Existence and Regularity of Solutions to a Variational Problem of Mumford and Shah: a Constructive Approach (1991)
| Citations: | 1 - 0 self |
BibTeX
@MISC{Wang91existenceand,
author = {Yang Wang},
title = {Existence and Regularity of Solutions to a Variational Problem of Mumford and Shah: a Constructive Approach},
year = {1991}
}
Bookmark
 |
 |
 |
 |
 |
 |
OpenURL
Abstract
We study a variational problem arising in the approach of Mumford and Shah to the image segmentation problem of computer vision. Given f 2 L 1 (D) for a domain D in R 2 the simplified Mumford--Shah energy associated to a decomposition D =\Omega 1 [\Delta \Delta \Delta[\Omega N is E 0 [\Gamma; ff] = N X i=1 Z \Omega i (f(x) \Gamma c\Omega i ) 2 dx + ffj\Gammaj; where ff ? 0 is a constant, c\Omega i is the average of f(x) on\Omega i and where j\Gammaj is the length of the boundary of the regions\Omega i not in @D. Mumford and Shah showed, using geometric measure theory, that for a continuous f a minimizing \Gamma exists which is piecewise C 2 . We prove this result constructively, and also extend it to show for general bounded measurable f that a minimizer exists. Furthermore, we prove that every minimizer must be piecewsie C 1;1 . Our approach is to study E 0 [\Gamma; ff] on the class of piecewise linear \Gamma. Key words: image segmentaion, Mumford--S...
