Bayesian Level Set Approach for Inverse Problems with Piecewise Constant Reconstructions

TL;DR

This paper introduces a Bayesian level set method for inverse problems involving piecewise constant fields, combining efficient optimization, posterior approximation, and uncertainty visualization techniques demonstrated on synthetic and real data.

Contribution

It develops a novel Bayesian framework with a Gauss-Newton approach, Laplace approximation, and Monte Carlo variance estimation for piecewise constant inverse problems.

Findings

Effective reconstruction of piecewise constant fields in synthetic tests

Accurate uncertainty quantification via posterior variance estimation

Successful application to real X-ray imaging data

Abstract

There are several challenges associated with inverse problems in which we seek to reconstruct a piecewise constant field, and which we model using multiple level sets. Adopting a Bayesian viewpoint, we impose prior distributions on both the level set functions that determine the piecewise constant regions as well as the parameters that determine their magnitudes. We develop a Gauss-Newton approach with a backtracking line search to efficiently compute the maximum a priori (MAP) estimate as a solution to the inverse problem. We use the Gauss-Newton Laplace approximation to construct a Gaussian approximation of the posterior distribution and use preconditioned Krylov subspace methods to sample from the resulting approximation. To visualize the uncertainty associated with the parameter reconstructions we compute the approximate posterior variance using a matrix-free Monte Carlo diagonal estimator, which we develop in this paper. We will demonstrate the benefits of our approach and solvers on synthetic test problems (photoacoustic and hydraulic tomography, respectively a linear and nonlinear inverse problem) as well as an application to X-ray imaging with real data.