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The Application of Tomographic Reconstruction Techniques to Ill- Conditioned Inverse Problems in Atmospheric Science and Biomedical Imaging

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The Application of Tomographic Reconstruction Techniques to Ill- Conditioned Inverse Problems in Atmospheric Science and Biomedical Imaging Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 12-2012 The Application of Tomographic Reconstruction Techniques to Ill- Conditioned Inverse Problems in Atmospheric Science and Biomedical Imaging Vern Philip Hart II Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Physics Commons Recommended Citation Hart, Vern Philip II, "The Application of Tomographic Reconstruction Techniques to Ill-Conditioned Inverse Problems in Atmospheric Science and Biomedical Imaging" (2012). All Graduate Theses and Dissertations. 1354. https://digitalcommons.usu.edu/etd/1354 This Dissertation is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. THE APPLICATION OF TOMOGRAPHIC RECONSTRUCTION TECHNIQUES TO ILL-CONDITIONED INVERSE PROBLEMS IN ATMOSPHERIC SCIENCE AND BIOMEDICAL IMAGING by Vern Philip Hart II A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Physics Approved: Timothy E. Doyle Ludger Scherliess Major Professor Committee Member James T. Wheeler Joseph V. Koebbe Committee Member Committee Member Eric D. Held Mark R. McLellan Committee Member Vice President for Research and Dean of the School of Graduate Studies UTAH STATE UNIVERSITY Logan, UT 2012 ii Copyright © Vern Philip Hart II 2012 All Rights Reserved iii ABSTRACT The Application of Tomographic Reconstruction Techniques to Ill-Conditioned Inverse Problems in Atmospheric Science and Biomedical Imaging by Vern Philip Hart II, Doctor of Philosophy Utah State University, 2012 Major Professor: Dr. Timothy E. Doyle Department: Physics A methodology is presented for creating tomographic reconstructions from various pro- jection data, and the relevance of the results to applications in atmospheric science and biomedical imaging is analyzed. The fundamental differences between transform and iter- ative methods are described and the properties of the imaging configurations are addressed. The presented results are particularly suited for highly ill-conditioned inverse problems in which the imaging data are restricted as a result of poor angular coverage, limited detec- tor arrays, or insufficient access to an imaging region. The class of reconstruction algo- rithms commonly used in sparse tomography, the algebraic reconstruction techniques, is presented, analyzed, and compared. These algorithms are iterative in nature and their accu- racy depends significantly on the initialization of the algorithm, the so-called initial guess. A considerable amount of research was conducted into novel initialization techniques as a means of improving the accuracy. The main body of this paper is comprised of three smaller papers, which describe the application of the presented methods to atmospheric and iv medical imaging modalities. The first paper details the measurement of mesospheric air- glow emissions at two camera sites operated by Utah State University. Reconstructions of vertical airglow emission profiles are presented, including three-dimensional models of the layer formed using a novel fanning technique. The second paper describes the application of the method to the imaging of polar mesospheric clouds (PMCs) by NASA’s Aeronomy of Ice in the Mesosphere (AIM) satellite. The contrasting elements of straight-line and dif- fusive tomography are also discussed in the context of ill-conditioned imaging problems. A number of developing modalities in medical tomography use near-infrared light, which interacts strongly with biological tissue and results in significant optical scattering. In order to perform tomography on the diffused signal, simulations must be incorporated into the algorithm, which describe the sporadic photon migration. The third paper presents a novel Monte Carlo technique derived from the optical scattering solution for spheroidal particles designed to mimic mitochondria and deformed cell nuclei. Simulated results of optical diffusion are presented. The potential for improving existing imaging modalities through continual development of sparse tomography and optical scattering methods is discussed. (167 pages) v PUBLIC ABSTRACT Vern Philip Hart II Tomography is an imaging technique in which 3D models of objects are created from several 2D projections viewed at different angles. When the number of available projec- tions is limited, the resulting data are said to be sparse. This restriction is often a direct result of the imaging geometry used to acquire the data. One-sided views and a small number of receivers can reduce the range of available projections, which makes the ob- ject more difficult to reconstruct. Approximate solutions to difficult imaging problems can be obtained using a class of iterative algorithms known as the algebraic reconstruc- tion techniques (ARTs). The presented research examines some of the capabilities of these algorithms and results are presented, which demonstrate that under particular initializa- tion conditions, these algorithms are able to accurately reproduce structure in the imaging object. The presented technique is then applied to atmospheric science and biomedical imaging to further explore its capabilities and potential. vi DEDICATION To my loving wife, Janelle vii ACKNOWLEDGMENTS I wish to first and foremost thank my wife, Janelle, whose patience and tireless support have made all of this possible. I am fortunate to have such a wonderful family, my source of joy in all that I do. I have much to be thankful for. I owe an incalculable debt to my advisor, mentor, and friend, Dr. Timothy E. Doyle, who has inspired and made possible all that I have accomplished during my time in graduate school and all that I wish to accomplish throughout my career. Without him I would not be where I am today and would not be going where I am in the future. I would like to thank Dr. Michael J. Taylor, who has provided much of the funding for this project and has been a valuable collaborator and friend. He made this research possible and deserves a significant amount of recognition and praise for the resulting publications. I owe him much and I am grateful for his contributions to my graduate career. I wish to thank all of the faculty and staff in the Physics Department at Utah State University, as well as my fellow students. Each of you contributed in some way to my success and I owe you a great deal of gratitude. I thoroughly enjoyed working with you and will always be thankful for your influence in my life. I would like to thank Karalee Ransom for assisting me so selflessly during my initial enrollment at USU and for continuing to provide me and my family with the help that we needed. I am grateful for the grants and scholarships, which made this research possible. They include: NIH-NCI Grant No. 1 R21 CA131798-01A1, NSF Grant No. ATM 0536876, and the University-based RC Grant and Seeley-Hinckley Scholarship. My experience at USU has been valuable and I will always be thankful for the education I received here. It will always be a part of who I am and I will forever be grateful for it. Vern Philip Hart II viii CONTENTS Page ABSTRACT ....................................................................................................................... iii PUBLIC ABSTRACT ........................................................................................................ v ACKNOWLEDGMENTS ................................................................................................ vii LIST OF FIGURES ........................................................................................................... xi CHAPTER 1 INTRODUCTION ........................................................................................................ 1 2 TOMOGRAPHIC THEORY ........................................................................................ 6 2.1 Transform Methods ................................................................................................ 6 2.1.1 Radon Transforms ............................................................................................ 7 2.1.2 Fourier Transforms ........................................................................................... 9 2.2 Back Projection .................................................................................................... 13 2.3 Matrix Methods .................................................................................................... 14 3 CORRECTION ALGORITHMS ................................................................................ 18 3.1 The Kaczmarz Method ......................................................................................... 18 3.2 Algebraic Reconstruction Techniques .................................................................. 21 3.3 Initialization .......................................................................................................... 24 4 THREE-DIMENSIONAL TOMOGRAPHIC RECONSTRUCTION OF MESO- SPHERIC AIRGLOW STRUCTURES USING TWO-STATION GROUND-BASED IMAGE MEASUREMENTS ...................................................................................... 29 4.1 Introduction .......................................................................................................... 30 4.2 Methods ...............................................................................................................
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