Feng, Xiaobing; Prohl, Andreas
Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 2 , p. 291-320
Zbl 1074.65106 | MR 2069148 | 1 citation dans Numdam
doi : 10.1051/m2an:2004014
URL stable : http://www.numdam.org/item?id=M2AN_2004__38_2_291_0

Classification:  35K55,  65M12,  65M15,  68U10,  94A08
This paper studies the gradient flow of a regularized Mumford-Shah functional proposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L 2 ×L initial data possesses a global weak solution, and it has a unique global in time strong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H 1 ×H 1 L . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence) of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on 1 ε and 1 k ε only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation k=o(h 1 2 ). Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.


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