Inverse problem application in the field of biomedical image reconstruction Restricted; Files Only

Abstract

This thesis addresses the challenges of biomedical image reconstruction, a field that plays a crucial role in medical diagnostics and patient care. This work aims to give a complete explanation of this research topic by exploring the complexities of inverse problems in the field of image reconstruction, which are particularly difficult because of the ill-posedness property, and complicated data acquisition process of biomedical images. Through examining and comparing regularization techniques like ℓ1 and ℓ2 regularization, the thesis provides insights into stabilizing the linear least-square system, and efficiently solving it with different numerical algorithms, such as Linear Conjugate Gradient Descent (CG), and Alternating Direction Method of Multipliers (ADMM). It is helpful for individuals who are new to this topic since it gives them a thorough understanding of it and points them in the direction of more efficient research and problem-solving techniques in this important field of study.

Table of Contents

1 Introduction 1

1.1 Forward and Backward Problem . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Image Data Acquisition Process . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Computed Tomography (CT) . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 MRI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Challenges in Biomedical Image Reconstruction . . . . . . . . . . . . . . . . 13

1.3.1 Types and Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . 13

2 Inverse Problem and Image Reconstruction Techniques 18

2.1 Problem Setup: Image Reconstruction Overview . . . . . . . . . . . . . . . . 18

2.2 Ill-Posedness Illustration with SVD Decomposition . . . . . . . . . . . . . . 19

2.2.1 Sensitivity Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 L2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 L1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.3 Choice of D Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.4 Choice of the Regularization Parameter λ . . . . . . . . . . . . . . . 26

2.3.5 Compare and Contrast between L1 and L2 Regularization . . . . . . 30

3 Comparison Between Different Reconstruction Techniques 34

3.1 Image Processing Performance Metrics . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 MSE-NMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.2 PSNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Filtering Techniques for denoising . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Inverse Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.1 Linear Conjugate Gradient Descent (CG) . . . . . . . . . . . . . . . . 37

3.3.2 ADMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Conclusion 43

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics Statistics Engineering, Biomedical
Keyword	Image Denoising Optimization Regularization Inverse Problem
Committee Chair / Thesis Advisor	Chung, Matthias, Emory University
Committee Members	Ruthotto, Lars, Emory University McAlister, Kevin, Emory University

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