Discrete approximation of a free discontinuity problem
- 1 January 1994
- journal article
- research article
- Published by Taylor & Francis in Numerical Functional Analysis and Optimization
- Vol. 15 (3-4) , 201-224
- https://doi.org/10.1080/01630569408816562
Abstract
We approximate by discrete Г-convergence a functional proposed by Mumford-Shah for a variational approach to image segmentation. Such a functional is first relaxed with a sequence of nonconvex functionals, which in turn, are dis-cretized by piecewise linear finite elements. Under a suitable relation between the relaxation parameter ∊and the meshsize h, the convergence of the discrete functionals and the compactness of any sequence of discrete minimizers are proved. The proof relies on the techniques of Г-convergence and on the properties of the Lagrange interpolation and Clement operators.Keywords
This publication has 16 references indexed in Scilit:
- Elliptic Partial Differential Equations of Second OrderPublished by Springer Nature ,2001
- An Introduction to Γ-ConvergencePublished by Springer Nature ,1993
- A variational method in image segmentation: Existence and approximation resultsActa Mathematica, 1992
- Approximation of functional depending on jumps by elliptic functional via t‐convergenceCommunications on Pure and Applied Mathematics, 1990
- Numerical minimization of geometrical type problems related to calculus of variationsCalcolo, 1990
- Existence theory for a new class of variational problemsArchive for Rational Mechanics and Analysis, 1990
- Existence theorem for a minimum problem with free discontinuity setArchive for Rational Mechanics and Analysis, 1989
- Variational problems in SBV and image segmentationActa Applicandae Mathematicae, 1989
- Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of ImagesPublished by Institute of Electrical and Electronics Engineers (IEEE) ,1984
- Minimal Surfaces and Functions of Bounded VariationPublished by Springer Nature ,1984