Variable order smoothness priors for ill-posed inverse problems

Abstract

In this article we discuss ill-posed inverse problems, with an emphasis on hierarchical variable order regularization. Traditionally, smoothness penalties in Tikhonov regularization assume a fixed degree of regularity of the unknown over the whole domain. Using a Bayesian framework with hierarchical priors, we derive a prior model, formally represented as a convex combination of autoregressive (AR) models, in which the parameter controlling the mixture of the AR models can dynamically change over the domain of the signal. Moreover, the mixture parameter itself is an unknown and is to be estimated using the data. Also, the variance of the innovation processes in the AR model is a free parameter, which leads to conditionally Gaussian priors that have been previously shown to be much more flexible than the traditional Gaussian priors, capable, e.g., to deal with sparsity type prior information. The suggested method, the Weighted Variable Order Autoregressive model (WVO-AR) is tested with a computed example.