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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 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 -

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