A proximal iteration for deconvolving Poisson noisy images using sparse representations

TL;DR

This paper introduces a novel deconvolution algorithm for Poisson noisy images that employs the Anscombe transform, sparse representations, and a fast iterative method, demonstrating significant improvements in image restoration quality.

Contribution

It presents a new convex optimization framework with a specialized iterative algorithm for Poisson noise deconvolution using sparse domain regularization.

Findings

Effective handling of Poisson noise via Anscombe transform

Convex formulation with sparsity-promoting penalties

Fast convergence of the proposed iterative algorithm

Abstract

We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms. Our key contributions are: First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a {\it non-linear} degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, and a non-smooth sparsity-promoting penalties over the image representation coefficients (e.g. $\ell_1$-norm). Third, a fast iterative backward-forward splitting algorithm is proposed to solve the minimization problem. We derive existence and uniqueness conditions of the solution, and establish convergence of the iterative algorithm. Finally, a GCV-based model selection procedure is proposed to objectively select the regularization parameter. Experimental results are carried out to show the striking benefits gained from taking into account the Poisson statistics of the noise. These results also suggest that using sparse-domain regularization may be tractable in many deconvolution applications with Poisson noise such as astronomy and microscopy.