontal arms and the vertical part of the letter had a width of 4.4 mm. As shown in Table 5-3, given the chosen system parameters, the resolution element in the field of view was 2.6 mm.

Consequently, the letter had a height of 12 resolution elements, a breadth of 7 resolution elements, and was composed of segments having a width of 1.7 resolution elements. In Figure 5.3, each symbol corresponds to one resolution element of the reconstructed image. With the assumption that image points having an intensity less than

40% of maximum are background noise, the reconstructed image is 12 resolution elements in height and 7 resolution elements in breadth. The width of the arms of the "E" is less clearly defined but is approximately 2 resolution elements.

This lack of definition is predictable for an arm having a width less than several resolution elements.

With an arm width of 1.7 resolution elements, there are several ways in which an arm could be aligned with a grid of resolution elements, as shown in Figure 5.4 for three cases. In case (a), the middle resolution element should ideally result in reconstruction with maximum intensity 95

one 1.7 resolution element resolution elements

resolution one resolution elements element

one resolution resolution elements clement

Figure 5.4 Alignment of ME" arms with rows of resolution elements. The width of the "E" arm is 1.7 resolution elements. while the lower element is reconstructed with 70% intensity*

In case (b), both the upper and middle elements reconstruct with an intensity equal to 85% of maximum. In case (c),

the middle element has maximum intensity while each of

the upper and lower elements has 35% intensity. Of course, cases intermediate between these extremes are more probable

in practice. The upper arm of the "E" in Figure 5.3

appears to correspond to case (c) because of the symmetry present. The lower arm, composed of two horizontal rows

of elements having an intensity above background, corre

sponds closely to case (a) in that the upper row is near

maximum intensity while the lower row is near 60% intensity.

The middle arm appears to correspond to a situation

intermediate between cases (a) and (c). The vertical

part of the ME" in Figure 5.3 also corresponds closely

to case ( c ) .

Although these two positive results were established

by the experiment, the cause of the noise present in the

reconstruction was not unambiguously identified. The

noise either could be introduced in the illuminating,

developing and digitizing of the optical shadowgram or

could be an inherent feature of the reconstruction code.

Any of the difficulties with optical shadowgram formation

previously discussed could result in the unevenly distributed 97 noise observed. In addition, the cause of the uneven intensity of the image cannot be determined with certainty, although it may be due to an anomalous background density gradient which appeared in the shadowgram transparency.

In order to determine to what extent these difficulties may be due to the reconstruction code alone, noiseless input data is needed. Por this reason, the computer generation of noiseless, digital shadowgrams was under taken next.

5.3 Reconstruction of Computer-Generated Shadowgrams

The initial motivation for generating shadowgrams by computer was to provide idealized input data for testing

the reconstruction code. However, another advantage of

computer generation is that it allows some testing which would be difficult or impossible to perform accurately with optical shadowgrams, for example, the ability to

image eithor an extended source of nonuniform intensity

or a three-dimensional source. In addition, a time

limitation is removed because the procedure for exposing

and digitizing optical shadowgrams required nearly two

weeks to complete. Therefore, once the ability of the

code to reconstruct images from actual laboratory data

was established, computer generation of shadowgrams was

substituted for optical generation. 98

5.3.1 Generation of the Shadowgrams

The shadowgrams were generated so as to satisfy the maximum resolution criteria of Chapter 4. The number of zones in the zone plate, 16, was chosen equal to one-fourth the number of detector resolution elements per side, 64, in accordance with Equation 4-10. The diameter of the zone plate shadow was chosen equal to the length of the side of the square detector; hence the equality of Equation

4-8 holds, and Equation 4-7 is automatically satisfied.

A source at the midpoint of one side of the square field of view will cast exactly half a zone plate shadow onto the detector. No values for the parameters s^, or a need be specified in order to achieve these optimal conditions.

To form a shadowgram, the source distribution was divided into an assembly of point sources. The shadowgram generation code provided an option for allowing each object field resolution clement to contain one, four, or sixteen point sources; in most studies one point source per object resolution clement was used. The intensity of each point source could be specified individually.

Each point source cast a zone plate shadow pattern of proportional intensity on the detector. A point source not at the exact center of the field of view cast only a 99 partial zone plate shadow because the shadow center is displaced and only part of the complete shadow should fall on the detector. The shadowgram was formed from the distribution of point sources by adding in each detector resolution element the contributions of the overlapping zone plate shadows.

5.3.2 Point Sources

The ability of the computer code to reconstruct the shadowgrams of point sources without noise was tested first. For a point source in the center of the field of view, the code resulted in a single point image at the center of the image field. When displayed using the line printer code of Table 5-2, the reconstruction appeared perfect in that it consisted of a single point of maximum intensity with all other points coded as blank.

A point source positioned halfway between the center and the right edge of the field of view was then imaged.

The resulting shadowgram, consisting of a partial zone plate shadow, had symmetry about a horizontal axis but not about a vertical axis. As a result, the image of the point source lacked symmetry. Because the shadowgram had a full complement of zones in the vertical direction, the vertical position of the image point was well defined, 100 that is, confined to a single horizontal set of points.

The image consisted of a correctly positioned point of maximum intensity flanked on each horizontal side by one point of intensity between 10% and 20% of maximum. In effect, the image was slightly out of focus in the hori zontal direction because the zone plate shadow did not have a full complement of zones in the horizontal direction.

Similarly, when a point source was displaced verti cally, the reconstructed image consisted of a single brightest point, slightly out of focus in the vertical direction. That is, both.above and below the brightest point was another point having much less intensity.

Next a point source positioned at the center of one quadrant of the square detector was imaged; that is, the source was displaced from the detector center both verti cally and horizontally. The reconstructed image was a point of maximum intensity surrounded by four nearest neighbor points of intensity between 10 and 20%.

These results indicate that when system parameters are chosen for maximum resolution, a source near the center of the field of view is imaged with the best resolution.

The resolution is degraded as the source moves away from the center of the field of view. It should be noted that a better uniformity of response can be achieved at the expense of overall resolution. By not choosing the zone 101 plate large enough to fill the detector, that is, by choosing it small enough so that as the source point moves around in the field of view, a small but complete circular shadow moves about on the detector, the recon structed image point will have a more nearly constant resolution throughout the image field. However, with this smaller zone plate shadow, few zones can be resolved by a detector of given resolution. By employing a zone plate of fewer zones, less resolution would be achieved

at all image points including those at the center of

the field of view.

A special problem is encountered when imaging a

point source exactly at one edge of the field of view.

It is an unavoidable fact that the digitized input data which is Fourier transformed behaves as one period of

an infinite array of periodic input data. Likewise the

calculated output is one period of an infinite array of

periodic output data. Hence when an image point on the

edge of the field of view is reconstructed, because it

is slightly out of focus it will be flanked on either side

by low intensity points. The low intensity points which

should appear just outside the edge of the image field

will appear on the opposite edge of the image field because

of the periodic nature of the output. This feature is an

inherent feature of the code. The problem can be dealt 102 with in one of two ways. First, the frame for defining the

source field of view (cf. Section 4.2.5 and Figure 4.12) can be designed to limit the field of view just enough

so that edge points are not imaged. Secondly, the input data matrix can be increased in dimension by the code from

x to (Nj + 1) x (Nj + 1) so that the x matrix

of true input is surrounded by a field of zeros.

These difficulties notwithstanding, it must be concluded

that the imaging system successfully reconstructs the image of a point source.

Also, a small group of two or three separate point

sources is successfully imaged. The resulting image is

very close to a superposition of the images which would

result from the sources imaged individually. However, if

the individual sources are of the same intensity, the

resulting image points will not necessarily have the

same intensity. Sources farther from the center will

cast less complete zone plate shadows which will in effect

diffract less light to the image point, and hence the

images of these sources will have less intensity than

centrally located sources.

5.3.3 Small Extended Sources

The ability of the imaging system to reconstruct

images of small extended sources was tested next. A 103 small extended source means here a source confined to one region of the field of view and occupying no more than

100 resolution elements.

The results obtained for imaging the letter "E" are typical of results obtained for all small extended sources.

The source pattern for the "E," Figure 5.5, was used to digitally generate a noiseless shadowgram, Figure 5.6, from which the image, Figure 5.7, was computed. The source pattern is composed of 64 point sources located near the center of the field of view. All source points are of equal intensity. The image is reconstructed with correct size, shape and position. Background noise is present throughout most of the image field; however, its intensity at most points is less than 30% of the maximum image intensity. At a few remote points and at most points close to the "E," the noise is a much as 40% of maximum.

Similar studies were performed for other small extended sources, including squares, triangles, and curved lines of irregular shape. These were positioned in various parts of the field of view. In general the results were the following. The system is able to form a clearly recognizable image of a small extended object. The size,

shape and position of the object are correctly imaged.

The image exists in a large field of low background

intensity. The size and intensity of the background field .104

#»*«#»# *# M M

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Figure 5.S The source pattern for the "E". 105

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Figure 5.6 The shadowgram for the "E".

« 106

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Figure 5.7 The image of the "H". 107 increases with the number of resolution elements contained in the source. This result is consistent with the theory presented in Section 3.2.

5.3.4 Large Extended Objects

Large objects are not imaged well by the system.

When the number of source resolution elements making up the object becomes more than 100, the background noise intensity at some points becomes equal to the image inten sity, and the noise tends to obscure the image. For a source composed of several hundred resolution elements, the image was almost completely lost in the background.

An attempt was made to determine if the ability to image objects composed of many resolution elements depended on the spatial frequency content of the object. A series of computer simulations was carried out in which the object field was a bar pattern consisting of alternate strips of bright and dark bars. The spatial frequency varied from the brightest possible, in which alternate columns of resolution elements were bright bars, to the lowest, in which half the field of view was bright and half dark. In each case the number of source (that is, bright) resolution elements was 2,048. The bar pattern was not reproduced at any spatial frequency. However,

for the lowest frequency, the bright half of the source 108 field was imaged as a bright area of indefinite boundary 1 in one half of the image field.

The conclusion is that whenever the source occupies more than roughly 101 of the available source field resolu tion elements the image is lost in the background noise.

5.3.5 Extended Sources with Cold Spots

The ability of the system to image areas emitting no radiation within a larger area of high intensity was investigated next. The results obtained with large extended sources indicated that if the field of high intensity is too large the resultant noise would obscure the cold spot, just as the dark bars of the bar pattern were completely obscured by the noise from the bright bars. In studying the ability of the system to image cold spots therefore, the scope of the study was restricted to cases in which the high intensity area was composed of a small number of resolution elements.

The first class of objects to be studied was squares, composed of four line segments each of one resolution element thickness. A small square, four resolution ele ments on a side, was imaged well. All points on the square were reconstructed with maximum intensity. Inside the square were nine resolution elements, five of which imaged with less than 10$ of maximum intensity and four, 109 in the corners, with less than 20%. The noise outside the square was almost everywhere less than 10% of maximum and never more than 20%. However, a small amount of noise appeared in a regular pattern far from the image of the square. Hence the system is able to reconstruct cold spots in very small objects.

However, as the size of the square increases this ability is less apparent. The small square above was outlined by 16 resolution elements. A similar square composed of 48 resolution elements was imaged. All points on the square reconstructed with maximum intensity, and

all points not on the square had less intensity. The

appearance of the background outside the square was very

similar to that around the letter "E" in Figure 5.7,

that is, it filled most of the field of view but was

generally under 50% of maximum intensity. Inside the

square the noise varied approximately bewteen 35 and

75% of maximum. The noise pattern was not random.

Because of the symmetry and absence of noise in the input

data, the reconstruction image noise was symmetrically

distributed in a regular pattern inside the square.

Another similar 80-element square was imaged with

similar results. Background was greatest near the line

segments making up the square and extended for an average 110 of eight resolution elements on each side of the line segments with a fairly uniform intensity of 60%.

With the imaging of a 112-element square, some change in response begins to appear. The size of the square has become large and its corners are distant from the center of the field of view and hence they recon struct with intensity as low as 70% of maximum. The noise level outside the square is low, lower in fact than with the 80-element square. Inside the image square, however, there is considerable background noise which is no longeT of greatest intensity closest to the line segments of the square. Instead, the noise is greatest in an approximately annular region inside the square. At a few points the noise intensity exceeds the intensity of the reconstructed corners of the square.

These new features were more pronounced in a 144-element square. The points of maximum intensity in the recon struction were in a centrally located, approximately circular group of background points; the diameter of this circle was equal to half the diagonal of the square. The image of the square was clearly visible amid the low

(up to 40%) background in the neighborhood of the line segments. But because the line segments had intensity values of only 40 to 70% of maximum, the image contrast H i was low. It should be pointed out here that although the number of elements in the square was not very large, the positions of the source points making up the square were well removed from the center of the field of view.

An interesting response was obtained for two larger squares. A 176-element square was not imaged at all.

The reconstruction consisted of a small bright noise formation having radial symmetry, and no hint of the line segments of the square appeared at the correct positions even though the noise level at these places was not excessive. However, a larger square composed of 208 resolution elements resulted in a very low contrast, but nevertheless visible, reconstruction of the square. Because the inverse of the distance between opposite sides of the square is the fundamental spatial frequency of the reconstruction, this result suggests that the imaging system may have a spatial frequency response which is not monotonically decreasing with increasing spatial frequency.

