Volume 63, pp. 33-62, 2025.

Variational Poisson denoising via augmented Lagrangian methods

Christian Kanzow, Fabius Krämer, Patrick Mehlitz, Gerd Wachsmuth, and Frank Werner

Abstract

In this paper, we denoise a given noisy image by minimizing a smoothness-promoting function over a set of local similarity measures which compare the mean of the given image and some candidate image on a large collection of subboxes. The associated convex optimization problem possesses a huge number of constraints which are induced by extended real-valued functions stemming from the Kullback–Leibler divergence. Alternatively, these nonlinear constraints can be reformulated as affine ones, which makes the model seemingly more tractable. For the numerical treatment of both formulations of the model (i.e., the original one as well as the one with affine constraints), we propose a rather general augmented Lagrangian method which is capable of handling the huge amount of constraints. A self-contained, derivative-free, global convergence theory is provided, allowing an extension to other problem classes. For the solution of the resulting subproblems in the setting of our suggested image denoising models, we make use of a suitable stochastic gradient method. Results of several numerical experiments are presented in order to compare both formulations and the associated augmented Lagrangian methods.

Full Text (PDF) [769 KB], BibTeX , DOI: 10.1553/etna_vol63s33

Key words

augmented Lagrangian method, nonsmooth optimization, Poisson denoising

AMS subject classifications

49M37, 90C30, 90C48, 90C90