Analysis of Iterative Deblurring: No Explicit Noise

TL;DR

This paper analyzes iterative deblurring methods, especially Richardson-Lucy, in high-energy physics contexts without explicit noise, focusing on null space issues and the effects of regularization on image restoration.

Contribution

It provides a detailed analysis of deblurring without explicit noise, highlighting the role of null space and regularization in high-contrast and low-contrast images.

Findings

Null space can be managed with nonnegativity constraints.

Regularization helps control null-space content in low-contrast images.

Deblurring can restore images to a less blurred state without explicit noise modeling.

Abstract

Iterative deblurring, notably the Richardson-Lucy algorithm with and without regularization, is analyzed in the context of nuclear and high-energy physics applications. In these applications, probability distributions may be discretized into a few bins, measurement statistics can be high, and instrument performance can be well understood. In such circumstances, it is essential to understand the deblurring first without any explicit noise considerations. We employ singular value decomposition for the blurring matrix in a low-count pixel system. A strong blurring may yield a null space for the blurring matrix. Yet, a nonnegativity constraint for images built into the deblurring may help restore null-space content in a high-contrast image with zero or low intensity for a sufficient number of pixels. For low-contrast images, control over null-space content can be achieved through regularization. When regularization is applied, the blurred image is, in practice, restored to one that is still blurred but less than the starting image.