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 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 . 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 and only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.
Keywords: image segmentation and inpainting, Mumford-Shah model, elliptic approximation, gradient flow, a priori estimates, finite element method, error analysis
@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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {291--320}, publisher = {EDP-Sciences}, volume = {38}, number = {2}, year = {2004}, doi = {10.1051/m2an:2004014}, mrnumber = {2069148}, zbl = {1074.65106}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2004014/} }
TY - JOUR AU - Feng, Xiaobing AU - Prohl, Andreas TI - Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2004 SP - 291 EP - 320 VL - 38 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2004014/ DO - 10.1051/m2an:2004014 LA - en ID - M2AN_2004__38_2_291_0 ER -
%0 Journal Article %A Feng, Xiaobing %A Prohl, Andreas %T Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting %J ESAIM: Modélisation mathématique et analyse numérique %D 2004 %P 291-320 %V 38 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2004014/ %R 10.1051/m2an:2004014 %G en %F 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: 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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