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.

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