Studies performed on the hollow squares described above indicate that the smaller sources containing relatively large cold areas are well imaged. Consider next small sources with cold spots covering only a few resolution elements. 112

A hollow 12 by 12 resolution element square was gradually filled in from the outside so that a cold area of diminishing size was at the center of the square.

Initially the square was composed of sides two resolution elements thick and so contained a cold area measuring

8 by 8. In the image this cold area was reconstructed with a noise level ranging from 40% to 70% of the true image. The highest noise level occurred in the neighbor hood of the center of the cold area, and the lowest toward the outside.

As the square was filled in with sources, not only did the size of the cold area decrease, but also the number of point sources in the object increased and hence background noise increased. With a 6 by 6 cold area the noise level in the cold area of the image grew to 50-80% of maximum image intensity. For a 4 by 4 cold area, the range was 60-80%. A 3 by 3 cold area was imaged with a uniform intensity equal to 70% of maximum. The same result was obtained for a 3 by 2 cold area and also for a 2 by 2 cold area. However, with an increase in the number of resolution elements in the source, the intensity of the reconstructed square decreased at the edges to

80% and at the corners to 70%. Hence although the cold area was visible with low contrast, the reconstructed intensity of the cold spots inside the square equaled the 113 intensity of reconstructed corners. For cold spots con sisting of three, two, and one resoltuion element, the results were the same: the cold spot images were recon structed with less than 70% of the intensity of the surrounding true image but the corner points of the square had an equally low intensity. However, even with no cold spots in the 12 by 12 square, the decrease in intensity at the edges occurred.

Hence small cold spots in sources can be imaged, but the contrast obtained is poor. It is difficult to dis tinguish a true cold spot of zero intensity from an area of reduced intensity.■ All the studies described to this point have dealt with sources of uniform intensity.

5.3.6 Sources of Nonuniform Intensity

Studies were undertaken to examine the ability of

the system to reconstruct sources of nonuniform intensity with correct relative intensities.

Initially two point sources of different intensities were used. The sources were spaced 10 resolution elements apart and each had an equal displacement from the center of the field of view. Several relative intensities were used, and the results are presented in Table 5-4 in terms

of the intensity of the less intense source relative to

that of the more intense source. The table shows that 114

Table 5-4. Relative Intensity of Weaker

Source and Weaker Image

Relative Intensity of Relative Intensity of i, Less Intense Source Less Intense Image (4 of more intense source) (4 of more intense image)

90 80

80 65

70 SO

60 37

50 27

40 18

30 11

20 6

*These results are average values derived from a graph of computer-generated data. 115 the less intense image is always reconstructed with less than the correct intensity value. This effect occurs because the strong noise introduced by the presence of the stronger source interferes with the reconstruction of the weaker image more than the weak noise of the weaker source interferes with the reconstruction of the stronger image. In imaging two sources of unequal intensity, the imaging system will tend to suppress the image of the weaker source.

This result was checked in tests involving two extended sources of uniform, but unequal intensities. The less intense extended source was usually imaged with correct relative intensity at its centermost image points but with less than correct intensity at those image points along its border. The relative intensity of the less

intense image was correct and most nearly uniform when the two extended sources were continuous at the center of the field of view. As the sources separated, and especially when they moved toward diagonally opposite corners of

the field of view, the uniformity of intensity in each of the images was lost. These results were obtained for

a wide range of values of relative intensity. The results

appear to be inconsistent with the low intensity suppression

observed for point sources. However, interpretation of

the results is complicated by the fact that the system does 116 not produce uniform image intensity even for a single uniform extended Lource, and by the fact that the back ground of the stronger image adds to the intensity of the weaker image.

5.3.7 Small Strong Sources in a Weak Source Field

The ability of the system to image a small but relatively intense source in the presence of a much weaker source field was examined for two cases. First, the entire object field of view was filled with a back* ground source field of constant, low-level intensity, I .

Secondly, the entire object field was filled with back ground sources such that each source intensity was chosen randomly between zero and a given upper limit, Iu , this limit being of low intensity compared to the strong source. For both cases the source was a hollow four by four square.

The case for which the weak background source was uniform was examined for various values of I. For an

Ic value equal to 25% of the strong source intensity, no image of the square was formed. For I equal to 10%, an extremely low contrast image of the square was formed; the image background intensity in the region of the image of the square was 75% of the square's image intensity.

For I_ B 5%, the corresponding background value was 65%. 117

Similarly, for Ic values of 2.51 and II, the image back ground values were 551 and 251, respectively.

The case for which the weak background sources were of random intensity was tested for various values of Iu .

As in the previous case, for an Iu value equal to 251 of the strong source intensity, no image of the source was formed. For Iu a 101, the image was present amid an image background intensity equal to 651 of image intensity.

For Iu equal to 51, 2.51 and 1%, the image background values were 451, 351 and 151, respectively.

No significant background occurred inside the small square in the absence of the background source field.

These results show that for a Fresnel zone plate imaging system an extremely high contrast between a strong source and widely distributed, weak background sources is needed to achieve acceptable contrast between image and image background. This difficulty can be lessened by reducing the viewing area defined by the mechanical frame (Figure

4.12) to a smaller value than that required by the para metric considerations of Chapter 4.

5.3.8 Sources in Different Tomographic Planes

The ability of the system to perform tomographic studies was tested next. Objects in different tomographic planes cast shadow patterns of different radii. As shown in Appendix D, the image plane in focus in the reconstruc tion depends on the choice of the parameter Af in Equation

D-44, which in turn depends on the choice of the parameter

R^ in Equation D-26. R^ corresponds to the radius of

the first zone of the Fresnel zone plate which would focus

light in the chosen image plane. Therefore, if a source

is positioned so that it casts a shadow of first zone

radius Rs , and if R^ in the reconstruction code is chosen equal to Rs, then that source's image will be in focus

in the reconstruction. A source in another plane will

appear out of focus to the extent that this other plane

is distant from the first. In the following studies the

source plane and the image plane will be specified by the

value of Rg and Rj, respectively, measured in units of number of resolution elements in the detector plane. In

all the studies discussed above, Rg and R^ were both

equal to 8.0.

Initial tomographic studies were performed on point

sources. One point source was positioned in plane 7.5

(that is, it cast a zone plate shadow having a first

zone radius of 7.5 resolution elements in the detector

plane), and another source was positioned in plane 9.5.

Reconstructions were computed for planes 7.0, 7.5, 8.0,

8.5, 9.0, 9.5 and 10.0. In all planes except 7.5 and 9.5, 119 the images were severely out of focus, and background

"noise" at a level generally below 201 of maximum was * distributed throughout most of the image field. This

"noise" was actually just the diffraction pattern calculated at an out-of-focus plane. One point source was reconstructed in each of the planes, 7.5 and 9.5. The resolution was good and the noise level was everywhere well below 10% of maximum.

Another experiment placed one point source in each of the seven planes mentioned above. In this case also each point was reconstructed in the correct image plane by being the only point of maximum intensity in that plane.

However, in each reconstruction, noise was present due to the out-of-focus sources. The reconstruction in plane

8.0 had the least noise; the out-of-focus sources of planes 7.5 and 8.5 appeared as small rings of intensity less than 20% of the focussed image point intensity. For other reconstruction planes the noise was greater, becoming considerable in planes 7.0 and 10.0. The reason is that the plane 8.0 is the source plane for which the system is designed to give the best resolution; it is the plane for which the zone plate shadow has parameters satisfying the maximum resolution criteria of Chapter 4.

Hence the system is fully able to reconstruct tomo graphic images of point sources. However, out-of-focus 120 images of points in other planes can appear; and without ' a priori knowledge that all sources are points and are of equal intensity, the true positions of these out-of-focus images is not well defined.

Images of small extended sources in different tomo graphic planes were considered next. A 16-resolution element triangle in plane 8.0 and a 16-resolution element square in plane 9.0 were used as sources to form a shadow gram. Image planes 7.0, 7.5, 8.0, 8.5, 9.0, 9.5 and 10.0 were reconstructed. This procedure was followed for various separation distances of the triangle and square, where separation distance means the center-to-center distance when the two sources are projected onto a single source plane. The results obtained were virtually inde pendent of separation distance, including the case where the two source patterns overlapped. The results illustrated in Figures 5.8, 5.9 and 5.10 are typical. Figure 5.8 shows plane 8.0. As expected, the low intensity background noise associated with small extended objects exists in most of the image plane. The triangle is in focus and lies to the upper left of center. Its intensity varies from maximum to about 551 of maximum. To the lower right of center is the out-of-focus square. Its intensity at several points is as high as approximately 65% of maximum. 121

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Figure 5.10 Image of triangle and square focussed on plane 9-. 0. 124

Figure 5.9 shows plane 8.5. Neither source is in focus.

In Figure 5.10 the square is in focus in image plane 9.0.

However, only a portion of the complete square is imaged with high intensity. The intensity of the out-of-focus triangle becomes at one point as high as about 55% of maximum. These results show that an extended object does not disappear from the reconstruction if the plane it exists in is not precisely in focus. Instead, the image of the object may appear, somewhat out of focus and at somewhat reduced intensity.

In further experiments the distance between the tomo graphic planes containing extended sources was varied.

The ability of the system to distinguish tomographic planes increases as the distance between the planes increases.

This is because any figure, a triangle for example, if only slightly out of focus, still appears as a triangle in the reconstruction. However, the ability of the system to reconstruct a high resolution image of a source decreases as the source becomes more distant from the source plane for whcih the system is designed to achieve its best resolution.

Attempts to image small cold spots at the center of three-dimensional sources were not successful, with the exception of the simplest case. A small cube, three resoltuion elements on a side, with hollow center was 125 imaged. When the image plane passed through the center of the cube image, the resulting two-dimensional image was a three by three square with a central resolution element having only S0% the intensity of the points making up the square.

5.4 Summary of Results

The optical simulation experiments yielded two basic results. First, the reconstruction code is capable of reconstructing an image of an extended optical source using digitized data obtained by scanning an actual experi mental shadowgram. Second, the parametric relations of Chapter 4 for determining the resolution characteris tics of a Fresnel zone plate imaging system correctly relate source field parameters to image field parameters and correctly predict the zone plate parameters required for maximum system resolution.

The computer simulation experiments examined the ability of the reconstruction code to reconstruct shadow grams of a variety of idealized source distributions. The system reconstructs images of point sources with good resolution, but the resolution and intensity of the image point depend on the position of the point source in the field of view. The system forms images of small extended 126 sources but not of large ones. Image background intensity fills most of the image field except for the smallest f sources. As source size increases, the background inten sity also increases. When the source covers 10% of the resolution elements in the source field the image becomes totally obscured by the background.

The ability to image cold spots in an otherwise intense source depends on the size of that source. The image background, which increases with source size, tends to fills in the cold spots in the image. This makes it difficult to distinguish a true cold spot of zero intensity from a source region of reduced intensity.

On the other hand, with separated sources of unequal intensity, the imaging system will tend to suppress the image of the weaker source.

When a strong source is imaged in the presence of a weaker, but widespread, field of background sources, this background intensity must be no more than a few percent of the strong source intensity in order to obtain an image with acceptable contrast.

The ability of the system to distinguish tomographic planes increases with the distance between the planes.

However, the plane to be imaged cannot be too far distant from the plane for which the system gives maximum resolution without seriously degrading image quality. CHAPTER 6

IMAGE ENHANCEMENT STUDIES

The computer studies of the previous chapter were intended to simulate the reconstruction results which would be obtained with ideal optical reconstruction in the absence of noise introduced by photographic and optical irregularities. In the present chapter, methods for improving the image by digital processing are considered.

Although these methods can in theory be employed with optical reconstruction, in practice they require such time and painstaking attention to detail that they could not be used routinely in a nuclear medicine clinic.

6.1 Introduction

Numerous image enhancement techniques have been developed for application in such fields as television reception, electron microscopy, military reconnaissance, and space exploration. Application of these techniques to the present study is limited by four considerations.

First, a technique may not apply because the noise source

is not found in the Fresnel zone plate system. Second,

127 128 the amount of computer time and storage may be excessive • in relation to the expected image improvement. Third, enhancement processes intended to improve image resolution can sharpen features only at the expense of the overall signal~to~noise ratio (SNR). Such processes may have a net disadvantage in a system with a low SNR. Lastly, the linearity and spatial variance characteristics of the

Fresnel zone plate system may differ from those of other systems.

Many image enhancement techniques were not considered for the reason that the noise source was not present in the Fresnel zone plate system. For example, in vidicon tubes there is often significant frame-to-framc image retention which requires removal. Space satellite images require removal of periodic noise arising from the coupling of periodic signals related to the rastcr-scan and data sampling devices. In some systems the noise pattern varies only slightly from one image to the next. This allows identification and removal of the noise pattern. In the

Fresnel zone plate imaging system, as discussed in Chapter

5 and Appendix B, the noise pattern depends on the size and shape of the object to be imaged.

A digital masking technique was studied but was found to be of questionable value because it often degraded the SNR. In the Fresnel zone plate imaging system, image' points near the edge of the field of view are reconstructed with about half the intensity of points near the center because the corresponding point sources cast half and full zone plate shadows, respectively, onto the detector face. A computer code was written to correct for this situation. Although the code worked well with a few isolated point sources, it has little practical advantage

for extended sources. Large extended sources were not

imaged well by the system with or without the code.

