Title: Reconciling Bayesian and Total Variation Methods for Binary Inversion,
Abstract: A central theme in classical algorithms for the reconstruction of
discontinuous functions from observational data is perimeter regularization. On
the other hand, sparse or noisy data often demands a probabilistic approach to
the reconstruction of images, to enable uncertainty quantification; the
Bayesian approach to inversion is a natural framework in which to carry this
out. The link between Bayesian inversion methods and perimeter regularization,
however, is not fully understood. In this paper two links are studied: (i) the
MAP objective function of a suitably chosen phase-field Bayesian approach is
shown to be closely related to a least squares plus perimeter regularization
objective; (ii) sample paths of a suitably chosen Bayesian level set
formulation are shown to possess finite perimeter and to have the ability to
learn about the true perimeter. Furthermore, the level set approach is shown to
lead to faster algorithms for uncertainty quantification than the phase field
approach.