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.

翻译：科学与技术的快速发展推动了对能够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。数据科学中的许多应用需要从一系列具有不连续性或奇异性的时变对象（例如具有边缘的动态计算机断层扫描（CT）图像）中进行时空重建。基于高斯过程（GP）的传统方法可能无法提供令人满意的解决方案，因为它们倾向于提供过度平滑的先验候选模型。近年来，通过小波展开并带有随机系数定义的贝索夫过程（BP）已被提出作为此类贝叶斯反问题中更合适的先验。虽然BP在图像分析中优于GP，能够产生边缘保持的重建结果，但它并未自动纳入动态变化图像中固有的时间相关性。本文通过将级数展开中的随机系数替换为遵循Q指数过程（该过程控制时间相关强度）的随机时间函数，将BP推广到时空域（STBP）。我们详细研究了STBP的数学和统计性质。还提出了STBP的白噪声表示，以通过最大后验（MAP）进行点估计，并通过后验采样进行不确定性量化（UQ）。通过两个有限角CT重建实例和一个涉及纳维-斯托克斯方程的高度非线性反问题，展示了所提出的STBP在保留空间特征的同时考虑时间变化的优势，并与经典STGP及时间无相关方法进行了比较。