The Image Deblurring Problem: Matrices, Wavelets, and Multilevel Methods

TL;DR

This paper reviews numerical linear algebra techniques for image deblurring, including regularization methods, matrix structures, and multilevel algorithms, supported by numerical examples.

Contribution

It provides a comprehensive overview of regularization, matrix structures, and multilevel methods specifically tailored for the image deblurring problem.

Findings

Regularization methods effectively restore images from blurred and noisy data.

Matrix structures can be exploited for efficient deblurring algorithms.

Multilevel methods improve computational efficiency while preserving structure.

Abstract

The image deblurring problem consists of reconstructing images from blur and noise contaminated available data. In this AMS Notices article, we provide an overview of some well known numerical linear algebra techniques that are use for solving this problem. In particular, we start by carefully describing how to represent images, the process of blurring an image and modeling different kind of added noise. Then, we present regularization methods such as Tikhonov (on the standard and general form), Total Variation and other variations with sparse and edge preserving properties. Additionally, we briefly overview some of the main matrix structures for the blurring operator and finalize presenting multilevel methods that preserve such structures. Numerical examples are used to illustrate the techniques described.