  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., 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 over-smooth
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. A white-noise representation
of STBP is also proposed to facilitate the inference. Simulations, 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.
