Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting
Feng, Xiaobing ; Prohl, Andreas
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004), p. 291-320 / Harvested from Numdam

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.

Publié le : 2004-01-01
DOI : https://doi.org/10.1051/m2an:2004014
Classification:  35K55,  65M12,  65M15,  68U10,  94A08
@article{M2AN_2004__38_2_291_0,
     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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {38},
     year = {2004},
     pages = {291-320},
     doi = {10.1051/m2an:2004014},
     mrnumber = {2069148},
     zbl = {1074.65106},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2004__38_2_291_0}
}
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) pp. 291-320. doi : 10.1051/m2an:2004014. http://gdmltest.u-ga.fr/item/M2AN_2004__38_2_291_0/

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