The Mumford-Shah functional for image segmentation is an original approachof the image segmentation problem, based on a minimal energy criterion. Itsminimization can be seen as a free discontinuity problem and is based onΓ-convergence and bounded variation functions theories. Some newregularization results, make possible to imagine a finite element resolutionmethod. In a first time, the Mumford-Shah functional isintroduced and some existing results are quoted. Then, adiscrete formulation for the Mumford-Shah problem is proposedand its Γ-convergence is proved. Finally, somenumerical results, computed from both artificial and realimages are presented and discussed.