# Fast roughness minimizing image restoration under mixed Poisson-Gaussian noise

## TL;DR

This paper introduces a fast ADMM-based method for image restoration under mixed Poisson-Gaussian noise, overcoming previous convergence speed limitations of primal-dual algorithms by developing a novel proximal computation technique.

## Contribution

We develop a new ADMM algorithm with a novel proximal computation for PG log-likelihood, enabling faster and more effective image restoration under mixed noise.

## Key findings

- The proposed method achieves faster convergence than primal-dual approaches.
- Restoration quality is high under mixed Poisson-Gaussian noise conditions.
- The algorithm is demonstrated effective through experimental examples.

## Abstract

Image acquisition in many biomedical imaging modalities is corrupted by Poisson noise followed by additive Gaussian noise. While total variation and related regularization methods for solving biomedical inverse problems are known to yield high quality reconstructions, such methods mostly use log-likelihood of either Gaussian or Poisson noise models, and rarely use mixed Poisson-Gaussian (PG) noise model. There is a recent work which deals with exact PG likelihood and totalariation regularization. This method is developed using the log-likelihood of PG model along with total variation regularization adapts the primal-dual splitting algorithm, whose step size is restricted to be bounded by the inverse of the Lipschitz constant of PG log-likelihood. This leads to limitations in the convergence speed. On the other hand, ADMM methods do not have such step size restrictions; however, ADDM has never been applied for this problem, for the possible reason that PG log-likelihood is quite complex. In this paper, we develop an ADMM based optimization for total variation minimizing image restoration under PG log-likelihood. We achieve this by first developing a novel iterative method for computing the proximal solution of PG log-likelihood, deriving the termination conditions for this iterative method, and then integrating into a provable convergent ADMM scheme. The effectiveness of the proposed methods is demonstrated using restoration examples.