Small extended sources were imaged best near the center of the field of view, in which case the code would reduce

the intensity of the central image and increase the

intensity of peripheral noise. Although the code could

result in image improvement in some cases, in practice

the clinician will normally attempt to align the organ

of interest with the center of the field of view. Hence

this code does not appear suitable for use as part of a

standard image enhancement package to be applied to every

image.

6.2 A Fourier Transform Technique

Several image enhancement techniques rely on the

existence of a transfer function for the system. Components 130 of a possible transfer function for shadowgram formation can be identified. The following definitions will be useful.

circ(r) * (6 -1)

rect(x) (6-2)

r ^ comb(x) » ) 6 (x - n) (6-3)

n*»-»

s m n x sinc(x) (6-4) TTX

A uniform field of gamma radiation is transmitted by the zone plate according to the transmittance function, g(r), given by

g(r) - \ - 3- Y, p ©xpUnpr2/^2) (6_s) p»*® 131

For a finite zone plate of radius R^ the transmittance is

gfy/r) “ g(r)circ(r/RN) (6-6)

The gamma distribution arriving at the plane of the detector face is given by

gd (r) - ** . . J 2 S r n Cr) (6-7)

This distribution is sampled by the rectangular array of resolution elements, each of dimension AxQ , giving the following sample values

8 s Cx,y) - gd (x,y)comb comb (j£-) (6 -8)

where gd is transformed to rectangular coordinates.

Finally, because the detector is a finite square, with the length of each side equal to D, a*finite.number of sample values results, given by

gD (x,y) - gs (x,y)rect (jj ) rect . (6-9) 132

The spatial frequency content of the sampled shadow gram is related to the spatial frequency content of source distribution by the Fourier transforms of these functions. In the frequency domain, the functions in

Equations (6 -6) through (6-9) must be replaced by their

Fourier transforms, and each multiplication must be replaced by a convolution integral. The Fourier transform of g(r), given in Appendix B, involves an infinite summa tion of terms, each containing integrals of exponentials.

The Fourier transform of circ(r/R^) is

JjUnRjjf) v where Jj is a Bessel function of the first kind, order one, and f is the two-dimensional spatial frequency equal to

(fx 2 + fy2)1^2 . The ratio (Sj + s2)/s1 is simply a multi plicative scaling factor. The Fourier transform of comb ( g~-) *comb (^T") is (Ax0 )2comb(Axofx)*comb(Axofy).

Finally, the Fourier transform of rect ( jj ^ rect ^ jy ) is sinc(Df^)sinc(Dfy). The expression resulting from successive convolutions of these functions is at best difficult to work with. Such an expression would relate 133 the Fourier transform of the source intensity distribution to the Fourier transform of the shadowgram intensity distribution.

Relation of shadowgram intensity to image intensity involves more than simple multiplication of functions. In

Appendix D it is shown that each sampled shadowgram intensity value must be assigned a phase value depending on the position of the resolution element recording that value and on the distance to the image plane. The resulting array of complex values is Fourier transformed, and the image intensity is found by taking the square modulus of each element in the transformed array. These considerations lead to the conclusion that relating source and image spatial frequency contents in the frequency domain for the purpose of deriving a system transfer function in closed form is not possible.

It was shown in Chapter 3 that a transfer function exists only for a system which is linear and spatially

invariant. A system is invariant if the image of a point

source changes only in location, not in functional form, as the point source moves about in the field of view. In

Chapter 5, we found that this condition does not hold for

the Fresnel zone plate system. For example, a point source moving vertically up from center casts a partial zone 134

plate shadow, and the moving image point loses definition

in the vertical direction. Hence the functional form of

the image point is position dependent, and the system is

spatially variant.

A system is linear if the principle of superposition

holds. We found in Chapter 5 that a low intensity image

point is suppressed by the system in the presence of a

high intensity point. Then the image resulting from two

point sources of different intensities is not simply the

superposition of images for the two points taken separately.

Hence the system is nonlinear.

These considerations indicate that no transfer

function exists for the Fresnel zone plate imaging system.

Computer studies confirm this result.

The method for determining the transfer function

of a system, discussed in Chapter 3, was applied to the

computer model described in Chapter 5. This method

consists of taking the Fourier transform of the image

.resulting from a point source input. If a transfer function

exists for the system, the result is independent of the

point source position. With the computer model used in

this study, the result was found to vary with the position

of the source.

With the source in the center of the field of view,

the results from the transfer function calculation show a relatively flat frequency response, as depicted in

Figure 6.1. The complete result is a 64 by 64 array covering positive and negative frequencies and having rough radial symmetry. The solid line shows the response to a test pattern with test lines parallel to the rows

(or columns) of resolution elements in the detector array.

The dashed line is for the case in which test lines are oriented diagonally. For intermediate orientation, the frequency response is between the two lines shown. There is a small spike at zero frequency indicating that the mean value of the transmitted intensity is nonzero. Other spatial frequency components are transmitted with good response: the response to the highest frequency is never less than one-third that of the lowest nonzero frequency.

With the source in the corner of the field of view, the result is different, as shown in Figure 6.2. There is considerably less radial symmetry. The response to diagonal test lines is much poorer at all frequencies than the response to parallel lines, decreasing essentially to zero for the two highest frequencies. For parallel test lines the response at the highest frequency is one- tenth that at the lowest nonzero frequency.

As the source point moves around in any given neighbor hood of the field of view, the results of the calculation 136 10 Modulus of the Frequency Response

10 20Af 30Af spatial frequency

Figure 6.1 Results of transfer function calculation for a point source at the center of the field of view. Test lines parallel to resolution elements (solid line) and diagonal (dashed line). 137

10 Modulus of the Frequency Response

10 f 20 f 30 f spatial frequency

Figure 6.2 Results of transfer function calculation for a point source in a corner of the field of view. Test lines parallel to resolution elements (solid line) and diagonal (dashed line). 138 change very little. This suggests that image enhancement ‘ may be possible for small extended objects confined to one neightborhood. Although it is possible to calculate an "average" transfer function for each neighborhood of the field of view, only the result for the neighborhood of the center is useful in practice. For other neighborhoods the frequency response is so low at high frequencies that division by the transfer function results in great ampli fication of the high frequency content of the image. In a neighborhood where the system has very poor ability to transmit high frequencies, the high frequency content of the image consists mostly of noise. Then the result is to amplify greatly the noise content.

The applicability of the "average" transfer function, derived for a central point source, to zone plate system image enhancement was studied. The technique was discussed in Chapter 3. Briefly, this technique consists of calcu lating the Fourier tranform of the image and dividing it by the system transfer function to obtain the Fourier transform of the source distribtuion, and then taking the

inverse Fourier transform of the latter to recover the source distribution. The study began with an attempt to

improve the image of a corner-located source using the transfer function for a central source. A significant 139

increase in the resolution of the image resulted. This

result suggested that the method may be useful for improving

the resolution of any image formed by the system and particularly useful for small extended sources in the center of the field of view.

The image enhancement technique was tested for various

small extended objects including curved lines and hollow

squares. In all cases, excellent results were obtained.

A typical result is the improvement of the image of the

letter "E", which was shown previously in Figure 5.7. The

processed image is shown in Figure 6.3. The absence of

significant noise around the image is striking. However,

the intensity of the various parts of the image is not

uniform. These results applied to most processed images

of smaller extended objects. For the smallest objects,

the intensities were also uniform. For intermediate sized

objects, some noise was present.

Images of somewhat larger objects were also improved.

Although significant improvement occurred in all cases

tested, the quality of the improved image still decreased

with increasing object size. This effect was demonstrated

by studying the processed images of hollow squares of

various sizes. The unprocessed images were described in

Chapter 5. For a square containing four resolution ele

ments on a side, processing removed the low-level, periodic #*

Figure 6,3 Enhanced image of the letter 141 noise appearing around the image. The final image was a clean image of a square of uniform intensity. The image of a square of side nine units showed excellent improvement. Whereas before processing almost the whole field of view contained background noise at intensities up to 30% of maximum, after processing only one point

(inside the square) had a noise level above 101. For a hollow square of length 17 (having 68 point sources altogether), some low-level, isolated, periodic noise appeared after image enhancement. In the case of a 25- element length, much of the inside of the square was filled with noise, usually under 201 of maximum, but occasionally up to 30%; nevertheless, the image was quite distinct and the unit width of the sides was well resolved.

Before processing the thickness of sides of the image square was uncertain because of the noise level.

For larger squares the results were less acceptable, even though some improvement was obtained. The image of the square of side 37 was almost completely lost’in background noise before enhancement. After enhancement the image was clearly present but of very low intensity, and the corners of the square did not appear. The most

intense portion of the reconstruction was the noise, consisting entirely of an intense round artifact located 142 at the center of the hollow square. Virtually identical results were obtained for all larger squares.

For very large objects the system formed no recog nizable image, and use of this technique to enhance the nonexistent image produces no result. It may be appro priate to reiterate here a point discussed in Chapter 3.

Image enhancement for nonlinear and spatially variant imaging systems is a relatively new field of study.

Although the image enhancement technique discussed above leads to good results in some cases, further research in this area could lead to better results for more, and possibly for all cases.

The image enhancement processing described here leads to excellent results for small objects and improved results for intermediate sized objects. In no case is the image degraded by the processing. Hence this technique should be included as a standard part of the digital reconstruction code. CHAPTER 7

CLINICAL ADVANTAGES AND LIMITATIONS OF THE SYSTEM

In this chapter the potential clinical applications of the Fresnel zone plate, germanium gamma camera are discussed. The applicability of the system depends on the size of the radioisotope distribution to be imaged.

In nuclear medicine images are formed either of the radio isotope distribution in a whole organ of the body or of the radioisotope uptake by a lesion in an organ. The applicability of the system for each of these studies is discussed.

7.1 Limitation on Source Size

The digital simulations of Chapters 5 and 6 demon strated that the Fresnel zone plate system forms good images of small sources, noisy images of intermediate sized sources, and no images of large sources.

In the nuclear medicine clinic a germanium camera with a Fresnel zone plate aperture would have an advantage over one with a conventional collimator for imaging small

143 144 structures. The advantage consists of better spatial resolution and the choice of either a greater signal-to- noise ratio (SNR) or a shorter collection time.

On the other hand, the system is limited to imaging structures of small size. For larger structures, a conventional collimator is to be preferred. In addition, a conventional collimator is better for imaging a cold spot within a large area of higher activity.

The size of a structure must be considered relative to the area of the camera detector. Because of the present cost of high purity germanium, the first germanium gamma cameras produced for clinical use will be relatively small. However, as the technology of high purity german

ium crystal growth advances and as the market for germanium increases, the cost is expected to decline and larger germanium cameras may become available. Then the size of a structure imageable with a zone plate system would

increase.

The size of the detector in the system described in

Appendix C is a square with a length of about 15 cm on

each side, containing 4,096 resolution elements. In

Chapter 5, we found that this system produces no useful

image if the number of resolution elements contained in

the source exceeds 101 of the total number contained in 145 the field of view, that is, about 400 resolution elements.'

Another limitation on the size of the source is that size at which the SNR for the zone plate system falls below the SNR obtainable with a conventional aperture. Appli cation of Equation 3-S gives the result that for point source imaging the system considered here has a SNR 20 times that of a pinhole aperture. The SNR's for the two apertures become equal when the number of resolution elements in the source equals 400, by coincidence, the same result as before.

The size of a resolution element in the source field of view depends on the magnification parameter, a.

Assuming that a is chosen to obtain optimum system reso lution, the theoretical resolution of the system, given in Table 4-4, is 1.7 mm. Then 400 resolution elements would be contained in any source distribution having a frontal area of 11.6 cm ; for example, a square of side

3.4 cm or a circle of diamter 3.8 cm. Structures smaller than these would be imaged better with a zone plate system than with a conventional collimator. For larger struc tures, the conventional collimator is better. A larger structures can be imaged by the zone plate system by decreasing a, but the spatial resoltuion of the system is degraded so that less detail of the larger structure can be imaged. 146

7.2 Imaging Whole Organs

In the nuclear medicine clinic, gamma cameras are used to form images of the distribtuion of a radiopharma ceutical in many organs of the body for the purpose of assisting in the diagnosis of disease. The organs normally

imaged are the brain, thyroid, lungs, heart, liver, pan

creas, spleen, kidneys and bone.

Various kinds of studies can be performed. For

example, to study the ability of a kidney to filter waste

products from the blood, a pharmaceutical normally

filtered efficiently with three or four passes through the

kidneys is tagged with a radioisotope and injected intra

venously. The field of view of the gamma camera is set

to include the kidney and the bladder. Sequential images

show the efficiency of the kidney in collecting the radio

pharmaceutical and delivering it to the bladder. In this

example, the clinician is interested in examining the

functioning of the organ as a whole.

In other cases, it is desirable to image the whole

organ in order to find a pathology in part of the organ.

For example, because the normal lung is richly supplied

with blood, the lung can be imaged by adding a radiotracer

to the blood. If some part of the lung is receiving

little or no blood, a cold spot appears in the lung image. 147

The cold spot shows the existence of disease, and the appearance of the cold spot may indicate the nature of the disease. Typically, a large cold area of the lung is associated with pulmonary edema, an effusion of fluid into lung tissue; and a small cold spot indicates a lesion, an abnormal discontinuity in the tissue of an organ.

