Bayesian decomposition using Besov priors

TL;DR

This paper develops Bayesian methods with Besov priors to decompose inverse problems into smooth and rough components, improving reconstruction quality in imaging tasks with mixed regularities.

Contribution

It introduces two novel prior models combining Besov and hierarchical Gaussian priors, along with hyperparameter inference and Gibbs sampling for better component separation.

Findings

Improved reconstruction quality over single-prior methods.

Successful hyperparameter estimation for balanced decomposition.

Effective application to 1D and 2D deconvolution problems.

Abstract

In many inverse problems, the unknown is composed of multiple components with different regularities, for example, in imaging problems, where the unknown can have both rough and smooth features. We investigate linear Bayesian inverse problems, where the unknown consists of two components: one smooth and one piecewise constant. We model the unknown as a sum of two components and assign individual priors on each component to impose the assumed behavior. We propose and compare two prior models: (i) a combination of a Haar wavelet-based Besov prior and a smoothing Besov prior, and (ii) a hierarchical Gaussian prior on the gradient coupled with a smoothing Besov prior. To achieve a balanced reconstruction, we place hyperpriors on the prior parameters and jointly infer both the components and the hyperparameters. We propose Gibbs sampling schemes for posterior inference in both prior models. We demonstrate the capabilities of our approach on 1D and 2D deconvolution problems, where the unknown consists of smooth parts with jumps. The numerical results indicate that our methods improve the reconstruction quality compared to single-prior approaches and that the prior parameters can be successfully estimated to yield a balanced decomposition.