Spatiotemporal Besov Priors for Bayesian Inverse Problems
针对传统高斯过程先验过于平滑、难以捕捉数据中突变或边缘等特征的问题，本文提出一种新的时空Besov过程先验，通过将随机系数替换为Q-指数过程来引入时间相关性，并在CT图像重建、Navier-Stokes方程反问题等实例中验证了其优势。
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, for example, 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 article, 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 nonlinear 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.