The smallest organ normally imaged is the thyroid.

Although there is considerable variation in thyroid size and weight, the average human thyroid weights 20 to 25 grams and has a thickness of 1 cm and a frontal area of 2 20 cm . The largest structure which can be imaged with the zone plate system considered here, when that system

is set for optimum spatial rcsoltuion, has a frontal area of 12 cm . Assuming the distribution of the radiopharma ceutical is approximately uniform throughout the organ, the thyroid is too large for imaging with this system.

In addition, the tomographic capability of the system within a one-centimeter thick organ is not impressive.

Hence a gamma camera with a 15 cm by 15 cm germanium detector using a conventional collimator would have a performance superior to one using a zone plate aperture

in all applications requiring imaging of a whole organ.

However, this conclusion might not apply to a camera with a larger germanium detector. 148

7.3 Imaging Small Lesions

Gamma cameras are also used to detect lesions within organs. Two types o£ lesions important in nuclear medicine are neoplasms and infarctions. A neoplasm is any abnormal growth such as a tumor. An infarction con sists of dead tissue resulting from the cutoff of the blood supply due to either a blood clot or a hemorrhage.

The size of both types of lesions can vary from micro scopic size up to 3 cm or more in diameter. The location of a lesion within an organ, as well as its size, will determine whether a patient notices symptoms of his condi tion. As a result, there is not a standard size at which lesions may produce those clinical symptioms which bring the patient to the nuclear medicine clinic. Lesions of any size may exist in the diseased organ. In practice, more large lesions are detected because they are more imageable. One important goal of nuclear medical research is to increase the probability of detection of the smaller lesions.

Lesions can appear on gamma camera images either as cold spots or hot spots, depending on whether the abnormal tissue has a greater or lesser affinity for the radio pharmaceutical than the surrounding normal tissue. In some organs such as the lungs, liver and kidneys, lesions must alwyas appear as cold spots because pharmaceuticals 149 with greater affinity for the abnormal tissue of these

organs have not yet been developed. For the brain, the

heart and bone, such pharmaceuticals have been developed

and are preferred for lesions detection because hot spots

are imaged with better resolution than cold spots.

The ability of the Fresnel zone plate system to detect

small lesions may be superior to the ability of a system

with a conventional collimator for some applications.

Three possible applications have been identified.

The first application is the detection of tumors in

bone tissue. Patients with diagnosed malignant tumors in

any internal organ are routinely given a bone scan to

determine if the disease has spread to the skeletal

system. The radiopharmaceutical, 99m -technecium pyro

phosphate, is administered to the patient because it is

absorbed faster by neoplastic bone tissue than normal

bone tissue. The ratio of neoplastic to normal tissue

activity is high, rougly an order of magnitude. Gamma

camera images are then taken of the various parts of the

skeletal structure. Tumors appear as intense hot spots

against a weak background. Small tumors, detected early,

may be successfully treated. The Fresnel zone plate

system could offer the improved spatial resoltuion needed

to increase the probability of detection of such small

tumors. 150

In addition, if the germanium camera were used to scan the whole skeleton system, the reduced collection time of the zone plate would minimize the total time required per patient.

The second application is the detection and location of myocardial (heart muscle) infarction. Once again,

99m technecium pyrophosphate is used because it tends to localize in the infarcted tissue. A major problem in locating the infarction results from the uptake of the same radiopharmaceutical by the sternum and rib cage overlying the heart. Fortunately, the uptake by the infarction is faster. Detection of a small infarction may be easier with a zone plate system not only because of its better spatial resolution, but also because of the

tomographic ability of the system. If the infarcted tissue

is separated from the overlying bone by several centimeters, such separation is well within the tomographic resolving power of the system.

A third application involves the detection of small

intracranial lesions. At present such lesions are very

difficult to detect unless they lie close to the skull,

for two reasons. First, images of deeper lesions are

degraded because the source lies deeper in a scattering

medium. Secondly, with a conventional, parallel-hole 151 collimator, radiation from a small source far from the collimator passes through many of the collimator holes.

As a result, the size and location of the lesion is not well defined. This combination of the size effect and the collimator effect often makes detection of small, deep lesions impossible. A germanium detector would greatly reduce the scatter effect. A Fresnel zone plate aperture would eliminate this collimator effect and would identify the tomographic level of the lesion.

Although the germanium camera is relatively small, several images could be taken to image the entire intra cranial region. In practice, however, the physician can often associate a patient's symptom, such as loss of feeling in the left arm, with a localized region of the brain, so that only a single image might be needed to con firm the presence of a lesion.

7.4 Conclusions

Although the particular zone plate system design described in this dissertation appears inferior to a con ventional system in clinical applications requiring the

imaging of whole organs, it may be superior for those clinical applications requiring the imaging of small lesions. 152

A nuclear medicine clinic often has several gamma cameras. For example, the clinic at The, Ohio State Uni versity Hospital has six scintillation cameras. There fore, it is not necessary for one camera to be useful for all applications. A germanium camera would complement, rather than replace, the scintillation camera.

If the scintillation camera image of a large organ indicates the possible existence of a small lesion, the germanium camera could then obtain a high resolution image of the region of interest to confirm the existence of the lesion. The framing device defining the field of view of the zone plate system, discussed in Chapter 4, can be adjusted so that only activity from the small region of interest is imaged. If this region has a frontal area 2 as small as 12 cm , then a very high resolution image of the region would be obtained. Such an image would not only increase the probability of confirming the existence of a lesion, but might also provide a much greater ability to diagnose the nature of the lesion. CHAPTER 8

SUMMARY OF RESULTS AND CONCLUSIONS

This study has examined the feasibility of using an on-axis Fresnel zone plate aperture and digital reconstruc tion with a high purity germanium gamma camera in nuclear medicine. In this chapter we summarize the research per formed and the results obtained. Possible extensions of the research are suggested and the major conclusions of the study are presented.

8.1 Summary of the Research

The major areas of the research are listed here.

1. A theory was developed for determining the optimum position and parameters of the zone plate in terms of the camera characteristics.

2. The physical system required to implement the zone plate concept was designed.

3. A computer code was developed for reconstructing an image from a shadowgram, the coded data received by the camera.

4. An optical system was designed and constructed for

forming shadowgrams with ordinary light in order to simulate

153 154 the forming of shadowgrams with gamma radiation and a germanium camera. A method of sampling the shadowgram data for digital processing was developed.

5. A computer code for generating shadowgrams was developed to obtain idealized data for the reconstruction code and to generate shadowgrams of sources difficult to simulate optically.

6 . Digital enhancement of Fresnel zone plate system images was investigated.

7. Clinical applications of the system were investi gated.

8.2 Summary of the Results

The major results of the research are summarized here.

1. When the parameters of the Fresnel zone plate system are chosen optimally, the theoretical spatial resolution of the system exceeds the theoretical spatial resolution of the camera. The validity of the theory for optimally choosing system parameters was established by optical simulation of shadowgrams.

2. A shielding frame must define the field of view to maximize the signal-to-noise ratio.

3. Images found by digital reconstruction, when they are not lost in obscuring noise, correctly reproduce the size, shape and three-dimensional position of the source. 155

4. Digital simulation of shadowgrams established

the kinds of source distributions which are imaged well by the system. Images of point sources are excellent.

Small objects are imaged well with widespread, but low

intensity noise. The images of larger objects are

obscured by noise when the number of resolution elements

in the object exceeds 10% of the total number in the field

of view. Cold spots, although they can be imaged under

special circumstances, are not imaged well in general. For

sources of nonuniform intensity, the relative intensities

are not well preserved in the reconstruction. Strong

sources in the presence of a background source field are

not imaged with good contrast unless the background field

is very weak. The system has tomographic capability, but

the ability to distinguish the tomographic levels of two

sources depends on their size, shape, and the degree to

which they overlap.

5. Image enhancement by digital processing is possi- » .ble. The results for small sources are excellent because

much of the low.level background is removed. The technique .

which removes noise for small sources docs not work for

Inrgc sources. Few standard image enhancement techniques

are applicable because the system is nonlinear and spatially

variant. 156

6 . In clinical application the system cannot be used to image whole organs because of the limited frontal area of the germanium detector. The system has signifi cant potential application for imaging small lesions.

8.2 Intensions of the Research

In the study of coded aperture imaging of gamma ray sources with a germanium camera and computer reconstruction,

further research is possible in the following areas.

1. The clinical applicability of a system with a

germanium detector of large frontal area may be suitable

for future study when the cost of high purity germanium is

reduced.

2. Research in the area of nonlinear, spatially

variant optical systems could lead to an image enhancement

technique useful for large sources.

3. The present study was aimed at maximizing the

spatial resolution of gamma camera images by use of an

on-axis zone plate. .A valuable future study would examine

the clinical usefulness of an off-axis zone plate with a

germanium camera, a system which would sacrifice spatial

resolution in order to image larger sources.

4. Other coded apertures recently introduced, espe

cially the nonredundant array and the time-varying array, 157 deserve future consideration if their initial apparent

disadvantages are overcome.

8.4 Conclusion

The use of an on-axis Fresnel zone plate and digital

reconstruction with a germanium gamma camera is both

possible and practical. Although conventional collimators

are more suitable in the nuclear medicine clinic for

imaging whole organs, the Fresnel zone plate system

provides superior resolution in imaging small lesions. APPENDIX A

OPTICAL THEORY OF FRESNEL ZONE PLATES AND HOLOGRAMS -

AN OVERVIEW

A.1 The Fresnel Zone Plate

A.1.1 Introduction

The zones of an advancing wavefront were first described by Fresnel, and the zone plate for spatially filtering the advancing wavefront was first constructed in 1871 (39). In 1875 Sorct published the first paper describing experimental work with zone plates. The literature in English refers to Fresnel zone plates; the literature in French refers to the same devices as Soret zone plates.

The first extensive investigation of zone plates was carried out by R. W. Wood (40). Wood constructed zone plates by various methods and performed a series of experiments using visible.light. Following Wood's work, which began in 1898, zone plates became a standard topic in optics texts (4j_,42^) and were used in optics courses to demonstrate the diffractive properties of light. It appears that only one paper (43) was published on zone

158 1S9 plates in the 30-year period before 1950, and no practical applications for them were found*

Fresnel conceived the theory of zones to aid in the understanding of wave diffraction. Diffraction effects are * classified as either occurring because of the interference of plane wavefronts from a source at infinity (Fraunhofer diffraction) or occurring because of the interference of spherical wavefronts from a source at a finite distance from the diffracting aperture (Fresnel diffraction). The latter kind of diffraction was originally described in terms of the somewhat obscure method of lunes and the

Cornu spiral. The method of zones, introduced by Fresnel, is an easier approach to Fresnel diffraction theory.

A.1.2 Basic Zone Plate Theory

Let 0 be a point source of waves (such as light waves) of wavelength X; see Figure A.I. Any sphere about

0 is an instantaneous wavefront, all points of which are in phase. Let us determine the amplitude of the light at the point P under various circumstances.

Let Mq be the intersection of the wavefront with OP.

Let a » OMq and b « Choose the point Mj on the wavefront such that

MjP = rt » b + j . (A-l) 160

Figure A.l Spherical wavefront expanding from point 0 .

% 161

If the line M,P is rotated about the axis ht P then M, 1 o 1 describes a circle on the wavefront. The area inside the circle is the first Fresnel zone. Choose the points Mn so that

(A-2)

then the area between the n-th circle and the (n-l)-th circle is the n-th Fresnel zone.

If a and b arc much greater than X, then these zones occur on a small enough portion of the wavefront so that the approximation R >> e will hold; see Figure A.2. Then

a2 - (a - e)2 + R2 s a2 - 2ac + R2 (A-3)

and

(A-4)

Let MP » r ° b + 6 . Then

r2» R2 + (b + e )2

(A-5)

s R2 + b2 + 2be

Substituting the above expression for e, and rearrang

ing, gives * Figure A.2 Wavefront geometry 163

R = 2ab 6 \fa + b (A-6)

Since 6 ** n and defining

abl a + b * (A-7) wc obtain

Rn B K /n* (A-8)

where Rn is the radius of the outer boundary of the n-th

Fresnel zone. Hence = K. The area of the n-th zone is

(A-9)

- *Ri

Hcncc all zones havo ,the same area.

Consider now the light amplitude at P. The light passing through the first zone may be considered of posi tive phase in that every point within the zone has a phase within 180° of the phase of light at the point MQ .

In the second zone the path length r differs from the 164 path length b by more than a half wavelength. The waves arriving at P from the first zone interfere destructively with the waves arriving at P from the second zone. Since the zones are of equal area, the cancellation is exact, provided the zones are small enough with respect to r so that the obliquity factor and the 1/r factor may be neglected. Let In be the light amplitude at P due to light passing through the n-th zone only, and consider an opaque screen placed between the points 0 and P at

Mq and having a pinhole of radius R centered at MQ . If

R B Rj the light amplitude at P is 1^. If R ** R2 then the amplitude at P is

I » Ix - I2 ** 0 (A-10)

If R a R3 then

1 ’ h ’ h * h = H ' h ( A - 1 1 J and so on.

It can be easily shown that if the opaque screen is entirely removed, the amplitude at P is 1/2 1^, or half the value due to the first zone alone.

