Region of Interest Image Reconstruction using IR Tools Open Access

Abstract

In this thesis, we consider large-scale, ill-posed inverse problems that arise in image processing applications. These problems can be modeled as linear systems, where the matrix that models the forward problem is extremely ill-conditioned. In addition, the observed data is contaminated by noise. Due to the ill-conditioning of the matrix and the presence of noise in the observed data, it is necessary to employ regularization to compute a meaningful approximation of the solution.

Although there has been some very effective but expensive image reconstruction algorithm, those algorithms cannot be applied to large images because of their cost. This thesis focuses on using inexpensive, fast methods to obtain an initial image reconstruction first, combined with more expensive methods to improve specific subimages, called region of interest (ROI) areas in the image.

Table of Contents

1. Introduction ---------------------------------------------------------------- 1

1.1 Mathematical Background -------------------------------------------------2

1.2 Conditioning of the Problem ---------------------------------------------- 4

1.3 Computational Difficulties ------------------------------------------------ 6

2. Regularization --------------------------------------------------------------9

2.1 Truncated SVD -----------------------------------------------------------10

2.2 Tikhonov Regularization -------------------------------------------------10

3. Image Deblurring ----------------------------------------------------------13

3.1 Spacially Invariant Blur --------------------------------------------------14

3.2 Spacially Variant Blur ----------------------------------------------------20

4. Iterative Methods for Image Deblurring -----------------------------------22

4.1 Iterative Solvers in IR Tools ----------------------------------------------23

4.2 Solve 2D Image Deblurring Problems ------------------------------------ 25

5. Region of Interest Computations ------------------------------------------35

5.1 Extract ROI subimages ---------------------------------------------------36

5.2 ROI Experiments ---------------------------------------------------------45

6. Concluding Remarks ------------------------------------------------------ 57

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Region of Interest Image Reconstruction Regularization
Committee Chair / Thesis Advisor	James Nagy, Emory University
Committee Members	Shun Yan Cheung, Emory University Yuanzhe Xi, Emory University

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