Cauchy difference priors for edge-preserving Bayesian inversion

Abstract We consider inverse problems in which the unknown target includes sharp edges, for example interfaces between different materials. Such problems are typical in image reconstruction, tomography, and other inverse problems algorithms. A common solution for edge-preserving inversion is to use total variation (TV) priors. However, as shown by Lassas and Siltanen 2004, TV-prior is not discretization-invariant: the edge-preserving property is lost when the computational mesh is made denser and denser. In this paper we propose another class of priors for edge-preserving Bayesian inversion, the Cauchy difference priors. We construct Cauchy priors starting from continuous one-dimensional Cauchy motion, and show that its discretized version, Cauchy random walk, can be used as a non-Gaussian prior for edge-preserving Bayesian inversion. We generalize the methodology to two-dimensional Cauchy fields, and briefly consider a generalization of the Cauchy priors to Lévy α-stable random field priors. We develop a suitable posterior distribution sampling algorithm for conditional mean estimates with single-component Metropolis–Hastings. We apply the methodology to one-dimensional deconvolution and two-dimensional X-ray tomography problems.