This work describes a Bayesian framework for reconstructing functions that represents the targeted features with uncertain regularity, i.e., roughness vs. smoothness. The regularity of functions carries crucial information in many inverse problem applications, e.g., in medical imaging for identifying malignant tissues or in the analysis of electroencephalogram for epileptic patients. We characterize the regularity of a function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates a function and its regularity. In addition, we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating functions in different types of inverse problems. Furthermore, this is a robust method under various noise types, noise levels, and incomplete measurement.

翻译：本文描述了一个贝叶斯框架，用于重构表示具有不确定规律性（即粗糙度与光滑度）的目标特征函数。在许多逆问题应用中，函数的规律性承载着关键信息，例如在医学成像中识别恶性组织，或在癫痫患者的脑电图分析中。我们通过函数的分数阶可微性来刻画其规律性。提出了一种分层贝叶斯公式，能够同时估计函数及其规律性。此外，我们还量化了估计中的不确定性。数值结果表明，所提方法是一种可靠的手段，适用于不同类型逆问题中的函数估计。同时，该方法对多种噪声类型、噪声水平及不完整测量均具有稳健性。