Linear inverse problems with noise: primal and primal-dual splitting

TL;DR

This paper introduces primal and primal-dual algorithms for solving noisy linear inverse problems, incorporating noise statistics and sparsity priors, with theoretical analysis and experimental validation on deconvolution, inpainting, and denoising tasks.

Contribution

It presents novel proximal splitting algorithms tailored for noisy inverse problems with specific noise models and sparsity priors, along with theoretical guarantees.

Findings

Algorithms effectively handle noise in inverse problems.

Experimental results show competitive performance with existing methods.

Theoretical analysis confirms well-posedness and convergence.

Abstract

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is introduced to reflect the statistics of the noise (e.g. Gaussian, Poisson). On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution, inpainting and denoising with some comparison to prior methods are also reported.