Estimation and uncertainty quantification for piecewise smooth signal recovery

TL;DR

This paper introduces a Bayesian learning algorithm for high order total variation sparsity priors, enabling uncertainty quantification in recovering piecewise smooth signals from inverse problems.

Contribution

It formulates HOTV sparsity priors via synthesis and develops a Bayesian method, broadening Bayesian inverse problem applications to piecewise smooth signal recovery.

Findings

More accurate estimates than l1 regularization methods

Provides full posterior density for uncertainty quantification

Effectively applied to inverse problems with piecewise smooth signals

Abstract

This paper presents a sparse Bayesian learning (SBL) algorithm for linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that rely on l1 regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then develops a corresponding Bayesian learning method. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.