Transportation Distances on the Circle

  • Julien RabinEmail author
  • Julie Delon
  • Yann Gousseau


This paper is devoted to the study of the Monge-Kantorovich theory of optimal mass transport, in the special case of one-dimensional and circular distributions. More precisely, we study the Monge-Kantorovich problem between discrete distributions on the unit circle S 1, in the case where the ground distance between two points x and y is defined as h(d(x,y)), where d is the geodesic distance on the circle and h a convex and increasing function. This study complements previous results in the literature, holding only for a ground distance equal to the geodesic distance d. We first prove that computing a Monge-Kantorovich distance between two given sets of pairwise different points boils down to cut the circle at a well chosen point and to compute the same distance on the real line. This result is then used to obtain a dissimilarity measure between 1-D and circular discrete histograms. In a last part, a study is conducted to compare the advantages and drawbacks of transportation distances relying on convex or concave cost functions, and of the classical L 1 distance. Simple retrieval experiments based on the hue component of color images are shown to illustrate the interest of circular distances. The framework is eventually applied to the problem of color transfer between images.


Optimal mass transportation theory Earth Mover’s Distance Circular histograms 


  1. 1.
    Ambrosio, L., Caffarelli, L.A., Brenier, Y., Buttazzo, G., Villani, C.: Optimal Transportation and Applications. Lecture Notes in Mathematics, vol. 1813. Springer, Berlin (2003) zbMATHGoogle Scholar
  2. 2.
    Burkard, R., Dell’Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia (2009) zbMATHCrossRefGoogle Scholar
  3. 3.
    Belongie, S., Malik, J., Puzicha, J.: Shape matching and object recognition using shape contexts. IEEE Trans. Pattern Anal. Mach. Intell. 24(4), 509–522 (2002) CrossRefGoogle Scholar
  4. 4.
    Cabrelli, C.A., Molter, U.M.: The Kantorovich metric for probability measures on the circle. J. Comput. Appl. Math. 57(3), 345–361 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cabrelli, C.A., Molter, U.M.: A linear time algorithm for a matching problem on the circle. Inf. Process. Lett. 66(3), 161–164 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cha, S.-H., Srihari, S.N.: On measuring the distance between histograms. Pattern Recognit. 35(6), 1355–1370 (2002) zbMATHCrossRefGoogle Scholar
  7. 7.
    Cullen, M.J.P.: A Mathematical Theory of Large-Scale Atmospheric-Ocean Flow. Imperial College Press, London (2006) CrossRefGoogle Scholar
  8. 8.
    Delon, J., Salomon, J., Sobolevskii, A.: Fast transport optimization for Monge costs on the circle. SIAM J. Appl. Math. 70(7), 2239–2258 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dvir, G.: Context-based image modelling. In: Proceedings of the 2002 IEEE International Conference on Pattern Recognition (ICPR), vol. 4, p. 40162. IEEE Comput. Soc., Los Alamitos (2002) Google Scholar
  10. 10.
    Frisch, U., Matarrese, S., Mohayaee, R., Sobolevski, A.: A reconstruction of the initial conditions of the universe by optimal mass transportation. Nature (2002) Google Scholar
  11. 11.
    Grauman, K., Darrell, T.J.: Fast contour matching using approximate earth mover’s distance. In: Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’04), pp. 220–227 (2004) CrossRefGoogle Scholar
  12. 12.
    Greenspan, H., Dvir, G., Rubner, Y.: Region correspondence for image matching via emd flow. In: CBAIVL ’00: Proceedings of the IEEE Workshop on Content-Based Access of Image and Video Libraries (CBAIVL’00), p. 27. IEEE Computer Society, Washington, DC (2000) CrossRefGoogle Scholar
  13. 13.
    Gangbo, W., McCann, R.J.: The geometry of optimal transportation. Acta Math. 177(2), 113–161 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Gurwitz, C.: Weighted median algorithms for L1 approximation. BIT Numer. Math. 30(2), 301–310 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hurtut, T., Gousseau, Y., Schmitt, F.: Adaptive image retrieval based on the spatial organization of colors. Comput. Vis. Image Underst. 112(2), 101–113 (2008) CrossRefGoogle Scholar
  16. 16.
    Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Int. J. Comput. Vis. 60(3), 225–240 (2004) CrossRefGoogle Scholar
  17. 17.
    Indyk, P., Thaper, N.: Fast image retrieval via embeddings. In: 3rd International Workshop on Statistical and Computational Theories of Vision, Nice, France (2003) Google Scholar
  18. 18.
    Jalba, A.C., Wilkinson, M.H.F., Roerdink, J.B.T.M.: Shape representation and recognition through morphological curvature scale spaces. IEEE Trans. Image Process. 15(2), 331–341 (2006) CrossRefGoogle Scholar
  19. 19.
    Kantorovich, L.: On the transfer of masses. Dokl. Akad. Nauk 37(2), 227–229 (1942) (in Russian) Google Scholar
  20. 20.
