Wasserstein Barycenter and Its Application to Texture Mixing

  • Julien Rabin
  • Gabriel Peyré
  • Julie Delon
  • Marc Bernot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6667)


This paper proposes a new definition of the averaging of discrete probability distributions as a barycenter over the Monge-Kantorovich optimal transport space. To overcome the time complexity involved by the numerical solving of such problem, the original Wasserstein metric is replaced by a sliced approximation over 1D distributions. This enables us to introduce a new fast gradient descent algorithm to compute Wasserstein barycenters of point clouds.

This new notion of barycenter of probabilities is likely to find applications in computer vision where one wants to average features defined as distributions. We show an application to texture synthesis and mixing, where a texture is characterized by the distribution of the response to a multi-scale oriented filter bank. This leads to a simple way to navigate over a convex domain of color textures.


Optimal transport texture synthesis and mixing barycenter of distributions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Villani, C.: Topics in Optimal Transportation. American Math. Society (2003)Google Scholar
  2. 2.
    Rubner, Y., Tomasi, C., Guibas, L.J.: The Earth Mover’s Distance as a Metric for Image Retrieval. International Journal of Computer Vision 40, 99–121 (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Pitié, F., Kokaram, A.: The Linear Monge-Kantorovitch Colour Mapping for Example-Based Colour Transfer. In: Proc. of CVMP 2006 (2006)Google Scholar
  4. 4.
    Dominitz, A., Tannenbaum, A.: Texture mapping via optimal mass transport. IEEE Transactions on Visualization and Computer Graphics 16, 419–433 (2009)CrossRefGoogle Scholar
  5. 5.
    Delon, J.: Movie and video scale-time equalization application to flicker reduction. IEEE Trans. Image Proc. 15, 241–248 (2006)CrossRefGoogle Scholar
  6. 6.
    Ambrosio, L., Caffarelli, L.A., Brenier, Y., Buttazzo, G., Villani, C.: Optimal Transportation and Applications. Mathematics and statistics edn. Lecture Notes in Mathematics, vol. 1813. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Efros, A.A., Leung, T.K.: Texture synthesis by non-parametric sampling. In: Proc. of ICCV 1999, p. 1033 (1999)Google Scholar
  8. 8.
    Wei, L.Y., Levoy, M.: Fast texture synthesis using tree-structured vector quantization. In: Proc. Siggraph 2000, pp. 479–488 (2000)Google Scholar
  9. 9.
    Efros, A., Freeman, W.: Image quilting for texture synthesis and transfer. ACM Trans. on Graphics, 341–346 (2001)Google Scholar
  10. 10.
    Ashikhmin, M.: Synthesizing natural textures. In: SI3D 2001: Proceedings of the 2001 Symposium on Interactive 3D Graphics, pp. 217–226 (2001)Google Scholar
  11. 11.
    Lefebvre, S., Hoppe, H.: Parallel controllable texture synthesis. ACM Trans. on Graphics 24, 777–786 (2005)CrossRefGoogle Scholar
  12. 12.
    Kwatra, V., Essa, I., Bobick, A., Kwatra, N.: Texture optimization for example-based synthesis. ACM Trans. on Graphics 24, 795–802 (2005)CrossRefGoogle Scholar
  13. 13.
    Kwatra, V., Schdl, A., Essa, I., Turk, G., Bobick, A.: Graphcut textures: Image and video synthesis using graph cuts. ACM Trans. on Graphics 22, 277–286 (2003)CrossRefGoogle Scholar
  14. 14.
    Perlin, K.: An image synthesizer. In: Proc. Siggraph 1985, pp. 287–296. ACM Press, New York (1985)Google Scholar
  15. 15.
    Bonet, J.S.D.: Multiresolution sampling procedure for analysis and synthesis of texture images. In: Proc. Siggraph 1997, pp. 361–368. ACM Press, New York (1997)Google Scholar
  16. 16.
    Paget, R., Longstaff, I.D.: Texture synthesis via a noncausal nonparametric multiscale markov random field. IEEE Trans. Image Proc. 7, 925–931 (1998)CrossRefGoogle Scholar
  17. 17.
    Mumford, D., Gidas, B.: Stochastic models for generic images. Q. Appl. Math. LIV, 85–111 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Heeger, D.J., Bergen, J.R.: Pyramid-Based texture analysis/synthesis. In: Proc. Siggraph 1995. Annual Conference Series, ACM SIGGRAPH, pp. 229–238 (1995)Google Scholar
  19. 19.
    Cook, R., DeRose, T.: Wavelet noise. ACM Trans. on Graphics 24, 803–811 (2005)CrossRefGoogle Scholar
  20. 20.
    Portilla, J., Simoncelli, E.P.: A parametric texture model based on joint statistics of complex wavelet coefficients. Int. Journal of Computer Vision 40, 49–70 (2000)CrossRefzbMATHGoogle Scholar
  21. 21.
    Bar-Joseph, Z., El-Yaniv, R., Lischinski, D., Werman, M.: Texture mixing and texture movie synthesis using statistical learning. IEEE Transactions on Visualization and Computer Graphics 7, 120–135 (2001)CrossRefGoogle Scholar
  22. 22.
    Peyré, G.: Texture synthesis with grouplets. IEEE Trans. Patt. Anal. and Mach. Intell. 32, 733–746 (2010)CrossRefGoogle Scholar
  23. 23.
    Hertzmann, A., Jacobs, C.E., Oliver, N., Curless, B., Salesin, D.H.: Image analogies. In: ACM (ed.) Proc. Siggraph 2001, pp. 327–340. ACM Press, New York (2001)Google Scholar
  24. 24.
    Liu, Z., Liu, C., Shum, H.Y., Yu, Y.: Pattern-based texture metamorphosis. In: Proc. Pacific Graphics 2002, pp. 184–193. IEEE Computer Society, Los Alamitos (2002)Google Scholar
  25. 25.
    Tonietto, L., Walter, M.: Texture metamorphosis driven by texton masks. Computers and Graphics 29, 697–703 (2005)CrossRefGoogle Scholar
  26. 26.
    Tal, A., Elber, G.: Image morphing with feature preserving texture. Comput. Graph. Forum 18, 339–348 (1999)CrossRefGoogle Scholar
  27. 27.
    Matusik, W., Zwicker, M., Durand, F.: Texture design using a simplicial complex of morphable textures. ACM Trans. on Graphics 24, 787–794 (2005)CrossRefGoogle Scholar
  28. 28.
    Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  29. 29.
    Rabin, J., Peyré, G., Cohen, L.D.: Geodesic shape retrieval via optimal mass transport. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6315, pp. 771–784. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  30. 30.
    Dowson, D.C., Landau, B.V.: The Fréchet distance between multivariate normal distributions. Journal of Multivariate Analysis 12, 450–455 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Agueh, M., Carlier, G.: Barycenters in the Wasserstein space. To appear in SIAM Journal on Mathematical Analysis (2011) Google Scholar
  32. 32.
    Simoncelli, E.P., Freeman, W.T., Adelson, E.H., Heeger, D.J.: Shiftable multiscale transforms. IEEE Trans. Info. Theory 38, 587–607 (1992)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Julien Rabin
    • 1
  • Gabriel Peyré
    • 2
  • Julie Delon
    • 3
  • Marc Bernot
    • 4
  1. 1.CMLA, ENS de CachanFrance
  2. 2.Ceremade, Univ. Paris-DauphineFrance
  3. 3.LTCI, Telecom ParisTechFrance
  4. 4.Thales Alenia SpaceFrance

Personalised recommendations