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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
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.
Keywords
- Type
- Research Article
- Information
- ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 2 , March 2004 , pp. 291 - 320
- Copyright
- © EDP Sciences, SMAI, 2004
References
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