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Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices

Brian Sweeney, Malena I. Español, Rosemary Renaut

TL;DR

It is proved that framelet and wavelet operators are efficient for these decomposition-based algorithms in the context of ℓ1-regularized image deblurring problems for cases involving column-orthogonal regularization matrices.

Abstract

We consider an $\ell_1$-regularized inverse problem where both the forward and regularization operators have a Kronecker product structure. By leveraging this structure, a joint decomposition can be obtained using generalized singular value decompositions. We show how this joint decomposition can be effectively integrated into the Split Bregman and Majorization-Minimization methods to solve the $\ell_1$-regularized inverse problem. Furthermore, for cases involving column-orthogonal regularization matrices, we prove that the joint decomposition can be derived directly from the singular value decomposition of the system matrix. As a result, we show that framelet and wavelet operators are efficient for these decomposition-based algorithms in the context of $\ell_1$-regularized image deblurring problems.

Efficient Decomposition-Based Algorithms for $\ell_1$-Regularized Inverse Problems with Column-Orthogonal and Kronecker Product Matrices

TL;DR

It is proved that framelet and wavelet operators are efficient for these decomposition-based algorithms in the context of ℓ1-regularized image deblurring problems for cases involving column-orthogonal regularization matrices.

Abstract

We consider an -regularized inverse problem where both the forward and regularization operators have a Kronecker product structure. By leveraging this structure, a joint decomposition can be obtained using generalized singular value decompositions. We show how this joint decomposition can be effectively integrated into the Split Bregman and Majorization-Minimization methods to solve the -regularized inverse problem. Furthermore, for cases involving column-orthogonal regularization matrices, we prove that the joint decomposition can be derived directly from the singular value decomposition of the system matrix. As a result, we show that framelet and wavelet operators are efficient for these decomposition-based algorithms in the context of -regularized image deblurring problems.
Paper Structure (16 sections, 4 theorems, 65 equations, 9 figures, 3 tables, 2 algorithms)

This paper contains 16 sections, 4 theorems, 65 equations, 9 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Iterative techniques for solution of the sparsity constrained problem
  3. Split Bregman
  4. Majorization-Minimization
  5. Parameter Selection Methods
  6. Solving Generalized Tikhonov with Kronecker Product Matrices
  7. Theoretical Properties of the GSVD of Column-Orthogonal Matrices
  8. Family of Regularization Operators
  9. Framelets
  10. Wavelets
  11. Numerical Results
  12. Blurring Examples
  13. Example 1
  14. Example 2
  15. Example 3
  16. ...and 1 more sections

Key Result

Lemma 1

\newlabellem:shortgsvd0 For ${\bf A} \in \mathbb{R}^{m \times n}$ and ${\bf L} \in \mathbb{R}^{p \times n}$, let ${\bf K} = \left[{\bf A};{\bf L}\right]$. Let $t = \text{rank}({\bf K})$, $r = \text{rank}({\bf K}) - \text{rank}({\bf L})$, and $s = \text{rank}({\bf A}) + \text{rank}({\bf L}) - \text where $\tilde{\boldsymbol{\Upsilon}} \in \mathbb{R}^{m \times t}$ and $\tilde{\bf M} \in \mathbb{R}^

Figures (9)

  • Figure 1: True image ${\bf x}$ ($128 \times 128$ pixels), PSF ($17 \times 17$ pixels), blurred image ${\bf b}_{true}$, and the blurred and noisy image ${\bf b}$ with $\text{BSNR}=10$ for Example 1.
  • Figure 2: Solutions to Example 1 in \ref{['Ex1']}. The solutions use either SB or MM to solve the problem and either framelets or wavelets for regularization. The parameter is either fixed optimally or selected with GCV or the non-central $\chi^2$ test.
  • Figure 3: RE and ISNR by iteration for the methods applied to Example 1 in \ref{['Ex1']}.
  • Figure 4: True image ${\bf x}$ ($512 \times 512$ pixels), PSF ($39 \times 39$ pixels), blurred image ${\bf b}_{true}$, and the blurred and noisy image ${\bf b}$ with $\text{BSNR}=10$ for Example 2.
  • Figure 5: Solutions to Example 2 in \ref{['Ex2']}. The solutions use either SB or MM to solve the problem and either framelets or wavelets for regularization. The parameter is either fixed optimally or selected with GCV or the non-central $\chi^2$ test.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Proof 1
  • Lemma 3
  • Proof 2
  • Corollary 4