Accelerated Alternating Minimization for X-ray Tomographic Reconstruction Open Access

Abstract

While Computerized Tomography (CT) images can help detect disease such as Covid-19, regular CT machines are large and expensive. Cheaper and more portable machines suffer from errors in geometry acquisition that downgrades CT image quality. The errors in geometry can be represented with parameters in the mathematical model for image reconstruction. To obtain a good image, we formulate a nonlinear least squares problem that simultaneously reconstructs the image and corrects for errors in the geometry parameters. We develop an accelerated alternating minimization scheme to reconstruct the image and geometry parameters.

Table of Contents

1 Introduction 1

1.1 Motivation................................. 1

1.2 Mathematics of Computed Tomography................. 3

1.3 InverseProblem.............................. 6

2 Alternating Minimization Scheme 10

2.1 Block Coordinate Descent ........................ 10

2.2 Linear LeastSquares Problem...................... 13

2.2.1 Normal Equations ......................... 14

2.2.2 QR Factorization and Least Squares Problem . . . . . . . . . 14

2.2.3 Regularization........................... 18

2.2.4 Parameter Choice Methods ................... 24

2.3 The LSQR Algorithm........................... 26

2.3.1 Hybrid LSQR........................... 31

2.4 Nonlinear Least Squares ......................... 32

2.4.1 Quasi-Newton and Gauss-Newton Method . . . . . . . . . . . 33

2.4.2 Implicit Filtering ......................... 36

3 Acceleration Schemes 38

3.1 Accelerated Block Coordinate Descent ................. 38

3.2 Anderson Acceleration .......................... 39

4 Numerical Experiments 42

4.1 BCD Exploiting Separability vs BCD.................. 43

4.2 Number of Angles............................. 46

4.3 Acceleration................................ 48

4.3.1 Anderson acceleration ...................... 49

4.3.2 imABCDS............................. 50

4.3.3 imABCDS and BCDS ...................... 52

4.4 Regularization............................... 53

4.5 Imfil Budget................................ 54

5 Conclusion 56

Bibliography 58

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Keyword	numerical linear algebra tomography medical imaging nonlinear least squares
Committee Chair / Thesis Advisor	James Nagy, Emory University
Committee Members	Gordon Berman , Emory University Yuanzhe Xi, Emory University

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