Image restoration using sparse approximations of spatially varying blur operators in the wavelet domain

TL;DR

This paper introduces a method for restoring images affected by spatially varying blur by approximating the blur operator with a sparse matrix in the wavelet domain, enabling efficient computation and application to blind deconvolution.

Contribution

It proposes a novel wavelet domain sparse approximation of spatially varying blur operators, justified mathematically and validated numerically, facilitating practical image restoration tasks.

Findings

Sparse wavelet domain matrices effectively approximate spatially varying blur operators.

Pre-defined sparsity patterns enhance blind deconvolution performance.

Numerical experiments confirm the approximation quality and computational efficiency.

Abstract

Restoration of images degraded by spatially varying blurs is an issue of increasing importance in the context of photography, satellite or microscopy imaging. One of the main difficulty to solve this problem comes from the huge dimensions of the blur matrix. It prevents the use of naive approaches for performing matrix-vector multiplications. In this paper, we propose to approximate the blur operator by a matrix sparse in the wavelet domain. We justify this approach from a mathematical point of view and investigate the approximation quality numerically. We finish by showing that the sparsity pattern of the matrix can be pre-defined, which is central in tasks such as blind deconvolution.