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

TL;DR

This paper introduces efficient algorithms for solving $ ext{l}_1$-regularized inverse problems by exploiting Kronecker product structures in the operators, enhancing image deblurring techniques.

Contribution

It develops a joint decomposition approach using generalized SVDs for $ ext{l}_1$ problems with Kronecker structures, integrating it into Split Bregman and Majorization-Minimization methods.

Findings

Joint decomposition improves computational efficiency.

Framelet and wavelet operators are effective in the proposed algorithms.

The approach is particularly effective for image deblurring with 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.