Least Squares Updating for Kronecker Products Open Access

Abstract

In this thesis, we consider an application within image processing where Kronecker products naturally arise. In image deblurring, we attempt to reconstruct the original image from a given blurred image. Due to the ill-conditioned nature of this matrix, along with the large size of the blurring matrix, it becomes difficult to use direct techniques to ï¬nd the solution to this problem. Therefore, we can use iterative methods and in particular, we used LSQR. However, in order for us to use LSQR efficiently, we constructed a preconditioner using the rank-one updating problem extended to Kronecker products. We showed that this method can provide us with a good preconditioner and through some deblurring examples, the solution is as accurate while reducing the average run time.

Table of Contents

1 Introduction 1

1.1 QR Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Givens Rotations . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Updating problem . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Kronecker Product . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Kronecker Product 11 2.1 Deï¬nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 New Preconditioner Based on QR Updating 17 3.1 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . .18 3.2 Rank-One Updating Problem . . . . . . . . . . . . . . . . . .20 3.3 Rank-One Update for Kronecker Products . . . . . . . . . 21 3.4 LSQR in Image Deblurring . . . . . . . . . . . . . . . . . . . 23 4 Numerical Results 27 5 Conclusion 32

About this Honors Thesis

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School	Emory College
Department	Mathematics and Computer Science
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Applied Mathematics
Keyword	kronecker product image processing preconditioner image deblurring
Committee Chair / Thesis Advisor	Nagy, James, Emory University
Committee Members	Rogers, Michael K, Emory University Seitaridou, Effrosyni, Emory University

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