Least Squares Updating for Kronecker Products Pubblico

Platero, Raul Enrique (2017)

Permanent URL: https://etd.library.emory.edu/concern/etds/p2676w28h?locale=it
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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

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