Spatiotemporal Besov Priors for Bayesian Inverse Problems

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 applications in the data science seek spatiotemporal reconstruction from a sequence of time-dependent objects with discontinuity or singularity, e.g. dynamic computerized tomography (CT) images with edges. Traditional methods based on Gaussian processes (GP) may not provide satisfactory solutions since they tend to offer over-smooth prior candidates. Recently, Besov process (BP) defined by wavelet expansions with random coefficients has been proposed as a more appropriate prior for this type of Bayesian inverse problems. While BP outperforms GP in imaging analysis to produce edge-preserving reconstructions, it does not automatically incorporate temporal correlation inherited in the dynamically changing images. In this paper, we generalize BP to the spatiotemporal domain (STBP) by replacing the random coefficients in the series expansion with stochastic time functions following Q-exponential process which governs the temporal correlation strength. Mathematical and statistical properties about STBP are carefully studied. A white-noise representation of STBP is also proposed to facilitate the point estimation through maximum a posterior (MAP) and the uncertainty quantification (UQ) by posterior sampling. Two limited-angle CT reconstruction examples and a highly non-linear inverse problem involving Navier-Stokes equation are used to demonstrate the advantage of the proposed STBP in preserving spatial features while accounting for temporal changes compared with the classic STGP and a time-uncorrelated approach.