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.

翻译：科学与技术的快速发展推动了对捕捉突变或锐利对比等特殊数据特征的恰当统计工具的需求。数据科学中的众多逆问题要求从一系列随时间变化且具备这些空间特征的对象中推导出时空解，例如带边缘的计算机断层扫描（CT）图像动态重建。基于高斯过程（GP）的传统方法往往无法提供令人满意的解，因为它们倾向于给出过平滑的先验。近年来，由具有随机系数的小波展开定义的Besov过程（BP）已成为处理此类贝叶斯逆问题的更合适先验。尽管BP在处理空间非均匀性方面表现出色，但它无法自动纳入动态变化对象中固有的时间相关性。在本文中，我们通过将级数展开中的随机系数替换为控制时间相关结构的随机时间函数（即Q-指数过程，Q-EP），将BP推广为一种新颖的时空Besov过程（STBP）。我们深入研究了STBP的数学和统计学性质。还提出了STBP的白噪声表示以促进推理。通过模拟实验、两个有限角度CT重建示例以及一个涉及纳维-斯托克斯方程的高度非线性逆问题，展示了与经典STGP及时间不相关方法相比，所提出的STBP在保留空间特征的同时考虑时间变化方面的优势。