The Mumford-Shah functional for image segmentation is an original approach of the image segmentation problem, based on a minimal energy criterion. Its minimization can be seen as a free discontinuity problem and is based on Γ-convergence and bounded variation functions theories. Some new regularization results, make possible to imagine a finite element resolution method. In a first time, the Mumford-Shah functional is introduced and some existing results are quoted. Then, a discrete formulation for the Mumford-Shah problem is proposed and its Γ-convergence is proved. Finally, some numerical results, computed from both artificial and real images are presented and discussed.
Publication Source (Journal or Book title)
Mathematical Modelling and Numerical Analysis
Bourdin, B. (1999). Image segmentation with a finite element method. Mathematical Modelling and Numerical Analysis, 33 (2), 229-244. https://doi.org/10.1051/m2an:1999114