A Gaussian hypermodel to recover blocky objects
Daniela Calvetti, Erkki Somersalo · Inverse Problems · 2007
The paper discusses inverse problems in which the unknown is a function that is assumed to be piecewise smooth with discontinuities of unknown location and size. If the locations of the discontinuities were known a priori, this prior information could be encoded into a structural smoothness prior. Since the locations are unknown, the smoothness prior itself can be considered as part of the unknown to be determined. A natural framework for this type of problem is provided by hierarchical Bayesian models. In this paper, we propose a hierarchical model suitable for this type of inverse problems, and we show that a single estimate based on this model can be effectively calculated. Being a fully Bayesian solution, it also allows sampling-based studies of the stability and reliability. We also demonstrate that the approach is closely related to bootstrap priors previously investigated by the authors in connection with image inpainting and deblurring. Computed examples with a deconvolution problem are presented.