# Reconciling Bayesian and perimeter regularization for binary inversion

## TL;DR

This paper explores the connection between Bayesian inversion methods and perimeter regularization, demonstrating how Bayesian approaches can incorporate perimeter concepts for reconstructing discontinuous functions with uncertainty quantification.

## Contribution

It establishes a theoretical link between Bayesian phase-field and level set methods and perimeter regularization, enhancing understanding of their regularization properties.

## Key findings

- MAP objective function relates to least squares plus perimeter regularization
- Bayesian level set samples have finite perimeter
- Bayesian methods can learn about the true perimeter

## Abstract

A central theme in classical algorithms for the reconstruction of discontinuous functions from observational data is perimeter regularization via the use of the total variation. 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, which itself introduces a form of regularization, is a natural framework in which to carry this out. In this paper the link between Bayesian inversion methods and perimeter regularization is explored. In this paper two links are studied: (i) the maximum a posteriori (MAP) objective function of a suitably chosen Bayesian phase-field 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.