A generalized model for optimal transport of images including dissipation and density modulation

Handle URI:
http://hdl.handle.net/10754/597278
Title:
A generalized model for optimal transport of images including dissipation and density modulation
Authors:
Maas, Jan; Rumpf, Martin; Schönlieb, Carola; Simon, Stefan
Abstract:
© EDP Sciences, SMAI 2015. In this paper the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouvé and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects.
Citation:
Maas J, Rumpf M, Schönlieb C, Simon S (2015) A generalized model for optimal transport of images including dissipation and density modulation. ESAIM: Mathematical Modelling and Numerical Analysis 49: 1745–1769. Available: http://dx.doi.org/10.1051/m2an/2015043.
Publisher:
EDP Sciences
Journal:
ESAIM: Mathematical Modelling and Numerical Analysis
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Nov-2015
DOI:
10.1051/m2an/2015043
Type:
Article
ISSN:
0764-583X; 1290-3841
Sponsors:
The authors acknowledge support of the Collaborative Research Centre 1060 funded by the German Science foundation.This work is further supported by the King Abdullah University for Science and Technology (KAUST) Award No. KUK-I1-007-43 and the EPSRC grant Nr. EP/M00483X/1.
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Full metadata record

DC FieldValue Language
dc.contributor.authorMaas, Janen
dc.contributor.authorRumpf, Martinen
dc.contributor.authorSchönlieb, Carolaen
dc.contributor.authorSimon, Stefanen
dc.date.accessioned2016-02-25T12:29:40Zen
dc.date.available2016-02-25T12:29:40Zen
dc.date.issued2015-11en
dc.identifier.citationMaas J, Rumpf M, Schönlieb C, Simon S (2015) A generalized model for optimal transport of images including dissipation and density modulation. ESAIM: Mathematical Modelling and Numerical Analysis 49: 1745–1769. Available: http://dx.doi.org/10.1051/m2an/2015043.en
dc.identifier.issn0764-583Xen
dc.identifier.issn1290-3841en
dc.identifier.doi10.1051/m2an/2015043en
dc.identifier.urihttp://hdl.handle.net/10754/597278en
dc.description.abstract© EDP Sciences, SMAI 2015. In this paper the optimal transport and the metamorphosis perspectives are combined. For a pair of given input images geodesic paths in the space of images are defined as minimizers of a resulting path energy. To this end, the underlying Riemannian metric measures the rate of transport cost and the rate of viscous dissipation. Furthermore, the model is capable to deal with strongly varying image contrast and explicitly allows for sources and sinks in the transport equations which are incorporated in the metric related to the metamorphosis approach by Trouvé and Younes. In the non-viscous case with source term existence of geodesic paths is proven in the space of measures. The proposed model is explored on the range from merely optimal transport to strongly dissipative dynamics. For this model a robust and effective variational time discretization of geodesic paths is proposed. This requires to minimize a discrete path energy consisting of a sum of consecutive image matching functionals. These functionals are defined on corresponding pairs of intensity functions and on associated pairwise matching deformations. Existence of time discrete geodesics is demonstrated. Furthermore, a finite element implementation is proposed and applied to instructive test cases and to real images. In the non-viscous case this is compared to the algorithm proposed by Benamou and Brenier including a discretization of the source term. Finally, the model is generalized to define discrete weighted barycentres with applications to textures and objects.en
dc.description.sponsorshipThe authors acknowledge support of the Collaborative Research Centre 1060 funded by the German Science foundation.This work is further supported by the King Abdullah University for Science and Technology (KAUST) Award No. KUK-I1-007-43 and the EPSRC grant Nr. EP/M00483X/1.en
dc.publisherEDP Sciencesen
dc.subjectFlow of diffeomorphismen
dc.subjectMetamorphosisen
dc.subjectOptimal transporten
dc.subjectVariational time discretizationen
dc.titleA generalized model for optimal transport of images including dissipation and density modulationen
dc.typeArticleen
dc.identifier.journalESAIM: Mathematical Modelling and Numerical Analysisen
dc.contributor.institutionInstitute of Science and Technology Austria, Klosterneuburg, Austriaen
dc.contributor.institutionUniversitat Bonn, Bonn, Germanyen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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