An Estimation Theoretical Approach to Ambrosio-Tortorelli Image Segmentation

Abstract

This paper presents a novel approach for Ambrosio-Tortorelli (AT) image segmentation, or, more exactly, joint image regularization and edge-map reconstruction.We interpret the AT functional, an approximation of the Mumford-Shah (MS) functional, as the energy of a posterior probability density function (PDF) of the image and smooth edge indicator. Previous approaches consider AT or MS segmentation as a deterministic optimization problem by minimizing the energy functional, resulting in a single point estimate, i.e. the maximum-a-posteriori (MAP) estimate. We adopt a wider estimation theoretical view-point, meaning we consider images to be random variables and investigate their distribution. We derive an effective block-Gibbs-sampler for this posterior PDF based on the theory of Gaussian Markov random fields (GMRF). The merit of our approach is multi-fold: First, sampling from the posterior PDF allows to apply different types of estimators and not only the MAP estimator. Secondly, sampling allows to estimate higher order statistical moments like the variance as a confidence measure. Third, our approach is not prone to get trapped into local minima as other AT image reconstruction approaches, but our approach is asymptotically statistical optimal. Several experiments demonstrate the advantages of our block-Gibbs-sampling approach.