Spatiotemporal Besov Priors for Bayesian Inverse Problems

TL;DR

This paper introduces a spatiotemporal Besov process (STBP) as a novel prior for Bayesian inverse problems, effectively capturing sharp spatial features and temporal correlations in dynamic data, outperforming traditional Gaussian process models.

Contribution

The paper develops the STBP by integrating wavelet-based Besov priors with stochastic time functions, providing a new framework for spatiotemporal Bayesian inverse problems.

Findings

STBP outperforms STGP and uncorrelated models in CT and temperature imputation tasks.

Mathematical properties of STBP are thoroughly analyzed.

Simulations demonstrate STBP's effectiveness in nonlinear inverse problems.

Abstract

Fast development in science and technology has driven the need for proper statistical tools to capture special data features such as abrupt changes or sharp contrast. Many inverse problems in data science require spatiotemporal solutions derived from a sequence of time-dependent objects with these spatial features, e.g., the dynamic reconstruction of computerized tomography (CT) images with edges. Conventional methods based on Gaussian processes (GP) often fall short in providing satisfactory solutions since they tend to offer oversmooth priors. Recently, the Besov process (BP), defined by wavelet expansions with random coefficients, has emerged as a more suitable prior for Bayesian inverse problems of this nature. While BP excels in handling spatial inhomogeneity, it does not automatically incorporate temporal correlation inherited in the dynamically changing objects. In this paper, we generalize BP to a novel spatiotemporal Besov process (STBP) by replacing the random coefficients in the series expansion with stochastic time functions as Q-exponential process (Q-EP) which governs the temporal correlation structure. We thoroughly investigate the mathematical and statistical properties of STBP. Simulations, two limited-angle CT reconstruction examples, a highly non-linear inverse problem involving Navier-Stokes equation, and a spatiotemporal temperature imputation problem are used to demonstrate the advantage of the proposed STBP compared with the classic STGP and a time-uncorrelated approach.