Image restoration and segmentation using the Ambrosio-Tortorelli functional and Discrete Calculus

Essential image processing and analysis tasks, such as image segmentation, simplification and denoising, can be conducted in a unified way by minimizing the Mumford-Shah (MS) functional. Although seductive, this minimization is in practice difficult because it requires to jointly define a sharp set of contours and a smooth version of the initial image. For this reason, various relaxations of the original formulations have been proposed, together with optimisation methods. Among these, the Ambrosio-Tortorelli (AT) parametric functional is of particular interest, because minimizers of AT can be shown to converge to a minimizer of MS. However this convergence is difficult to achieve numerically using standard finite difference schemes. Indeed, with AT, discontinuities need to be represented explicitly rather than implicitly. In this work, we propose to formulate AT using the full framework of Discrete Calculus (DC), which is able to sharply represent discontinuities thanks to a more sophisticated topological framework. We present our proposed formulation, its resolution, and results on synthetic and real images. We show that we are indeed able to represent sharp discontinuities and as a result significantly better stability to noise, compared with finite difference schemes.