Suppose, however, a zone plate is constructed so that the odd numbered zones are transparent and the even numbered zones are opaque. The first few zones of such 165 a plate are shown in Figure A.3. Then only waves of positive phase arrive at P; the resulting light amplitude at P is much greater than if no zone plate were present.

The amplitude resulting from a plate with N zones (half of them transparent) is ^ 1^. The amplitude is not infinite for an infinite number of zones because the 1/r factor becomes significant for a zone of very large radius.

In effect, the zone plate acts as a lens to focus the light from 0 onto the point P.

Alternatively, the same effect can be achieved by making the odd zones opaque and the even zones transparent.

Although the above theory applies only to an on-axis source at 0, an off-axis wave source at O' will give rise to.an off'axis image at P ' , as shown in Figure A.4. A small object such as a light bulb filament centered at 0 may be considered a large number of point sources. Hence a white screen in the plane at P will receive an image of the filament.

Zone plates have multiple focal points. If the screen is moved closer to the zone plate, a position is found at which a second image occurs. At this position each zone of the plate allows three Fresnel zones of the wavefront to pass through to the new point P2 . Since two of these zones interfere destructively at P2 and only one contributes *

166

Figure A.3 A Fresnel zone plate I zone plate

Figure A.4 Imaging an off-axis point 168 to the image, this image is less intense than the primary* one. By moving the screen increasingly closer to the plate, more images of decreasing intensity can be found.

The primary image may be made more intense by a zone plate in which both odd and even zones are transparent but in which the even zones introduce an optical path length increase of one-half wavelength. The light from all zones interferes constructively at the image point. Such a plate is called a phase-reversal zone plate, or simply a phase zone plate.

To find the primary focal length of a zone plate, consider the radius of the first zone; it is

CA-12)

Rearranging gives

CA-13)

If we define f ■ -j- , m e n the zone plate conforms to (A-14) the standard lens formula

(A-IS)

The other focal lengths are f/3, f/5, f/7, .... 169

Zone plates behave as lenses even to the point of producing virtual images as well as real images. They have not replaced lenses in optical instruments because

1. The focal length is inversely proportional

to wavelength. The focal length for red is

half the focal length for violet, and good

images can be obtained only for monochromatic

light.

2. The primary image is degraded by the presence

of the defocusscd secondary images.

3. The primary image is degraded by the presence

of the virtual images.

4. Accurate construction of zone plates with

many zones is difficult. The ability of

the plate to obtain high resolution depends

on the number of zones in the plate.

In 1951, 0. E. Myers (44) published a study of the resolving power of zone plates and considered their possible use for focussing infrared and X-radiation. He considered the effect of approximations used in zone plate theory and concluded that for a point object on-axis the maximum allowable number of transparent zones for good focus (the exact optical path length should not differ from the cal culated path length by more than 1/4 X) is given by 170

N - \/& - VsxtS-tt (a'16)

This limitation becomes significant only when the object is very close to the zone plate.

The maximum allowable angle a between the principle axis and the parallel rays of light from an off-axis point object at infinity for a plate of N zones is given by

I ±/2fNX NX/f sin a - NX sin2 a /< 1/4X (A-17)

The resolving power of a zone plate may be defined analo gous to that of a lens: two distant point objects arc resolvable if the central maximum of the diffraction pattern of one falls at or beyond the first minimum of the diffraction pattern of the other. For a zone plate of diameter d the minimum angular resolution 3 is the same as for a lens of aperture d, viz,

0 « I-.,22 , radians (A-18)

This is also the angular resolution of image points with respect to the zone plate. The result is valid for large

Experiments by Myers show the existence of an image at

f/2 due in part to imporfections in plate construction. In 1967 Stigliani, et al. , (4S) extended Myers' work on the resolving power of a zone plate to consider the case of a small number of zones. The results for a positive zone plate (even zones being opaque) differ from the results for a negative zone plate (odd zones opaque). The resolv ing power of each converges to the resolving power of a lens as N increases. The results indicate that a nega tive plate has greater resolving power than a lens of the same aperture while a positive plate has less. How ever, the negative plate has a brighter ring of first order in the diffraction pattern of the image of a point source, while the positive plate has a less bright ring, when both are compared to the ring for a lens. Hence for closely clustered points there may be no real advantage to using a negative plate.

Studies of the diffraction pattern produced by a zone plate were begun by Boivin in 1952. He extended the classical theory of diffraction by a circular slit to the case of several concentric ring-shaped apertures, and he considered Fresnel zone plate diffraction as an example

(46). Boivin, et al., in 1956 published further theo retical and experimental work on zone plate diffraction patterns (47). In 1958, Dyson (48^) derived the light intensity distribution in the image formed by an optical 172 grating consisting of equally spaced concentric circles.

Apart from a proposal by Murty (49) to manufacture a zone-sphere, the zone plate diffraction problem was not discussed in the literature for some time.

In the latter half of the 1960's zone plate diffrac tion studies were renewed. An exact calculation of the diffraction pattern due to a single Fresnel zone was first carried out in 1965 (50_). Four years later the complete zone plate diffraction problem was solved for the case of an image of an on-axis point source. The solution to the problem of focal spot dimensions and energy distribution was derived by Childers and Stone

(51) as a function of wavelength and number of zones.

The diffraction pattern of off-axis images was determined by Bottema by summation of the patterns due to the first 10 zones of a zone plate (£2). A study of zone plate aberrations concluded that good images could be obtained only from highly monochromatic light but that the zone plate, like the pinhole aperture, produced images having no linear distortion, even for a very wide field of view (SjS).

Useful approximations to the exact solution of the diffraction problem were worked out by Lehovec and

Fedotowsky (54) in a study of image formation by a general ized zone-plate-sensor system. 173

A.1.3 Applications of Zone Plates

The focussing ability of zone plates has suggested their use for imaging infrared, ultraviolet and soft x-ray sources (55). The use of zone plates for ultraviolet and x-ray telescopes mounted on satellites above the earth's atmosphere seems practical. Sets of zone plates have been used to determine the locus of points having equal optical path lengths through an optical system (56); and zone plate can be used for precision alignment of points along an axis, with accuracy of a few microns over a 12 meter distance (57). The ability of a

zone plate to focus light of various wavelengths at

spatially well-separated points has suggested the use of

a zone plate in combination with a pinhole stop as a band-pass filter (58).

A.1.4 Manufacturing of Zone Plates

Manufacturing of a zone plate by precision ruling

is difficult if hundreds of zones are to be produced.

Other methods have been treated by Rogers (5_9) , Chau (60) ,

and New (61) . 174

A. 2 Holography

A. 2.1 Basic Holography

Holography is the recording of the interference pattern formed by the intersection of two wave trains.

Consider two wave trains intersecting at a photographic film, as in Figure A. S. Figure A. 5 shows the .interfer ence fringes that are formed at the film; in Figure A.S the solid lines represent wave peaks, and the dotted lines wave troughs. Constructive interference occurs along the locus of points at which peaks intersect peaks and troughs intersect troughs. Destructive interference occurs where troughs intersect peaks. Figures A. 5(a) and

(b) show that as the angle 0 between the two wave trains increases, the spacing between the fringes decreases.

The wave tTain perpendicular to the film surface can be called the reference beam and the other train may be called the signal. If the reference beam were not present the signal would uniformly expose the film; with the reference beam present the film records information about the angle of the signal beam. The information recordod cannot, however, distinguish between the signal of Figure A.5(c)

and that of Figure A.5(d): the two signals produce the

same interference pattern. ( C ) ( d )

Figure A.5 Reference beam and signal arriving at the film, (a) and (b) show signals at different angles; (c) and Cd) show two signals at the same angle but different directions. 176

The above discussion assumes that the two beams arc ‘ of the same wavelength and come from a single source of waves; otherwise each wave train would be a superposition of randomly emitted wave trains from many small sources.

Any single source of waves is made up of many atomic sources of waves and these must act in unison (i.e. coherently) to produce usable wave trains.

If the film is developed and the resulting trans parency is illuminated with a train of coherent waves, the interference pattern acts as a diffraction grating: part of the incident wave is undiffractcd and travels straight ahead, forming the zero-th order beam; two first- order beams are diffracted at the angle 0 as shown in

Figure A.6. In effect, the transparency reproduces the two beams of Figure A.5(c) used to expose the film and also another beam corresponding to the possible situation of Figure A.5(d).

A true diffraction grating with sharply defined edges would give rise to higher order diffraction beams.

However, because of the sinusoidal intensity variation of the interference fringes, only the zero-th and first order beams are formed.

Notice that it makes no difference whether a positive or a negative transparency of the film is used. 177

first order beam

incident beam O-th order reference beam

first order beam (signal)transparency

Figure A.6 The diffracted beams 178

Consider now a point scatterer S of waves at d as

shown in Figure A.7 (a). At the film the unscattered plane waves act as a reference beam and record information

about the signal which is the spherical waves scattered

by S. Interference fringes are formed on the film as in

Figure A.7(b). The fringe spacing decreases moving

away from the center of the pattern as the angle 6 between

the two interfering wavefronts increases. It is easy

to show that the radius of the n-th fringe is proportional

to /n.

When the transparency in this case is exposed to

coherent waves, the zero-th order beam goes straight

through as before. The first order beam corresponding to

the original signal leaves the transparency at such an

angle that it appears to come from a point at a dis

tance d behind the transparency. The point is the

point where the scatterer was when the film was exposed.

Hence a virtual image of the scatterer is seen through the

transparency. The other first-order beam is diffracted to

a point S2 in front of- the transparency. Since all fringes

on the transparency diffract waves to this point, the waves

converging at S2 form a real image of the original

scattering object. See Figure A.8.

A small object may be considered as a collection of

point scatters. The interference pattern formed on the 179

(a)

(b)

Figure A.7 Hologram of a point scatterer. (a) waves interfering at the film plate; (b) the holographic pattern. 180

first order (signal)

0-th order ^ (reference)

Figure A.8 Reconstructing a point image. 181

film may be an elaborate superposition of many such over- ‘

lapping circular fringes. In fact, for a complex object, even microscopic observation of the patterns of intensity variation in the transparency may reveal no clearly observ

able fringe pattern. Such a transparency is called a

hologram. When illuminated with coherent light the holo

gram forms a real and a virtual image of the original

object. The images formed by holograms are three-dimensional

since the position of any point in the image varies

according to the position of the original object with

respect to the film. If the scatterer S is far from the

recording film, the virtual image of S is formed far

behind the illuminated hologram, and vice versa.

Notice that any small segment of the hologram (so long

as the segment is much larger than the fringe spacing) can

produce the real and virtual image of S. But the image

produced by a segment of the hologram has less intensity

than that produced by the whole hologram.

A.2.2 A History of Holography

Dennis Gabor invented holography in 1947 in an

attempt to remove the spherical aberration of the images

formed by an electron microscope. Tho interference pattern

between a plane electron beam and a scattered electron beam

is recorded, but the scattered beam is not spherically 182 symmetric due to the aberration introduced by electron lenses. A transparence of the pattern is illuminated with coherent light and an image of the original object is formed. During the image formation an optical lens having a spherical aberration opposite to that of the electron lens produces an undistorted image of the original object.

Holography was only of limited success in improving the resolution capabilities of the electron microscope.

The main problem with the method was the deterioration of image quality caused by the out-of-focus real image being superimposed on the virtual image. During the early

1950's, Baez attempted to use the holographic method to form images with soft x-rays (62), and several other workers tried to separate the two overlapping images (63).

None of the early efforts to remove the extraneous image were very successful. The effort most worthy of mention was a spatial filtering technique which blocked out the unwanted image but serious degraded the quality of the remaining image (64).

One of the first successful applications of the holo graphic principle was the classified work on "side-looking radar" carried out at the University of Michigan beginning in 1956 (65). As an airplane flies past an object, the 183 interference pattern between a radar beam reflected from the object and a local oscillator is recorded. The pattern of maxima and minima recorded is a one-dimensional holo gram of the object: the pattern may be duplicated on a photographic transparency and a good image of the object

is obtained when the transparency is illuminated with coherent light from a mercury arc lamp. Hence up until

1960 all the work on holography consisted of either

(a) forming an interference pattern with electrons or

X-rays and reconstructing the image with visible light

having a much larger wavelength; or • (b) forming an inter

ference pattern with radar and reconstructing with visible

light having a much shorter wavelength.

