Nonlinear Analysis 64 (2006) 1908 – 1930 www.elsevier.com/locate/na An approximation of the Mumford–Shah energy by a family of discrete edge-preserving functionals Gilles Aubert a , Laure Blanc-Féraud b, ∗ , Riccardo March c a Laboratoire J.A. Dieudonné, Umr 6621 du CNRS, Université de Nice Sophia Antipolis, 06 108 Nice cedex 02, France b Projet Ariana, laboratoire I3S (CNRS/UNSA) et INRIA Sophia Antipolis, INRIA, 2004 route des Lucioles, BP 93 06902 Sophia Antipolis cedex, France c Istituto per le Applicazioni del Calcolo, CNR, Viale del Policlinico 137, 00161 Roma, Italy Received 17 December 2004; accepted 20 July 2005 Abstract We show the -convergence of a family of discrete functionals to the Mumford and Shah image segmentation functional. The functionals of the family are constructed by modifying the elliptic approximating functionals proposed by Ambrosio and Tortorelli. The quadratic term of the energy related to the edges of the segmentation is replaced by a nonconvex functional.  2005 Elsevier Ltd. All rights reserved. Keywords: -convergence; Finite elements; Image segmentation 1. Introduction Segmentation is an important task in image processing. The goal is to decompose an observed image into several homogeneous regions. Such a segmentation can be achieved by computing the regions or the edges limiting the regions. The most well-known segmentation functional in image processing is the one proposed by Mumford–Shah, which we write in ∗ Corresponding author. Tel.: +33 4 92 38 77 14; fax: +33 4 92 38 76 48. E-mail addresses: gaubert@math.unice.fr (G. Aubert), blancf@sophia.inria.fr (L. Blanc-Féraud), r.march@iac.cnr.it (R. March). URL: http://www.inria.fr/ariana 0362-546X/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.07.028