Alternating Direction Method of Multipliers for Negative Binomial Model with The Weighted Difference of Anisotropic and Isotropic Total Variation

TL;DR

This paper introduces an optimization method using ADMM for image recovery in low-photon, overdispersed Poisson noise scenarios, employing a weighted total variation regularizer to improve image quality.

Contribution

It proposes a novel ADMM-based optimization framework incorporating a weighted anisotropic-isotropic total variation regularizer for negative binomial noise models.

Findings

Effective in very photon-limited conditions

Reduces staircasing artifacts in image reconstruction

Demonstrates superior performance over existing methods

Abstract

In many applications such as medical imaging, the measurement data represent counts of photons hitting a detector. Such counts in low-photon settings are often modeled using a Poisson distribution. However, this model assumes that the mean and variance of the signal's noise distribution are equal. For overdispersed data where the variance is greater than the mean, the negative binomial distribution is a more appropriate statistical model. In this paper, we propose an optimization approach for recovering images corrupted by overdispersed Poisson noise. In particular, we incorporate a weighted anisotropic-isotropic total variation regularizer, which avoids staircasing artifacts that are introduced by a regular total variation penalty. We use an alternating direction method of multipliers, where each subproblem has a closed-form solution. Numerical experiments demonstrate the effectiveness of our proposed approach, especially in very photon-limited settings.