Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem

Handle URI:
http://hdl.handle.net/10754/599097
Title:
Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem
Authors:
Lellmann, Jan; Lenzen, Frank; Schnörr, Christoph
Abstract:
We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finite-dimensional problems. While for the latter several optimality bounds are known, to our knowledge no such bounds exist in the infinite-dimensional setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula. © 2012 Springer Science+Business Media New York.
Citation:
Lellmann J, Lenzen F, Schnörr C (2012) Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem. J Math Imaging Vis 47: 239–257. Available: http://dx.doi.org/10.1007/s10851-012-0390-7.
Publisher:
Springer Nature
Journal:
Journal of Mathematical Imaging and Vision
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
9-Nov-2012
DOI:
10.1007/s10851-012-0390-7
Type:
Article
ISSN:
0924-9907; 1573-7683
Sponsors:
This publication is partly based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorLellmann, Janen
dc.contributor.authorLenzen, Franken
dc.contributor.authorSchnörr, Christophen
dc.date.accessioned2016-02-25T13:52:49Zen
dc.date.available2016-02-25T13:52:49Zen
dc.date.issued2012-11-09en
dc.identifier.citationLellmann J, Lenzen F, Schnörr C (2012) Optimality Bounds for a Variational Relaxation of the Image Partitioning Problem. J Math Imaging Vis 47: 239–257. Available: http://dx.doi.org/10.1007/s10851-012-0390-7.en
dc.identifier.issn0924-9907en
dc.identifier.issn1573-7683en
dc.identifier.doi10.1007/s10851-012-0390-7en
dc.identifier.urihttp://hdl.handle.net/10754/599097en
dc.description.abstractWe consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finite-dimensional problems. While for the latter several optimality bounds are known, to our knowledge no such bounds exist in the infinite-dimensional setting. We provide such a bound by analyzing a probabilistic rounding method, showing that it is possible to obtain an integral solution of the original partitioning problem from a solution of the relaxed problem with an a priori upper bound on the objective. The approach has a natural interpretation as an approximate, multiclass variant of the celebrated coarea formula. © 2012 Springer Science+Business Media New York.en
dc.description.sponsorshipThis publication is partly based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSpringer Natureen
dc.subjectApproximation bounden
dc.subjectCombinatorial optimizationen
dc.subjectConvex relaxationen
dc.subjectLinear programming relaxationen
dc.subjectMulticlass labelingen
dc.subjectTotal variationen
dc.titleOptimality Bounds for a Variational Relaxation of the Image Partitioning Problemen
dc.typeArticleen
dc.identifier.journalJournal of Mathematical Imaging and Visionen
dc.contributor.institutionUniversitat Heidelberg, Heidelberg, Germanyen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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