One of the members of the group which developed this

technique was Emmett Leith who, along with Juris Upatnieks,

established holography as a major field of purely optical

research. Between 1961 and 1966, Leith and Upatnieks at

the University of Michigan overcame all the previous

stumbling blocks of holography. In 1961 they considered

holography from a communication theory viewpoint (66):

the photographic film is a square-law detection device

receiving a modulated carrier frequency which has under

gone a frequency dispersion; during the reconstruction,

an inverse dispersion operation is carried out in which the carrier frequency (zero-th order beam) is separated from . the upper and lower side bands (the real and virtual

images). Using the communication theory viewpoint, they determined the method of successfully separating the two

images, as shown in Figure A.9. The reference beam is made to meet the beam diffracted by the object at an angle

0 during the hologram formation. During the image recon

struction the two images will be separated by an angle

20. In 1962, Leith and Upatnieks introduced the laser as

the source of coherent light for holography. Up to this

time holograms had been made only of small scatterers mounted on a transparent background; in 1963 Leith and

Upatnieks formed holograms of transparent objects (e.g.

letters of the alphabet) on an opaque background, and also

developed the ability to form holograms of continuous -

tone objects, such as an ordinary black and white photo

graphic transparency (67). In 1964, "diffused" holograms

were first made by the same researchers; instead of

simply allowing light to be scattered by semi-transparent

objects, they formed holograms by recording the inter

ference pattern between the reference beam and the light

diffusely reflected from opaque objects, as shown in

Figure A.10. 185

incident beam prism

reference ^ beam

signal film

transparent object

virtual image incident beam 0-th order beam

hologram real image (b)

Figure A.9 Hologram of a transparent object. (a) Making the hologram; (b) reconstruction. 186

opaque incident beam A objects /// signal mirror

reference beam

film

Figure A.10 Hologram of an opaque object. Reconstruction is the same as in Figure A.9 (b). 187

After 1965, progress in laser technology and holo graphic image quality was so rapid and the applications of holography so many that it is not possible to cover them concisely. In 1966 an excellent summary of the differnt basic holographic techniques along with their advantages and disadvantages for various applications was given by G. L. Rogers (68).

A. 3 The Relationship of Fresnel Zone Plates to Holograms

Shortly after Gabor's original work on "the theory of reconstructed wavefronts," G. L. Rogers noted that the interference pattern formed by the waves from a point scatterer and a coaxial reference beam resembled the pattern of opaque and transparent regions of a Fresnel zone plate (69). If p^ is the radius of the first dark ring of the pattern then /n p^ is the radius of the n-th ring. The rings are alternately light and dark. Since any object may be considered as a large number of small

light scatterers, a hologram may be considered a large number of overlapping zone plates. Conversely, the zone plate may be considered a special kind of hologram. The

three-dimensional character of holograms and the existence of the two conjugate images is easily understood once the hologram is conceived of as a multiple-zone-plate (70). 188

When coherent light falls on the hologram, each zone plate forms one real and one virtual image point: the super position of all the virtual image points makes up the virtual image of the hologram, and similarly the real holographic image is formed by superposition of the many real image points.

The analogy between Fresnel zone plates and holograms is far from exact, however, because ideal zone plates consist of perfectly opaque and perfectly transparent regions with sharp edges, resulting in an infinite number of higher order images, both real and virtual, as wc have seen. The hologram of a point source consists ideally of a sinusoidal variation of intensity from one fringe to another; this results in only first order images. This ideal hologram of a point source has been called a Gabor zone plate by Horman and Chau (71). An actual hologram of a point source does not produce an exact sinusoidal film exposure because the transmittivity of exposed film does not vary linearly with the intensity of the light exposing the film. The actual hologram of a point source has been termed a binary zone plate; when illuminated with coheront light it also gives rise to multiple images.

The conceptual understanding of holograms as multiple zone-plates has made possible the theoretical calculation of the intensity variation patterns of the holograms of 189 some simple three-dimensional, geometrical objects. When' these patterns are plotted, photoreduced and illuminated with laser light, they give rise to the three-dimensional image of the intended object. The patterns have been plotted by computer point-by-point (72) (each point being fully opaque or fully transparent) and have also been created by sequential printing of overlapping zone plate patterns on a photographic plate (73). These studies were begun in the late 1960's.

The similarities and differences in the mathematics used to describe Fresnel and Gabor zone plates has been the subject of some discussion (7£,75), and the discussion has led to a new way of describing the efficiency of the

Fresnel zone plate for imaging a point source (76). A

good summary article on the relation of zone plates

and holograms is given by Ferrier (77). APPENDIX B

THEORY OF ZONE PLATE IMAGING OF GAMMA RAYS

The theory of zone plate imaging of gamma rays has

been developed primarily by Barrett (78), and is summar

ized here.

Consider a source distribution given by d(r), where

r is the position vector in the source plane. The

Fresnel zone plate is located at a distance z » s^ from

the source: coordinates in this plane are denoted by r'.

Let the transparency of the zone plate be denoted by

g(ir'). The detector plane is located at z « s^ + S2 » and

its coordinates are denoted by r". •+■ 2 -f Let h(r") d r" be the average number of gamma rays 2+ falling on a resolution element d rM in the detector

plane during time t . By a straightforward but tedious

application of geometric shadowcasting principles-, Barrett * shows that h(rM) is given by

h(?") - f (sl * s2 ' *)'ZS(r’)

Using Fourier transform techniques it is easily shown that

the source distribtuion is

190 191

d(r) » ^ exp[-2ni(ig.r)ld2l (B-2) c G(-f/b) where

a » Sj/CSj + s2) (B-3)

c = ( t / 4 it)/(S j + s2)2 (B-5)

and H and G are the Fourier transforms of h and g respec tively, and ? is the two-dimensional spatial frequency.

Equation B-2 shows that the source distribtuion can be recovered from a knowledge of h and g, provided that the

Fourier transform of g is nonzero for all spatial frequen’ cies contained in d(r).

For an on-axis Fresnel zone plate, g(r) is given by

1 , if sinOrr2/!^2) > 0

g(r) CB-6) 0 , if sin(Trr2/Rj2) < 0 where r * |r|, and Rj is the radius of the first Fresnel zone. The Fourier transform of g is given by 192

G(f) - I «(?) + Y, ( - T - ) ld2r e*p[2»i pCodd)'2* P / J

(?-r + j pr2/ ^ 2)] (B-7)

where 6 is the delta function and p is a dummy summation

index. Computer calculation of G(t) shows that its absolute value is reasonably constant and nonzero for all 2 frequencies up to a cutoff frequency given by fc B » where is the radius of the zone plate having N zones.

Beyond the cutoff frequency |G(?)| falls quickly to zero.

Maximum contrast occurs in the image when the scaled

spatial frequency of the source matches the cutoff fre

quency of the zone plate, that is,

f - — f . (B-8) source Sj c v *

Barrett also considered the effect of quantum noise

in Fresnel zone plate imaging (79). The only noise source

considered was the statistical variation of the source

radiation emission. The signal-to-noise ratio (SNR) in

the reconstructed image was found to depend on the size

and shape of the source.

A single point source will produce a single zone plate

shadow. A randomly sampled zone plate will, on the average, 193 focus as well as a full zone plate. However, noise repre sents deviations from average behavior. The ratio of the mean amplitude of the reconstruction wave at the image point to its standard deviation is given by

where is the number of gamma rays detected coming from

the point source.

For a collection of M point sources, the emissions

from each source are statistically independent. The mean

and variance at each image point arc then additive. Let

Nm denote the number of gamma rays detected from the m-th

source and denote the total number from all M sources.

Then the ratio of mean amplitude to standard deviation at

the corresponding m-th reconstruction point is

(B-10)

if all point sources are of equal intensity and all cast

full zone plate shadows on the detector, then this result

reduces to 194

(B-11)

where, on the average, Ramma rays are detected

from each point source.

In experimental work the observed quantity is not amplitude but intensity. The intensity SNR is given in each case above by

. < IAI > Z/(AA)2

For the case of M equal point sources, this reduces to

(B-13)

For purposes of comparison, the SNR for a pinhole

aperture with any source distribution is ^ N y ^ 1^2- For

a single point source, as an example, if the open area

of a zone plate is 500 times that for a pinhole, then for

equal exposure times, the zone plate SNR will be about

10 times better than that for the pinhole. On the other

hand, for a fixed SNR in both cases, the pinhole requires

an exposure time a factor of 100 greater than that for the

zone plate. However, for a source made up of M resolution ele ments, while the pinhole SNR remains constant, the zone -1/2 J plate SNR is reduced by a factor of M ' .

These results apply to both on-axis and off-axis zone plates. A more extensive study of the effect of quantum noise in off-axis zone plate systems has been reported by Joy and Houle (80),(81). APPENDIX C

THE GERMANIUM GAMMA CAMERA

The imaging system considered here employs a high purity germanium gamma camera. A camera as large as the

one described in this appendix has not been constructed

at the present time, but several developmental efforts

are underway. The camera design used in this study is

based on developmental research in our laboratories in

the Nuclear Engineering Department at The Ohio State

University. The camera consists of an array of orthogo

nal strip detectors with associated electronics and

mechanical equipment. A detailed description of this

design can be found elsewhere (82); an overview of the

design with emphasis on those features most relevant

for Fresnel zone plate imaging is presented here.

C.l The Orthogonal Strip Detector

The detection system is made up of an array of

orthogonal strip detectors, each of which is cut from a

single crystal of high purity germanium. The detector is

fabricated (85) from p-type germanium by depositing an

196 197 n-type lithium contact on one face and a p-type palladium * contact on the opposite face of a rectangular planar crystal. Electrically isolated electrode strips are formed by cutting grooves into each face of the crystal.

By cutting the strips on one face orthogonal to those on the opposite face, as shown in Figure C.l, an array of isolated detector elements is formed. Each strip is connected to a resistor network having a charge sensitive preamplifier at each end. This resistor network elec

tronically identifies the detector location of the burst of charge released by a gamma ray interaction: a charge pulse emerges from the electrode strip on each face nearest to the interaction site and divides into the

resistor network according to its location along that

resistor network. The amount of charge, or Q2» arriving

at either end of the upper network decreases proportion

ately with an increase in the distance of the interaction

from that end of the network. Similarly, Q3 and fix

the position of the interaction in the orthogonal direction.

The energy of the incident gamma ray is proportional to

the total charge released, + Q2*

The spatial resolution attainable by this detector

is limited by the minimum center-to-center spacing of the

electrode strips. Because of the practical problems of

accurately mounting the detector strips onto metal contacts 198

CHARGE* SPLITTING RESISTOR NETWORK

ELECTRODE STRIPS (n-TYPE) ^

p-TYPE r HIGH PURITY GERMANIUM

ELECTRODE STRIPS fp-TYPE)

Figure C.l Schematic of the orthogonal strip, charge splitting detector. 199

leading to the resistor networks, the minimum strip spacing achievable at present is considered to be about 1 mm (84).

The design considered here has a strip spacing of 1.2 mm.

C.2 Detector Array

In addition to the constraint imposed by the minimum

strip spacing, additional constraints are imposed by the

limited size of available germanium crystals and by the

cost of these crystals. The largest square detector that

can be cut from available high purity germanium crystals

is 3.78 cm on a side. Because the surface area of such

a detector is so small, an array of these detectors is

needed to achieve the capability of imaging the organs

of interest in nuclear medicine. However, the present

high cost of ultrapure germanium ($10-$18 per gram)

limits the number of detectors in the array.

The camera is designed to have 16 detectors in a

four-by-four square array, as shown in Figure C.2. Each

detector has 32 strips on each face, and hence the complete

array contains 128 by 128 resolution elements. It is

important that the center-to-center spacing of neighboring

strips on adjacent detectors be the same as that spacing

for neighboring strips on an individual detector. If it

is not the same and a conventional collimator is used,

then bright or dark lines will appear on the image 200 >. /— IU-I/Z10-1/2" DDIA. I* (USABLE)

\ ' \

12fl strips

l.490H (3.70 cm) \

15.3 cm . * .

AREA" 36 In2 ( 2 3 2 cm2 )

Figure C.2 Illustration of the 4 x 4 detector array. The array is shown inside of the usable area (dashed circle) of a scintillation camera. 201 corresponding to the detector borders; but when a Fresnel'

zone plate aperture is used, the result will be a decrease in the resolution for all points in the image.

The dashed circle of Figure C.2 indicates the usable area of the standard scintillation camera, 86 square

inches; the germanium camera area is 36 square inches.

When used with a conventional collimator, the germanium

camera can be used to image almost the entire vertex view

of the brain and somewhat greater than half of the posterior, anterior, or lateral views; about half of one

lung and one-third of the liver can also be imaged.

Each detector in the array has a thickness of 2 cm.

This thickness represents a compromise between high

detector efficiency and minimum germanium cost. For

140 keV photons the detector efficiency is 86%, whereas

the standard scintillation camera efficiency is about 95%.

However, for gamma ray energies above 500 keV this ger

manium camera is more efficient than the scintillation

camera.

C.3 Electronics

The array of 16 high purity germanium detectors is

divided into four electronically separate quadrants, each

containing four detectors. The four detectors in each 202 quadrant are interconnected electronically forming, in effect, a single detector having a single charge splitting resistor network for each of the two sets of orthogonal strips.

A complete set of filtering electronics is provided for each quadrant of the detector. The charge pulses from each quadrant, Q^, (J2, and as shown in Figure

C.l, are received by the instrumentation diagrammed in

Figure C.3. Each charge pulse is integrated by a separate preamplifier resulting in proportional voltage outputs

V V2, V3 and V^, respectively. A summing Gaussian filter adds and V 2 , and puts out a Gaussian shaped pulse having a peak value proportional to the energy of the incident gamma ray. If the energy is outside the preset energy window, the system is cleared and reset to receive another gamma event. If the energy is inside the window, the summed pulse is stored in the peak detector for future processing. A summation Gaussian trapezoidal filter puts out a pulse having a peak value proportional to - V2, and hence proportional to the x-position of the gamma event. Another such filter finds Vj - and defines the y-position.

