Variational Image Segmentation Models Involving Non-smooth Data-Fidelity Terms

Abstract

This article introduces a class of piecewise-constant image segmentation models that involves \(L^1\) norms as data fidelity measures. The \(L^1\) norms enable to segment images with low contrast or outliers such as impulsive noise. The regions to be segmented are represented as smooth functions instead of the Heaviside expression of level set functions as in the level set method. In order to deal with both non-smooth data-fitting and regularization terms, we use the variable splitting scheme to obtain constrained optimization problems, and apply an augmented Lagrangian method to solve the problems. This results in fast and efficient iterative algorithms for piecewise-constant image segmentation. The segmentation framework is extended to vector-valued images as well as to a multi-phase model to deal with arbitrary number of regions. We show comparisons with Chan-Vese models that use \(L^2\) fidelity measures, to enlight the benefit of the \(L^1\) ones.