Bayesian decomposition using Besov priors

TL;DR

The paper addresses linear inverse problems where the unknown is a sum of a smooth part and a piecewise-constant part, proposing two Bayesian decomposition schemes: (i) a two-Besov prior model combining Haar-based and smooth Besov priors, and (ii) a hierarchical Gaussian prior on the discrete gradient for the piecewise-constant part coupled with a smooth Besov prior for the smooth component. It analyzes identifiability issues arising from competing Besov bases and introduces a hierarchical Gaussian-Besov approach with hyperpriors on the gradient variance and a Besov prior on the smooth component, enabling efficient Gibbs sampling or RTO-based updates. Through 1D and 2D deconvolution experiments, the two-Besov model shows improved reconstruction over single-prior approaches but faces sampling coherence, while the hierarchical Gaussian-Besov method achieves a balanced decomposition with meaningful uncertainty while introducing somewhat slower mixing. The work advances uncertainty quantification for decomposed inverse problems and provides practical Gibbs/RTO schemes for inferring both components and hyperparameters, with demonstrated applications to deconvolution and image deblurring.

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.

Figures (13)

• Figure 1: The ground truth signal and the degraded data (in red, to the right), together with the true piecewise constant component (to the left) and the smooth component (in the middle).
• Figure 2: Posterior estimates using the Haar and DB(8) wavelets in the two-Besov prior decomposition model with the parameters $s_{g}=s_{h}=p_{g}=p_{h}=\lambda_{g}=\lambda_{h}=1$.
• Figure 3: Posterior estimates and autocorrelation functions (ACFs) computed from the chains at a representative grid point (marked with a red dot). Row 1: One-Besov prior using Haar, i.e., DB(1), wavelets. Row 2: One-Besov prior using DB(8) wavelets. Row 3: Two-Besov prior model combining Haar and DB(8) wavelets (as in Section \ref{['sec:post-bayes-decomp']}). All reconstructions use the same prior parameters $s=p=\lambda=1$ for single Besov priors and $s_{g}=s_{h}=p_{g}=p_{h}=\lambda_{g}=\lambda_{h}=1$ for the two-Besov model.
• Figure 4: Posterior estimates for the decomposition method with different choices of prior strength parameters.
• Figure 5: Ground truth (to the right) as a sum of the piecewise constant component (to the left) and the smooth component (in the middle) together with the degraded data (to the right).

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Remark 1