    Lv, Q., Charikar, M., Li, K.: Image similarity search with compact data structures. In: CIKM ’04: Proceedings of the Thirteenth ACM International Conference on Information and Knowledge Management, pp. 208–217. ACM, New York (2004) CrossRefGoogle Scholar
  21. 21.
    Ling, H., Okada, K.: An efficient Earth Mover’s distance algorithm for robust histogram comparison. IEEE Trans. Pattern Anal. Mach. Intell. 29(5), 840–853 (2007) CrossRefGoogle Scholar
  22. 22.
    Lowe, D.G.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vis. 60(2), 91–110 (2004) CrossRefGoogle Scholar
  23. 23.
    Liu, Y., Zhang, D., Lu, G., Ma, W.-Y.: Region-based image retrieval with high-level semantic color names. In: MMM ’05: Proceedings of the 11th International Multimedia Modelling Conference, Washington, DC, USA, pp. 180–187. IEEE Comput. Soc., Los Alamitos (2005) Google Scholar
  24. 24.
    McCann, R.J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(2), 309–323 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    McCann, R.J.: Exact solutions to the transportation problem on the line. In: Proceedings: Mathematical, Physical and Engineering Sciences, pp. 1341–1380 (1999) Google Scholar
  26. 26.
    Mokhtarian, F.: Silhouette-based occluded object recognition through curvature scale space. Mach. Vis. Appl. 10(3), 87–97 (1997) CrossRefGoogle Scholar
  27. 27.
    Monge, G.: Mémoire sur la théorie des déblais et des remblais. In: Histoire de l’Académie Royale des Sciences (1781) Google Scholar
  28. 28.
    Morovic, J., Sun, P.L.: Accurate 3d image colour histogram transformation. Pattern Recognit. Lett. 24(11), 1725–1735 (2003) CrossRefGoogle Scholar
  29. 29.
    Orlin, J.: A faster strongly polynomial minimum cost flow algorithm. In: STOC (1988) Google Scholar
  30. 30.
    Pele, O.: Source code for EMD.
  31. 31.
    Pele, O.: Source code for MKT2:
  32. 32.
    Pitié, F., Kokaram, A., Dahyot, R.: Automated colour grading using colour distribution transfer. Comput. Vision Image Underst., February 2007 Google Scholar
  33. 33.
    Pele, O., Werman, M.: A linear time histogram metric for improved sift matching. In: ECCV08 (2008) Google Scholar
  34. 34.
    Pele, O., Werman, M.: Fast and robust earth mover’s distances. In: ICCV (2009) Google Scholar
  35. 35.
    Rabin, J., Delon, J., Gousseau, Y.: Circular Earth Mover’s Distance for the comparison of local features. In: Proceedings of the IEEE International Conference on Pattern Recognition (ICPR). IEEE Computer Society, Los Alamitos (2008) Google Scholar
  36. 36.
    Rabin, J., Delon, J., Gousseau, Y.: A statistical approach to the matching of local features. SIAM J. Imaging Sci. (2009) Google Scholar
  37. 37.
    Rabin, J., Delon, J., Gousseau, Y.: Regularization of transportation maps for color and contrast transfer. In: Proceedings of the IEEE International Conference on Image Processing (ICIP). IEEE Computer Society, Los Alamitos (2010) Google Scholar
  38. 38.
    Rabin, J., Peyré, G., Cohen, L.D.: Geodesic shape retrieval via optimal mass transport. In: Proceedings of the European Conference on Computer Vision (ECCV’10) (2010) Google Scholar
  39. 39.
    Ruzon, M.A., Tomasi, C.: Edge, junction, and corner detection using color distributions. IEEE Trans. Pattern Anal. Mach. Intell. 23(11), 1281–1295 (2001) CrossRefGoogle Scholar
  40. 40.
    Rubner, Y., Tomasi, C., Guibas, L.J.: The Earth Mover’s distance as a metric for image retrieval. Int. J. Comput. Vis. 40(2), 99–121 (2000) zbMATHCrossRefGoogle Scholar
  41. 41.
    Rubner, Y.: Source code for EMD.
  42. 42.
    Shirdhonkar, S., Jacobs, D.W.: Approximate earth mover’s distance in linear time. In: CVPR08, pp. 1–8 (2008) Google Scholar
  43. 43.
    Shen, H.C., Wong, A.K.C.: Generalized texture representation and metric. Comput. Vis. Graph. Image Process. 23(2), 187–206 (1983) MathSciNetCrossRefGoogle Scholar
  44. 44.
    Villani, C.: Topics in Optimal Transportation. Am. Math. Soc., Providence (2003) zbMATHGoogle Scholar
  45. 45.
    Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2008) Google Scholar
  46. 46.
    Werman, M., Peleg, S., Melter, R., Kong, T.Y.: Bipartite graph matching for points on a line or a circle. J. Algorithms 7(2), 277–284 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Werman, M., Peleg, S., Rosenfeld, A.: A distance metric for multidimensional histograms. Comput. Vis. Graph. Image Process. 32(3), 328–336 (1985) CrossRefGoogle Scholar
  48. 48.
    Zheng, Q.-F., Wang, W.-Q., Gao, W.: Effective and efficient object-based image retrieval using visual phrases. In: MULTIMEDIA ’06: Proceedings of the 14th Annual ACM International Conference on Multimedia, pp. 77–80. ACM, New York (2006) CrossRefGoogle Scholar
  49. 49.
    Zhu, L., Yang, Y., Haker, S., Tannenbaum, A.: An image morphing technique based on optimal mass preserving mapping. IEEE Trans. Image Process. 16(6), 1481–1495 (2007) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.CMLAENS de CachanCachanFrance
  2. 2.CNRS LTCI, Télécom ParisTechParis Cedex 13France

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