article

Sliced and Radon Wasserstein Barycenters of Measures

Authors:

Nicolas Bonneel,

Gabriel Peyré,

Hanspeter PfisterAuthors Info & Claims

Journal of Mathematical Imaging and Vision, Volume 51, Issue 1

Pages 22 - 45

https://doi.org/10.1007/s10851-014-0506-3

Published: 01 January 2015 Publication History

Abstract

This article details two approaches to compute barycenters of measures using 1-D Wasserstein distances along radial projections of the input measures. The first method makes use of the Radon transform of the measures, and the second is the solution of a convex optimization problem over the space of measures. We show several properties of these barycenters and explain their relationship. We show numerical approximation schemes based on a discrete Radon transform and on the resolution of a non-convex optimization problem. We explore the respective merits and drawbacks of each approach on applications to two image processing problems: color transfer and texture mixing.

References

[1]

Agueh, M., Carlier, G.: Barycenters in the wasserstein space. SIAM J. Math. Anal. 43(2), 904---924 (2011)

[2]

Averbuch, A., Coifman, R., Donoho, D., Israeli, M., Shkolnisky, Y., Sedelnikov, I.: A framework for discrete integral transformations: II. The 2D discrete radon transform. SIAM J. Sci. Comput. 30(2), 785---803 (2008)

Digital Library

[3]

Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution of the monge-kantorovich mass transfer problem. Numer. Math. 84(3), 375---393 (2000)

[4]

Benamou, J.D., Froese, B.D., Oberman, A.M.: A viscosity solution approach to the Monge-Ampere formulation of the Optimal Transportation Problem. arXiv:1208.4873v2 (2013, unpublished)

[5]

Bertsekas, D.: The auction algorithm: a distributed relaxation method for the assignment problem. Ann. Operat. Res. 14, 105---123 (1988)

Digital Library

[6]

Bigot, J., Klein, T.: Consistent estimation of a population barycenter in the wasserstein space. Preprint arXiv:1212.2562v3 (2014)

[7]

Boman, J., Lindskog, F.: Support theorems for the radon transform and Cramèr---Wold theorems. J. Theor. Prob. 22(3), 683---710 (2009)

[8]

Bonneel, N., van de Panne, M., Paris, S., Heidrich, W.: Displacement interpolation using lagrangian mass transport. ACM Trans. Graph. (SIGGRAPH ASIA'11) 30(6), 1---12 (2011)

[9]

Brady, M.L.: A fast discrete approximation algorithm for the radon transform. J. Comput. 27(1), 107---119 (1998)

Digital Library

[10]

Cuturi, M., Doucet, A.: Fast computation of wasserstein barycenters. arXiv:1310.4375v1 (2013, unpublished)

[11]

Dellacherie, C., Meyer, P.A.: Probabilities and Potential Math. Stud. 29. North Holland, Amsterdam (1978)

[12]

Delon, J.: Movie and video scale-time equalization application to flicker reduction. IEEE Trans. Image Process. 15(1), 241---248 (2006)

Digital Library

[13]

Desolneux, A., Moisan, L., Ronsin, S.: A compact representation of random phase and Gaussian textures. In: Proc. the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 1381---1384 (2012)

[14]

Digne, J., Cohen-Steiner, D., Alliez, P., Goes, F., Desbrun, M.: Feature-preserving surface reconstruction and simplification from defect-laden point sets. J. Math. Imaging Vis. 48(2), 369---382 (2013)

Digital Library

[15]

Ferradans, S., Xia, G.S., Peyré, G., Aujol, J.F.: Optimal transport mixing of gaussian texture models. In: Proc. SSVM'13 (2013)

[16]

Galerne, B., Gousseau, Y., Morel, J.M.: Random phase textures: theory and synthesis. IEEE Trans. Image Process. 20(1), 257---267 (2011)

Digital Library

[17]

Galerne, B., Lagae, A., Lefebvre, S., Drettakis, G.: Gabor noise by example. ACM Trans. Graph. (Proceedings of ACM SIGGRAPH 2012) 31(4), 73.1---73.9 (2012)

[18]

Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Int. J. Comput. Vis. 60(3), 225---240 (2004)

Digital Library

[19]

Helgason, S.: The Radon Transform. Birkhauser, Boston (1980)

[20]

Kantorovich, L.: On the transfer of masses. Doklady Akademii Nauk 37(2), 227---229 (1942). (in russian)

[21]

Kuhn, H.W.: The Hungarian method of solving the assignment problem. Naval Res. Logist. Quart. 2, 83---97 (1955)

[22]

Matusik, W., Zwicker, M., Durand, F.: Texture design using a simplicial complex of morphable textures. ACM Trans. Graph. 24(3), 787---794 (2005)

Digital Library

[23]

McCann, R.J.: A convexity principle for interacting gases. Adv. Math. 128(1), 153---179 (1997)

[24]

Mérigot, Q.: A multiscale approach to optimal transport. Comput. Graph. Forum 30(5), 1583---1592 (2011)

[25]

Papadakis, N., Peyré, G., Oudet, E.: Optimal transport with proximal splitting. SIAM J. Imaging Sci. 7(1), 212---238 (2014)

Digital Library

[26]

Pitié, F., Kokaram, A.C., Dahyot, R.: N-Dimensional probability density function transfer and its application to color transfer. In: Computer Vision, 2005. ICCV 2005. Tenth IEEE International Conference on, vol. 2, pp. 1434---1439. (2005)

[27]

Rabin, J., Delon, J., Gousseau, Y.: Removing artefacts from color and contrast modifications. IEEE Trans. Image Process. 20(11), 3073---3085 (2011)

Digital Library

[28]

Rabin, J., Peyré, G., Delon, J., Bernot, M.: Wasserstein barycenter and its application to texture mixing. In: Scale Space and Variational Methods in Computer Vision (SSVM'11), vol. 6667, pp. 435---446 (2011).

