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
Rights statement
- Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
School |
|
Department |
|
Degree |
|
Submission |
|
Language |
|
Research Field |
|
关键词 |
|
Committee Chair / Thesis Advisor |
|
Committee Members |
|