Matrix Computations and Optimization for Spectral Computed Tomography Open Access

Abstract

In the area of image science, the emergence of spectral computed tomography (CT) detectors highlights the concept of quantitative imaging, in which not only reconstructed images are offered, but also weights of different materials that compose the object are provided. In this thesis, we focus on optimization, preconditioning and model development of spectral CT. For simple energy discriminating detectors, a nonlinear optimization framework is built on a Poisson likelihood estimator and bound constraints. A nonlinear interior-point trust region method is implemented to compute the solution. For energy-windowed spectral CT, a nonlinear least squares approach is proposed to describe the problem and under bound constraints, a two-step method using the projected line search and the trust region approach, incorporated with an adaptive preconditioner, is used to solve the problem. In addition, a weighted least squares formulation is derived from the Gaussian noise assumption and another preconditioner that is based on rank-1 approximation is introduced to obtain robust reconstruction. The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), along with a projection step, is used to calculate the solution iteratively. Compared with a direct solver, a two-step model is developed using an auxiliary variable. With this two-step model, a row-wise computational method is proposed, which further reduces memory requirements and improves solution accuracy. Numerous numerical experiments are conducted to indicate the strength of methods and real-life examples are presented to show possible applications.

Table of Contents

1 Introduction ... 1

1.1 Contributions of Work ... 2

1.2 Outline of Thesis ... 5

2 X-ray Computed Tomography ... 7

2.1 Physical Background ... 9

2.2 The Beer-Lambert’s Law ... 10

2.3 Classical Reconstruction Methods ... 12

2.3.1 The Radon Transform and the Inverse Radon Transform ... 12

2.3.2 Algebraic Reconstruction Technique ... 15

2.3.3 Statistical Reconstruction Methods ... 18

2.4 Ill-posed Inverse Problems ... 20

3 Nonlinear Optimization for the Energy Integrating Detector Model ... 22

3.1 The Energy Integrating Detector Model ... 24

3.2 Poisson Log-likelihood Function ... 28

3.3 Implementation of Nonlinear Interior Point Trust Region Method ... 33

3.4 Numerical Experiments ... 40

3.4.1 Full Angle Reconstruction ... 43

3.4.2 Limited Angle Reconstruction ... 46

3.5 Conclusions and Remarks ... 48

4 Nonlinear Optimization for Energy-windowed Spectral Computed Tomography ... 50

4.1 The Energy-windowed Spectral CT Model ... 52

4.2 Problem Set-up and Preconditioning ... 56

4.2.1 The Constrained Least Squares Problem ... 56

4.2.2 Preconditioning of the Hessian ... 58

4.3 Optimization and Regularization ... 63

4.3.1 Optimization with the Proposed Preconditioner ... 63

4.3.2 Regularization and Scaling ... 68

4.4 Numerical Experiments ... 70

4.5 Conclusions and Remarks ... 76

5 Preconditioning and Optimization for Energy-windowed Spectral Computed Tomography ... 78

5.1 The Weighted Least Squares Problem ... 81

5.2 Preconditioning and Regularization ... 87

5.2.1 Preconditioning ... 87

5.2.2 Regularization ... 91

5.3 FISTA and Projections ... 93

5.3.1 FISTA ... 93

5.3.2 Lipschitz Constant ... 94

5.3.3 Projections ... 95

5.4 Numerical Experiments ... 97

5.5 Conclusions and Remarks ... 103

6 A Two-Step Method for Energy-windowed Spectral Computed Tomography ... 105

6.1 The Two-step Method ... 107

6.1.1 The Framework of Two-step Model ... 107

6.1.2 A Solution to the Two-step Model ... 110

6.2 The Coupled Method ... 115

6.3 Numerical Experiments ... 119

6.4 Conclusions and Remarks ... 122

7 Conclusions and Future Works ... 124

Bibliography ... 126 

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Applied Mathematics Mathematics
Keyword	Spectral Computed Tomography Image Reconstruction Numerical Optimization Inverse Problems Ill-posedness Numerical Linear Algebra
Committee Chair / Thesis Advisor	Nagy, James G., Emory University
Committee Members	Ruthotto, Lars, Emory University Xi, Yuanzhe, Emory University

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