[29]

Reinhard, E., Pouli, T.: Colour spaces for colour transfer. In: Proceedings of the Third international conference on Computational color imaging. CCIW'11, pp. 1---15. Springer, Berlin (2011)

Digital Library

[30]

Rubner, Y., Tomasi, C., Guibas, L.: A metric for distributions with applications to image databases. In: IEEE International Conference on Computer Vision (ICCV'98), pp. 59---66 (1998)

[31]

Solodov, M.: Incremental gradient algorithms with stepsizes bounded away from zero. Comput. Optim. Appl. 11(1), 23---35 (1998)

Digital Library

[32]

Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics Series. American Mathematical Society (2003)

Cited By

Song YFellegara RIuricich FDe Floriani L(2024)Parallel Topology-aware Mesh Simplification on Terrain TreesACM Transactions on Spatial Algorithms and Systems10.1145/365260210:2(1-39)Online publication date: 13-Mar-2024
https://dl.acm.org/doi/10.1145/3652602
Chen YTruong QShen XLi JKing IBaeza-Yates RBonchi F(2024)Shopping Trajectory Representation Learning with Pre-training for E-commerce Customer Understanding and RecommendationProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining10.1145/3637528.3671747(385-396)Online publication date: 25-Aug-2024
https://dl.acm.org/doi/10.1145/3637528.3671747
Chen TLiu GTao MTheodorou EOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Deep multi-marginal momentum schrödinger bridgeProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668617(57058-57086)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668617
Show More Cited By

Sliced and Radon Wasserstein Barycenters of Measures
1. Computing methodologies
  1. Computer graphics

Recommendations

On locality of Radon to Riesz transform

In this paper we present a novel approach to locally compute the Riesz transform from the knowledge of the Radon transform. Previous implementations of the Riesz transform are based on the Fourier or the Radon transforms and their inversion formulae, ...
Learning to Generate Wasserstein Barycenters
Abstract
Optimal transport is a notoriously difficult problem to solve numerically, with current approaches often remaining intractable for very large-scale applications such as those encountered in machine learning. Wasserstein barycenters—the problem of ...
A Canonical Barycenter via Wasserstein Regularization

We introduce a weak notion of barycenter of a probability measure $\mu$ on a metric measure space $(X, d, {\bf m})$, with the metric $d$ and reference measure ${\bf m}$. Under the assumption that all optimal transport plans transporting ${\bf m}$ to any ...

Comments

Information & Contributors

Information

Published In

cover image Journal of Mathematical Imaging and Vision

Journal of Mathematical Imaging and Vision Volume 51, Issue 1

January 2015

228 pages

ISSN:0924-9907

Issue’s Table of Contents

Copyright © Copyright © 2015 Springer Science+Business Media New York.

Publisher

Kluwer Academic Publishers

United States

Publication History

Published: 01 January 2015

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

88
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 22 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Song YFellegara RIuricich FDe Floriani L(2024)Parallel Topology-aware Mesh Simplification on Terrain TreesACM Transactions on Spatial Algorithms and Systems10.1145/365260210:2(1-39)Online publication date: 13-Mar-2024
https://dl.acm.org/doi/10.1145/3652602
Chen YTruong QShen XLi JKing IBaeza-Yates RBonchi F(2024)Shopping Trajectory Representation Learning with Pre-training for E-commerce Customer Understanding and RecommendationProceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining10.1145/3637528.3671747(385-396)Online publication date: 25-Aug-2024
https://dl.acm.org/doi/10.1145/3637528.3671747
Chen TLiu GTao MTheodorou EOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Deep multi-marginal momentum schrödinger bridgeProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668617(57058-57086)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668617
Liu GChen TTheodorou ETao MOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Mirror diffusion models for constrained and watermarked generationProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667980(42898-42917)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3667980
Nguyen KRen THo NOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Markovian sliced Wasserstein distancesProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667852(39812-39841)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3667852
Fayad AIbrahim MOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)On slicing optimality for mutual informationProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667728(36938-36950)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3667728
Mahey GChapel LGasso GBonet CCourty NOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Fast optimal transport through sliced wasserstein generalized geodesicsProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3667658(35350-35385)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3667658
Nguyen KHo NOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)Energy-based sliced wasserstein distanceProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3666915(18046-18075)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3666915
Glaser PWidmann DLindsten FGretton AEvans RShpitser I(2023)Fast and scalable score-based kernel calibration testsProceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence10.5555/3625834.3625899(691-700)Online publication date: 31-Jul-2023
https://dl.acm.org/doi/10.5555/3625834.3625899
Zhang YZhang DLacoste-Julien SBurghouts GSnoek CKrause ABrunskill ECho KEngelhardt BSabato SScarlett J(2023)Unlocking slot attention by changing optimal transport costsProceedings of the 40th International Conference on Machine Learning10.5555/3618408.3620170(41931-41951)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.5555/3618408.3620170
Show More Cited By

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents