Convergence results for critical points of the one-dimensional Ambrosio-Tortorelli functional with fidelity term Nam Q. Le Department of Mathematics, Columbia University 2990 Broadway, MC 4406, New York, NY 10027, USA Email: namle@math.columbia.edu November 24, 2008 Abstract In this paper, we show that critical points of the one-dimensional Ambrosio- Tortorelli functional with fidelity term converge to those of the corresponding Mumford- Shah functional, a famous model for image segmentation. Equi-partition and con- vergence of the energy-density are also derived. MSC: 49Q20, 49J45, 35B38, 35J60 Keywords: Mumford-Shah functional, Ambrosio-Tortorelli functional, Gamma-convergence, critical point, image segmentation. 1 Introduction In this paper, we continue our previous study [7] on the convergence of critical points of the Ambrosio-Tortorelli functional [2, 3] to those of the Mumford-Shah functional [11]. Here, as opposed to the Dirichlet case in [7], the functionals we study contain the fidelity term linking the approximate images to the original image. These functionals were pro- posed as models for image segmentation in computer vision. Because real digital images are two-dimensional, the analysis of these functionals in 2D will be of greater interest and applications. However, carrying out this task is quite a challenge for the time being, and we only limit ourselves to the one-dimensional case. We expect that many statements will eventually carry over to the two-dimensional case. We now introduce the Ambrosio-Tortorelli and Mumford-Shah functionals used in the paper. Throughout, C stands for a generic positive constant (so that e.g. C =2C ) and L is the length of the interval under consideration. 1