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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.
Maas, Jan 1 ; Rumpf, Martin 2 ; Schönlieb, Carola 3 ; Simon, Stefan 2
@article{M2AN_2015__49_6_1745_0,
author = {Maas, Jan and Rumpf, Martin and Sch\"onlieb, Carola and Simon, Stefan},
title = {A generalized model for optimal transport of images including dissipation and density modulation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1745--1769},
publisher = {EDP-Sciences},
volume = {49},
number = {6},
year = {2015},
doi = {10.1051/m2an/2015043},
mrnumber = {3423274},
zbl = {1348.94009},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015043/}
}
TY - JOUR AU - Maas, Jan AU - Rumpf, Martin AU - Schönlieb, Carola AU - Simon, Stefan TI - A generalized model for optimal transport of images including dissipation and density modulation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1745 EP - 1769 VL - 49 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015043/ DO - 10.1051/m2an/2015043 LA - en ID - M2AN_2015__49_6_1745_0 ER -
%0 Journal Article %A Maas, Jan %A Rumpf, Martin %A Schönlieb, Carola %A Simon, Stefan %T A generalized model for optimal transport of images including dissipation and density modulation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1745-1769 %V 49 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015043/ %R 10.1051/m2an/2015043 %G en %F M2AN_2015__49_6_1745_0
Maas, Jan; Rumpf, Martin; Schönlieb, Carola; Simon, Stefan. A generalized model for optimal transport of images including dissipation and density modulation. ESAIM: Mathematical Modelling and Numerical Analysis , Optimal Transport, Tome 49 (2015) no. 6, pp. 1745-1769. doi: 10.1051/m2an/2015043
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