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

J. Maas, M. Rumpf, C. Schönlieb, S. Simon, ESAIM: Mathematical Modelling and Numerical Analysis 49 (2015) 1745–1769.


Journal Article | Published | English
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Abstract
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.
Publishing Year
Date Published
2015-11-01
Journal Title
ESAIM: Mathematical Modelling and Numerical Analysis
Acknowledgement
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.
Volume
49
Issue
6
Page
1745 - 1769
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Cite this

Maas J, Rumpf M, Schönlieb C, Simon S. A generalized model for optimal transport of images including dissipation and density modulation. ESAIM: Mathematical Modelling and Numerical Analysis. 2015;49(6):1745-1769. doi:10.1051/m2an/2015043
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(6), 1745–1769. https://doi.org/10.1051/m2an/2015043
Maas, Jan, Martin Rumpf, Carola Schönlieb, and Stefan Simon. “A Generalized Model for Optimal Transport of Images Including Dissipation and Density Modulation.” ESAIM: Mathematical Modelling and Numerical Analysis 49, no. 6 (2015): 1745–69. https://doi.org/10.1051/m2an/2015043.
J. Maas, M. Rumpf, C. Schönlieb, and S. Simon, “A generalized model for optimal transport of images including dissipation and density modulation,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 49, no. 6, pp. 1745–1769, 2015.
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(6), 1745–1769.
Maas, Jan, et al. “A Generalized Model for Optimal Transport of Images Including Dissipation and Density Modulation.” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 49, no. 6, EDP Sciences, 2015, pp. 1745–69, doi:10.1051/m2an/2015043.

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