# Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting

- Volume: 38, Issue: 2, page 291-320
- ISSN: 0764-583X

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topFeng, Xiaobing, and 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 38.2 (2004): 291-320. <http://eudml.org/doc/244655>.

@article{Feng2004,

abstract = {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\times L^\infty $ 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 \times H^1\cap L^\infty $. 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 $\frac\{1\}\{\{\varepsilon \}\}$ and $\frac\{1\}\{k_\{\varepsilon \}\}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^\{\frac\{1\}\{2\}\})$. Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.},

author = {Feng, Xiaobing, Prohl, Andreas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {image segmentation and inpainting; Mumford-Shah model; elliptic approximation; gradient flow; a priori estimates; finite element method; error analysis; image segmentation; ellpitic approximation; apriori estimates; convergence},

language = {eng},

number = {2},

pages = {291-320},

publisher = {EDP-Sciences},

title = {Analysis of gradient flow of a regularized Mumford-Shah functional for image segmentation and image inpainting},

url = {http://eudml.org/doc/244655},

volume = {38},

year = {2004},

}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2004

PB - EDP-Sciences

VL - 38

IS - 2

SP - 291

EP - 320

AB - 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\times L^\infty $ 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 \times H^1\cap L^\infty $. 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 $\frac{1}{{\varepsilon }}$ and $\frac{1}{k_{\varepsilon }}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac{1}{2}})$. Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.

LA - eng

KW - image segmentation and inpainting; Mumford-Shah model; elliptic approximation; gradient flow; a priori estimates; finite element method; error analysis; image segmentation; ellpitic approximation; apriori estimates; convergence

UR - http://eudml.org/doc/244655

ER -

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