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Gamma-Convergence for Beginners$
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Andrea Braides

Print publication date: 2002

Print ISBN-13: 9780198507840

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198507840.001.0001

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SEGMENTATION PROBLEMS

SEGMENTATION PROBLEMS

Chapter:
(p.85) 5 SEGMENTATION PROBLEMS
Source:
Gamma-Convergence for Beginners
Author(s):

Andrea Braides

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198507840.003.0006

The study of segmentation problems, i.e., variational problems where the unknown is a piecewise-constant function, allows the introduction of a new type of weak convergence, related to the weak* convergence of measures, and the corresponding variational concepts. In particular, lower semicontinuity of segmentation functionals is proved to be equivalent to sub-additivity of integrands. Correspondingly, the notions of relaxation and Gamma-convergence can be expressed through suitable sub-additive envelopes. The theory of Caccioppoli partitions in higher dimension is outlined.

Keywords:   segmentation problems, weak* convergence, subadditivity, Caccioppoli partitions

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