The Effects of Projection on Iterative Methods in Image Deblurring 公开

Abstract

Abstract
The Effects of Projection on Iterative Methods in Image Deblurring
By James L. Herring
Applications of iterative regularization methods encompass a broad spectrum, in-
cluding image deblurring. One unique feature in image deblurring problems is the
non-negativity of the solution. In light of this knowledge, this thesis explores the
effectiveness of three projected iterative methods for image deconvolution: projected
successive over-relaxation method (SOR), projected Landweber method, and an in-
terior point gradient method. Specifically, this thesis compares the effectiveness of
these methods to the standard un-projected SOR method, comparing quality of image
reconstruction and the cost of each method. The thesis begins with an introduction
to the field of image deblurring problems, iterative regularization, and the methods
tested, and follows with experiments and analysis designed to determine the useful-
ness of these methods for image deblurring.

Table of Contents

Contents
1 Introduction 1
1.1 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Structure and Sparsity . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Iterative Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Projection: Additional Regularization . . . . . . . . . . . . . . . . 11
1.5 Regularization Parameters and the Discrepancy Principle . . 12
2 Methods 14
2.1 Successive Over-Relaxation . . . . . . . . . . . . . . . . . . . . . 14
2.2 Projected SOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Projected Landweber Method . . . . . . . . . . . . . . . . . . . . 17
2.4 Interior-Point Gradient Method . . . . . . . . . . . . . . . . . . . 18
3 Filtering Properties 20
3.1 Filtering in Landweber . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 SOR Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Numerical Experiments 25
4.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Noise Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.3 Stopping Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
5 Experiments on Simulated Problems 35
5.1 Tests on Real Images . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39

About this Honors Thesis

Rights statement

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School	Emory College
Department	Mathematics and Computer Science
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Mathematics
关键词	Methods Regularization Iterative SOR Imaging Landweber Deconvolution Projection Deblurring
Committee Chair / Thesis Advisor	Nagy, James, Emory University
Committee Members	Roth, Connie, Emory University Benzi, Michele, Emory University

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