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

Xiaobing Feng; Andreas Prohl

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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 L2 x 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 H1 x H1 ∩ 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.

How to cite

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Feng, 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 38.2 (2010): 291-320. <http://eudml.org/doc/194215>.

@article{Feng2010,
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 L2 x 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 H1 x H1 ∩ 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 $\frac\{1\}\{\{\varepsilon\}\}$ and $\frac\{1\}\{k_\{\varepsilon\}\}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^\{\frac12\})$. 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},
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; finite element method; error analysis},
language = {eng},
month = {3},
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/194215},
volume = {38},
year = {2010},
}

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
DA - 2010/3//
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 L2 x 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 H1 x H1 ∩ 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 $\frac{1}{{\varepsilon}}$ and $\frac{1}{k_{\varepsilon}}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac12})$. 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; finite element method; error analysis
UR - http://eudml.org/doc/194215
ER -

References

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