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

Published online by Cambridge University Press:  15 March 2004

Xiaobing Feng  and
Andreas Prohl
Xiaobing Feng
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA.
Andreas Prohl
Affiliation:
Department of Mathematics, ETHZ, 8092 Zürich, Switzerland, apr@math.ethz.ch.
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Abstract

This paper studies the gradient flow of a regularized Mumford-Shah functionalproposed 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 timestrong 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.

Keywords

Image segmentation and inpaintingMumford-Shah modelelliptic approximationgradient flowa priori estimatesfinite element methoderror analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 2 , March 2004 , pp. 291 - 320
Copyright
© EDP Sciences, SMAI, 2004

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