Regularized Richardson-Lucy Algorithm for Sparse Reconstruction of Poissonian Images

TL;DR

This paper introduces a regularized Richardson-Lucy algorithm that leverages sparse representations in transformed domains to improve Poissonian image reconstruction, demonstrating enhanced accuracy and efficiency over existing methods.

Contribution

The paper proposes a novel modification of the Richardson-Lucy algorithm that recovers sparse transform coefficients, broadening applicability and improving performance in Poisson noise image restoration.

Findings

The modified algorithm converges to sparse solutions in transformed domains.

It outperforms traditional methods in estimation accuracy.

It has lower computational complexity.

Abstract

Restoration of digital images from their degraded measurements has always been a problem of great theoretical and practical importance in numerous applications of imaging sciences. A specific solution to the problem of image restoration is generally determined by the nature of degradation phenomenon as well as by the statistical properties of measurement noises. The present study is concerned with the case in which the images of interest are corrupted by convolutional blurs and Poisson noises. To deal with such problems, there exists a range of solution methods which are based on the principles originating from the fixed-point algorithm of Richardson and Lucy (RL). In this paper, we provide conceptual and experimental proof that such methods tend to converge to sparse solutions, which makes them applicable only to those images which can be represented by a relatively small number of non-zero samples in the spatial domain. Unfortunately, the set of such images is relatively small, which restricts the applicability of RL-type methods. On the other hand, virtually all practical images admit sparse representations in the domain of a properly designed linear transform. To take advantage of this fact, it is therefore tempting to modify the RL algorithm so as to make it recover representation coefficients, rather than the values of their associated image. Such modification is introduced in this paper. Apart from the generality of its assumptions, the proposed method is also superior to many established reconstruction approaches in terms of estimation accuracy and computational complexity. This and other conclusions of this study are validated through a series of numerical experiments.