After the peak value of each pulse is determined, the control circuit is designed to detect and reject HIGH VOLTAGE

ANTISYMMETRIC SUMMATION GAUSSIAN X - CHANNEL PREAMPLIFIER TRAPEZOIDAL DRIVER FILTER PEAK DETECTOR

PREAMPLIFIER SUMMING ENERGY CHANNEL GAUSSIAN FILTER DRIVER ENERGY DISCRIMINATOR PREAMPLIFIER ANTISYMMETRIC SUMMATION GAUSSIAN Y - CHANNEL TRAPEZOIDAL DRIVER FILTER CONTROL PILE-UP AND TO PROCESSOR PREAMPLIFIER OVERLAP REJECTION RESET T r OM PROCESSOR

Figure C.3 Schematic diagram of the amplifier, noise filtering and control circuitry

required to process signals Q., Q-, Q3 and Q* emerging from one quadrant 203 of the detector array. 204 two or more gamma events occurring so close in time that their individual energy and positional information is not accurate. The complete camera has a count rate capability of 100,000 counts per second.

The time constants selected for the filtering systems determine the energy and spatial resolutions of the

system. For time constants of interest, of the order of a microsecond, energy resolution is degraded as spatial

resolution is improved. The goal for system resolutions,

full width at half maximum, for 140 keV gamma rays is

2.4 mm spatial• resolution and 3.8 keV energy resolution (8j>).

C.4 Mechanical Components

The mechanical components of the camera consist of

a camera assembly, a cryogenic system and a vacuum system.

The camera assembly consists of a camera stand

supporting the camera head and adjustable counterweights

used to balance aperture systems of different weights.

Figure C.4 shows this camera assembly with no aperture

attached. The camera head, shown in cutaway view in

Figure C.5, contains the detector array, preamplifier

assembly, a cooling unit and a vacuum chamber maintained

by a vacuum pump. The lower vacuum chamber plate is a

thin aluminum window directly below the detector array. CAMERA HEAD R01ATI0N BRAKE

ADJUSTABLE COUNTER-WEIGHT

TWO-AXIS GIMBAL

VERTICAL BRAKE 205

Figure C.4 The camera assembly consisting of camera stand, camera head,and adjustable counter-weight. 206

ROUOHINO PORT

CLOSED CYCLE 30 L/S REFRIQERATION UNIT ION PUMP

JEt

PREAMPLIFIER MOUNTIN'! FLANGES SECTION FOR CAMERA STAND UPPER VACUUM 'CHAMSER PLATE

k w A W W S W A 1 ALUMINUM NINDOW

Figure C.5 Cutaway view of the camera head showing the the detector array, the preamplifier section, the cooling devices and vacuum equipment. 207

The cryogenic system maintains the detector array, the resistor networks and the preamplifier FET's at a temperature of approximately 130°K to minimize electronic noise. This refrigeration system is a closed cycle

system containing an air expander housed in the camera head connected by flexible tubing to a remotely located

compressor, not shown in the figure.

The vacuum system helps to maintain the cold tempera

ture and also prevents contamination of the high purity

germanium detectors. A roughing pump is used to pump down

to 10’^ Torr, and a 30 liter per second ion pump maintains

the chamber pressure at 10’** Torr or less. APPENDIX D

THEORY OF DIGITAL RECONSTRUCTION OF SHADOWGRAMS

Gamma rays from a distributed source of Tadintion passing through a Fresnel zone plate aperture cast a shadow pattern on the detector which is a hologram of the source distribution. The resulting shadowcast hologram, like the interference hologram, contains coded information about the source distribution. The image of the source can be reconstructed by passing a coherent beam of laser light through a transparency of the hologram. The image is the resulting diffraction pattern. The difficulty with this optical method has been the obscuring of the diffraction pattern by the undiffracted beam of light.

A solution to this difficulty is to calculate the diffraction pattern digitally. The theory of digital reconstruction is derived here.

D.l The Diffraction Integral

Suppose coherent radiation passes through a shadow gram of area S surrounded by an opaque screen, as shown

208 209

in FigUTe D.l. In the following development a vector valued function will be denoted by an arrow. Let the

electric field vector at any point P in the electromagnetic

field be given by

U(P) = U0(?)exp[-i

where the real valued function UQ (P) is the wave amplitude,

u is a constant unit vector, and <|>(P) is the phase.

Let UCPj) be the radiation distribution in the plane

of the shadowgram. The radiation amplitude and phase

at a point P is given by (86J)

1 UCPJ IT j j 9 ( ? l ) cos(n,rQ1)dS (D-2)

where X is the radiation wavelength, i is the unit imagi

nary number, n is the unit normal to the surface of the

volume in which the radiation propagates, and

and CD-3)

r01 ’ l?0ll 210

shadowgram

opaque screen

Figure D.l Coherent radiation incident on a shadowgram.

i 211

Several approximations can be made to change

Equation D-2 to a more readily usable form. Figure D.2 illustrates the reconstruction geometry for an illumi nated shadowgram in the x ^ t y^^ plane forming an image in the X q , yg plane. Let the distance between these two planes* be z. Let a^ be the maximum distance of a point

in the shadowgram from the origin in the x^, yj plane,

and let a^ be the maximum distance from the origin

in the Xq , y^ plane at which the image intensity is to be calculated. If rfll >> and rQ1 >> a^, then the

following approximations are valid;

cos (n, rQ1) = 1 (D-4)

r01 “ 2

Assume the shadowgram is illuminated by a plane wave.

Then all points in the shadowgram plane have the same

phase, taken arbitrarily to be zero, and hence

u(px) « ■ V xi*yi5" * (D_5)

Equation D-2 becomes

“ (x0 ,y0) " Uz JJ Uo(xX*yl) exp[2irir01/X]dx1dy1 . (D-6) Figure D.2 The reconstruction geometry 212 213

Notice that in the phase term cannot be replaced by z because it is divided by a small number, X. Hence small variations in r ^ produce large phase changes.

D.2 The Fresnel Approximation

The Fresnel approximation is often used to simplify diffraction calculations. It consists of replacing the spherical wavefront expanding from every point toward

Pq by a parabolic wavefront whose axis is the z axis.

The approximation is valid if Pj and Pq are close to the origins of their respective planes and if z is suffi ciently large. The value rni can be expressed as

Using the approximation that for e << 1,

(1 + e)1/2 = 1 + j e - g-e2+ ...

we obtain Substituting this expression into Equation D-6 yields

®cx0,y0> - / / V W * p | &

[(Xi - x0)2 + (y1 - y0)2]{ dxldyl (D'7)

This approximation is valid provided the next term in the expansion has a value much less than one radian, that is,

^l[(^)4*(^)4]«d c d -.j

Substituting a^ for x^ and y^, and a^ for Xq and yg, and rearranging, we obtain

^ (flj * a0)4 « zJ CD-9)

Assuming a factor of 5 means "much greater than," a minimum value for z is obtained for which the Fresnel approximation is valid. 215

To check whether Equation D-7 is valid for a typical holographic reconstruction, the. following typical values are substituted into Equation D-10. For a helium-neon laser, X = 0.6328 microns; and typicall a^ and aQ arc a centimeter; hence zm^n is about 17 cm. Since z is typically

100 cm, the Fresnel approximation is valid for holographic image reconstruction.

The shadowgram is a superposition of Fresnel zone plate patterns. In order to focus a plane wave at a distance z the focal length of the zone plate pattern

(sec Equation 3-3) must be equal to z, that is,

Z “ -j±-Ri (D-ll)

where R^ is the radius of the central Fresnel zone. When an on-axis zone plate pattern is used, the shadow pattern must be smaller than the dimensions of the shadowgram transparency. The smallest practical zone plate is one with four zones, that is two transparent zones. For this pattern to exist .within the dimensions of the hologram, the condition

2Rj < ax (D-12)

must hold. This condition places an upper bound on z, namely, 216

CD-13) zmax * VT

Because in practice a^ and ag are approximately equal,

let us replace both by the symbol a. Combining Equations

D-10 and D-13, we obtain

2.52 a4/3 X_1/3 CD-14)

This expression defines the limits on z values for which

the Fresnel approximation is useful for the calculation

of the image.

D.3 The Fraunhofer Approximation

The Fraunhofer approximation is often used to simplify

diffraction integrals, but it cannot be used for the

digital reconstruction of shadowgrams. The Fraunhofer

assumption is

<< z CD-15)

or equivalently

CD-16) 217

Because Equations D-13 and D-16 cannot both be satisfied simultaneously, the Fraunhofer approximation cannot be used for digital reconstruction of shadowgrams.

D.4 The Reconstructed Image as a Fourier Transform

Equation D-7 is a special case of a two-dimensional

Fourier transform. To see this, introduce a new unit of length, , defined by

(D-17)

Substituting this expression into Equation D-7 makes it independent of X and z and dependent only on their product.

The plane of reconstruction is now specified by the choice of Rj. The equation can be simplified by introducing the following dimensionlcss variables. The dimensionless image coordinates are

and (D-18)

The dimensionless shadowgram coordinates are

(D-19) 218

Then Equation D-7 becomes

8(x,y) = “°»Pj2»*?-/.U . / / u0 (£x ,« ).«p j «i [ffx - x)2 s (

+ (fy ' y)2] I dfxd£y • (D'20)

The intensity, I, of optical radiation detected (by the eye or by film) at a point in the image plane varies as

the square of the amplitude. That is,

I(x,y) - U(x,y)fi*Cx,y) (D-21)

where the asterisk denotes the complex conjugate. There

fore, the phase factor in front of the integral does not

contribute to the image intensity, and it will be ignored.

Expanding the exponential terms of Equation D-20 gives

fi(x,y) » u f f U0 (fx »fy)exp|ni(fx2 ♦ fy2 ) Jexp £ iri

(x2 + y2)J exp [-2ni(fxx + fyy) J d f xdfy (D-22) 219

The phase factor depending only on x and y can be brought * outside the integral; hence it will not contribute to the intensity and can be ignored. Define a new complex amplitude in the shadowgram plane to be

X(fx.fy) - U0Cfx,fy)exp[iil(fx2 + fy2 )]“ (D‘2S)

Then Equation D-22 becomes

S(x.y) - / J X(fx ,fy)exp[-2»i(fxx ♦ fyy)] dfxdfy . CD-24) s

Equation D-24 shows that the complex amplitude in the image plane is the two-dimensional Fourier transform of the complex amplitude in the shadowgram plane. With this

in mind, the shadowgram may be thought of as a spatial

frequency plane (f„»fv); and the image, which is actually x y given by the square of the amplitude, may be considered

a Fourier transform of the shadowgram.

D.5 Application of Sampling Theory

Sampling theory is a familiar part of communications

theory concerned with transforming digitized information

back and forth between the frequency and time domains 220 with minimal loss of information and minimal introduction of noise. The fundamental theorem of sampling theory is that a continuous waveform having no frequency content higher than F can be completely recovered from the inter polation of data sampled at a frequency greater than or equal to 2F.

Consider the spatial frequency content of a shadowgram

recorded by a Fresnel zone plate on an orthogonal strip

gamma ray camera. The output of the camera is digital:

the number of counts per detector resoltuion element is

determined. The detector is a square of side with

spatial resolution AXj and contains Nj by Nj resolution

elements. The shadowgram can contain no spatial frequency

higher than one cycle per pair of resolution elements:

then alternate resolution elements are irradiated and the

intervening ones are not. Then the highest frequency

resolvable in the shadowgram is

F - ■ (D-25>

In dimensionless units, the dimensionless spatial resolu

tion element in the shadowgram plane is

Ax Af B h . (D-26) K1 221

Now F is given by

F * • (D‘27)

We see that the sampling rate, or a number of samples per unit distance, is 1/CAfRj); and the sampling rate is double the highest frequency, F.

Consider next the spatial frequency content of the

image reconstructed from this shadowgram. According to

sampling theory, if samples of a frequency spectrum

are taken at intervals Af, up to a maximum frequency

F » NjAf, then the Fourier transform of the frequency

spectrum gives the waveform in the time domain with time

resolution At “ 1/F for a total time T ■ NAt. Let us

apply this theorem to the case of the orthogonal strip

detector. Equation D-24 gives the spatial distribution

of the image amplitude. It follows that the quantities

f and f in that equation must represent the spatial x y frequency content of the image in orthogonal directions.

