Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting
ESAIM: Modélisation mathématique et analyse numérique, Volume 38 (2004) no. 2, pp. 291-320.

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.

DOI: 10.1051/m2an:2004014
Classification: 35K55, 65M12, 65M15, 68U10, 94A08
Keywords: image segmentation and inpainting, Mumford-Shah model, elliptic approximation, gradient flow, a priori estimates, finite element method, error analysis
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     author = {Feng, Xiaobing and Prohl, Andreas},
     title = {Analysis of gradient flow of a regularized {Mumford-Shah} functional for image segmentation and image inpainting},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {291--320},
     publisher = {EDP-Sciences},
     volume = {38},
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     year = {2004},
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     mrnumber = {2069148},
     zbl = {1074.65106},
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     url = {http://www.numdam.org/articles/10.1051/m2an:2004014/}
}
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Feng, Xiaobing; Prohl, Andreas. Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting. ESAIM: Modélisation mathématique et analyse numérique, Volume 38 (2004) no. 2, pp. 291-320. doi : 10.1051/m2an:2004014. http://www.numdam.org/articles/10.1051/m2an:2004014/

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