Hence the quantity Af of F.quation D-26, which is the

dimensionless spatial resolution in the shadowgram plane,

is numerically equal to the dimensionless spatial frequency

resolution in the image plane. Therefore, by the sampling

theorem stated above,the dimensionless spatial resolution

in the image plane is or, in terms of dimensioned parameters,

(D-29)

The dimensioned spatial resolution of the image is given by

AXq * RjAx (D-30)

By substituting from Equation D-29 and using the relation

D1 * N xAxi» we obtain

CD-31) Ax0 " TJJ- •

This equation shows that the size of the image resolution element depends only on the shadowgram size. We show next that the size of the image depends only on the size of the shadowgram resolution element. The size of the

image is

CD-32)

Substituting Equations D-29 and D-30 gives 223

(D-33)

The ratio of image size to shadowgram size is

2 (D-34)

D.6 Digitizing the Reconstruction Integral

The integration in Equation D-24 must be replaced

by summation for digital reconstruction. For purposes of

numerical integration it is convenient to begin with the

integrand in the form given in Equation D-20. In this

integrand we see that the phase difference for coherent

radiation from the origin of the shadowgram plane to the

origin in the image plane is zero. It is necessary to

preserve this relation to avoid distortion of the image

in digital reconstruction, and so the values of f , f , x y x, and y should be zero at the ori'gin. Because computer

FORTRAN does not allow negative subscripts in an array,

which are normally used to denote points left of center,

an offset parameter s is introduced, defined to be the

number (not necessarily an integer) of resolution elements

between the center of the shadowgram and the center of the

outermost resolution element, that is, 224

N, + 1 s » -±~2---- . (D-35)

It is assumed that the resolution in the shadowgram plane, Af, is the same in the f and f directions and x y that the resolution in the image plane, Ax, is the same in the x and y directions. In order to digitize Equation

D-20, the following substitutions are used:

f ■ (m - s)Af ; m e 1, 2, ..., Nj

fy - (n - s)Af ; n » 1, 2, ..., Nx

(D-36) x ■ (j s)Ax i j a l, 2, ...,

y » (k - s)Ax ; k » 1, 2, ..., Nx

Ignoring phase factors outside the integral, Equation

D-20 is written as a double summation as follows

* 0 .1 0 - 5 : i uU0 (m,n)exp(i, is given by

■ irj£(m - s)Af - (j - s)AxJ2 + £(n - s)Af - (k - s)AxJ2 j

(D-38) 225

When this expression is expanded, terns not containing m or n nay be factored outside the double summation. Phase

factors outside the double sum may be ignored since they will not contribute to the radiation intensity; then the

remaining terms are

* - "Tcm2 + n 2) (Af)2 ♦ 2s(m + n) (ji- - CAf)2)J

(jm + kn) CD-39)

Define the digitized complex shadowgram amplitude to be

A(m,n) » uUQ(m,n)cxp iir[Cm2 ♦ n2) (Af)2

+ 2s(m ♦ n) - (Af)2 )] j . CD-40)

Then the digitized complex image amplitude is given by

Ni Ni U(j.k) = Y , Y . ^(m,n)exp I" - O m ♦ kn) 1 . (D-41) mal n*»l L 1 J 226

The digitized complex image amplitude is a two-dimensional* discrete Fourier transform (DFT) of the digitized complex shadowgram amplitude. The intensity of a point in the image plane is given by

I(jJO ■ U(jtk)u*(j,k) (D-42)

Hence the Nj by array of shadowgram intensity values,

U(m,n), is transformed into the by array of image intensity values, I(j,k).

D.7 Reconstructing the True Image

The model described above relies on the focussing ability of shadowgrams to form the real, conjugate image on a screen in the image plane. In addition to this conju gate image, the virtual, true image also exists in the diffraction pattern but requires a converging lens to focus it to a real image on a screen. Assume the shadowgram illuninated with a plane coherent beam of radiation forms the conjugate image at a distance, z. It is easily shown by geometric optics that placing a converging lens of focal length z/2 immediately after the shadowgram results in forming the true image at the same distance, z. 227

The effect of this lens, essentially in the shadow gram plane, is to modify the complex shadowgram amplitude with the phase factor Cxx * 8*ven by (®Z.)

UfcCxi.Xi) = exp[- t ^ ( x j 2 ♦ yi2 )] (D-43)

When substitutions are made using Equations D-17, D-19 and

D-36, and when this factor is included in Equation D-41,

the net result is merely to change the sign of the first

term in the exponential function of Equation D-40. That

is, the complex shadowgram amplitude becomes

A(m,n) ** uU(m,n)exp -iirl(m^ + n^)(Af)^

(D-44)

Equations D-41 and D-42 are unchanged.

Different tomographic planes in the image can be

brought into focus either by changing the position of the

screen or by changing the focal length of the lens. Because

we have related the lens' focal length to the image position, 228 these two alternatives are mathematically equivalent. The focal plane of the image is controlled by the choice of the parameter Af in Equation D-44. BIBLIOGRAPHY

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21. B. Macdonald, et al., "Gamma Ray Imaging Using a Fresnel Zone Plate Aperture, Multiwirc Proportional Chamber Detector, and Computer Reconstruction," IEEE Trans, on Nuc. Sci.. NS-21, Feb, 1974; p. 678.

22. H. H. Barrett, et al., "Apertures, Images and Nuclear Medicine," Optical Spectra, Feb. 1974; p. 31. 231

23. M. H. Farmelant, "Improved Anatomical Definition by a Fresnel Zone Plate Imager," J. Nuclear Medicinet 14, 6; p. 393.

24. H. H. Barrett, et al., "Fresnel Zone Plate Imaging in Radiology and Nuclear Medicine," Optical Engi neering , 1 2 , 8; 1973.

25. J. W. Goodman and R. W. Lawrence, "Digital Image Formation from F.lectronically Detected Holograms," App. Phys. Letters, 11, 3, Aug. 1, 1967; p. 77.

26. Y. Aoki and A. Boivin, "Computer Reconstruction of Images from a Microwave Hologram," Proc. IEEE, May 1970; p. 821.

27. Y. Aoki and Y. Suzuki, "Hologram Conversion by Computer," Proc. IEEE, May 1974; p. 643.

28. B. Macdonald, et al., op. cit.

29. E. 0. Brigham, The Fast Fourier Transform, Prentice- Hall, Inc.; Englewood Cliffs, N.J., 19/^7 p. 25.

30. ibid. , p. 25.

31. J. W. Goodman, Introduction to Fourier Optics, McGraw- Hill, New York, 1075; p. 18.

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33. T. C. Rindfleisch, et al., "Digital Processing of Mariner 6 and 7 Pictures," J. of Geophysical Research, 76 (2), 1971; p. 394.

34. A. W. Lohmann and D. P. Paris, "Space Variant Image Formation," J. Opt. Soc. Am., 55 (8), 196S; p. 1007.

35. "LSI Chip Handles Fourier Transforms," Electronics. March 6, 1976; p. 32.

36. A. H. Deutchman, personal communication, The Ohio State University, 19,75. 232

37. A. H. Deutchman, "Quantitative Radiotracer Imaging; The Development of a Clinically Practical Instru mentation and Analysis System," Ph.D. dissertation, The Ohio State University; June 197S.

38. ibid.

39. Lord Rayleigh, Theory and Sound, Vol. II, New York, Dover, 1945.

40. R. W. Wood, Physical Optics, Macmillan, New York; 1914, second ed., pp. 37-40, 217, 218.

41. C. F. Meyer, The Diffraction of Light, X-Rays and Material Particles, UniiT. of Chicago Press, Chicago, 1934.

42. F. A. Jenkins and H. E. White, Fundamentals of Optics, New York; McGraw-Hill, 1957, p. 360.

43. C. T. Lane, "The Theory of the Zone Plate Derived from Voight's Integral," Canadian Journal of Physics, 2, 26-30, 1930.

44. Ora E. Myers, Jr., "Studies of Transmission Zone Plates," Am. J. Phys. , 19, 359; 1951.

45. Stigliani, Mittra, and Semonin, "Resolving Power of a Zone Plate," J. Opt. Soc. of Am., 57 (5), 1967.

46. A. Boivin, "On the Theory of Diffraction of Concentric Arrays of Ring-Shaped Apertures," J. Opt. Soc. of Am., 42, 60; 1952.

47. A. Boivin, A. Dion and H. Koenig, "Etude Expdrimentale de la Diffraction des Micro-ondes par dcs Ouvcrtures a symdtrie de revolution," Canadian J. Phys., 34, 166; 1956. *“

48. J. Dyson, "Circular and Spiral Diffraction Gratings," Proc. Royal Society (London) Ser. A . 248, 93; 1958.

49. M. V. R. K. Murty, "Spherical Zone Plate Diffraction Grating," J. Opt. Soc. Am., 50, 923; 1969.

50. B. A. Lippman, "Exact Calculation of the Field Due to a Single Fresnel Zone," J. Opt. Soc. of Am., 55, 360; 1965. 233

51. H. M. Childers and D. E. Stone, ’’Solution to the Fresnel Zone Plate Problem” Am. J. Phys., 37, 721; 1969. “

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53. M. Young, "Zone Plates and Their Aberrations,” J. Opt. Soc. of Am., 62, 972; 1972.

54. K. Lehovec and A. Fcdotowsky, Theoretical Studies of Zone Plate Monolithically Integrated with Sensors-; AFCft'tTT2 - 0'49'ffT~l972: ~ 55. A. V. Baez, "Fresnel Zone Plate for Optical Image Formation Using Extreme Ultraviolet and Soft X-Radiation,” J. Opt. Soc. of Am., 51, 405; 1961.

56. M. V. R. K. Murty, "Common Path Interferometry Using Fresnel Zone Plates," J. Opt. Soc. of Am., 53, 568; 1962.

57. G. Boivin and R. Gagnon, "Methods d' Aligncment an Moycn D'une Mire a Foyers Multiples a Phase Inversee," Canadian J. Phys., 49 , 1284; 1971.

58. P. N. Keating, et al., "Fresnel-Zone-Plate Spectrometer with Center Stop," J. Opt. Soc. of Am., 62, 945; 1972. ” ”

59. G. L. Rodgers, "The manufacture of a stepped zone- plate," J. Sci. Instrum., 43, 328; 1966.

60. H. Chau, "Zone Plates Produced Optically," App. Opt., 8, 1209; 1969.

61. B. M. New, "Design, Production and Performance of Circular Fresnel Zone Plates," App. Opt., 10, 498; 1971.

62. A. V. Baez, "A Study in Diffraction Microscopy with Special Reference to X-rays," J. Opt. Soc. of Am., 42, 7S6; 1952.

63. P. Kirkpatrick and H. El-Sum, "Image Formation by Reconstructed Wavefronts, I. Physical Principles and Methods of Refinements," J. Opt. Soc. of Am., 46, 825; 1956. 234

64. A. Lohmann, "Optical Single-Sideband Transmission Applied to the Gabor Microscope," Opt. Acta, 3, 97; 1956.

65. L. Cutrova, E. Leith, C. Palermo and L. Porcello, "Optical Data Provessing and Filtering Systems," IRE Transactions on Information Theory IT-6, 386; i960.

66. E. Leith and J. Upatnieks, "Reconstructed Wavefronts and Communication Theory," J. Opt. Soc. of Am., 52, 1123; 1962. “

67. E. Leith and J. Upatnieks, "Wavefront Reconstruction and Continuous Tone Objects," J. Opt. Soc. of Am., 53, 1377; 1963.

68. G. L. Rogers, "The Design of Experiments for Recording and Reconstructing Three-dimensional Objects in Coherent Light (Holograph)," J. Sci. Instrum.,43, 677; 1966.

69. G. L. Rogers, "Gabor Diffraction Microscopy: the Hologram as a Generalized Zone Plate," Nature, 166, 237; 1950.

70. W. E. Kock, et al., "Holograms and Zone Plates," IEEE Proc. , 19, 1599; 1966.

71. M. H. Ilorman and H. H. Chau, "Zone Plate Theory and Holography," App. Opt., 6, 217; 1967.

72. K. Clifford and G. Waldman, "Comments on Zone Plate Theory Based on Holography," App. Opt.. 6, 1415; 1967.

73. M. H. Ilorman, "Reply to Comments on Zone Plate Theory Based on Holography," App. Opt., 6, 1415; 1967.

74. M. H. Horman, Efficiencies of Zone Plates and Phase Zone Plates," App. Opt., 6, 2011; 1967.

75. W. G. Ferrier, "The Zone Plate and Its Role in Holography," Contemporary Phsyics, 10, 413; 1969.

76. J. P. Waters, "Holographic Image Synthesis Utilizing Theoretical Methods," App. Phys. Letters, 9, 405; 1966.

77. W. J. Siemens-Wapniarski and M, Parker Givens, "The Experimental Production of Synthetic Holograms," App. Opt. , 7_, 535; 1968, 23S

78. H. H. Barrett and F. A. Horrigan, "Theory of Fresnel Zone Plate Imaging of Gamma Rays," Raytheon Technical Memorandum (T-926), 1973.

79. H. H. Barrett and G. D. De Meester, "Quantum Noise in Fresnel Zone Plate Imaging," Raytheon Technical Memorandum (T-972), 1973.

80. M. L, G. Joy and S. Houle, "The Potential Performance of Off-Axis Fresnel Zone Plate Imaging Systems on Arbitrary Objects," IRF.F, Transactions on Nuclear Science, Vol. NS-22, February, 197 5; p. 364.

81. S. Houle and M. L. G. Joy, "Small Signal Suppression in Coded Aperture Imaging in Nuclear Medicine," IEEE Transactions on Nuclear Science, Vol. NS-22, February 197 5; p. 369.

82. M. S. Gerber, "The Analysis and Development of Instru mentation for a Position Sensitive Semiconductor Gamma Ray Camera," Ph.D. dissertation, The Ohio State University; June 1975.

83. I. G. Zubal, "The Fabrication and Analysis of Orthogo nal Strip High Purity Germanium Detectors," M.Sc. thesis, The Ohio State University, August 1974.

84. P. A. Schlosser, personal communication, The Ohio State University, 1975.

85. M. S. Gerber, personal communication, The Ohio State University, 1975.

86. J. W. Goodman, Introduction to Fourier Optics, McGraw- Hill, New York, 1968; p. 45.

87. ibid.. p. 81