Transportation distances on the circle

Journal of Mathematical Imaging and Vision

Volumn 41, Issue 1-2, 2011, Pages 147-167

  Rabin, Julien ^a   Delon, Julie ^b   Gousseau, Yann ^b  

^a UNIVERSITÉ PARIS SACLAY   (France)

^b INSTITUT TELECOM   (France)

Author keywords

Circular histograms; Earth Mover's Distance; Optimal mass transportation theory

Indexed keywords

CIRCULAR HISTOGRAMS; COLOR IMAGES; COLOR TRANSFERS; DISCRETE DISTRIBUTION; DISCRETE HISTOGRAMS; DISSIMILARITY MEASURES; EARTH MOVER'S DISTANCE; GEODESIC DISTANCES; INCREASING FUNCTIONS; OPTIMAL MASS TRANSPORT; OPTIMAL MASS TRANSPORTATION THEORY; REAL LINE; TRANSPORTATION DISTANCE; TWO-POINT; UNIT CIRCLES;

COST FUNCTIONS; GEODESY; GRAPHIC METHODS; OPTIMIZATION;

MASS TRANSPORTATION;

EID: 80051547151     PISSN: 09249907     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10851-011-0284-0     Document Type: Article

References (49)

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)
2. Burkard, R., Dell'Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia (2009)
3. S. Belongie J. Malik J. Puzicha 2002 Shape matching and object recognition using shape contexts IEEE Trans. Pattern Anal. Mach. Intell. 24 4 509 522 10.1109/34.993558 (Pubitemid 34448769)
4. C.A. Cabrelli U.M. Molter 1995 The Kantorovich metric for probability measures on the circle J. Comput. Appl. Math. 57 3 345 361 1335789 0819.60001 10.1016/0377-0427(93)E0213-6
5. C.A. Cabrelli U.M. Molter 1998 A linear time algorithm for a matching problem on the circle Inf. Process. Lett. 66 3 161 164 1627995 0925.68197 10.1016/S0020-0190(98)00048-9 (Pubitemid 128401900)
6. S.-H. Cha S.N. Srihari 2002 On measuring the distance between histograms Pattern Recognit. 35 6 1355 1370 0997.68123 10.1016/S0031-3203(01)00118-2 (Pubitemid 34262340)
7. Cullen, M.J.P.: A Mathematical Theory of Large-Scale Atmospheric-Ocean Flow. Imperial College Press, London (2006)
8. J. Delon J. Salomon A. Sobolevskii 2010 Fast transport optimization for Monge costs on the circle SIAM J. Appl. Math. 70 7 2239 2258 2678036 05876536 10.1137/090772708
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)
10. Frisch, U., Matarrese, S., Mohayaee, R., Sobolevski, A.: A reconstruction of the initial conditions of the universe by optimal mass transportation. Nature (2002)
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)
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)
13. W. Gangbo R.J. McCann 1996 The geometry of optimal transportation Acta Math. 177 2 113 161 1440931 0887.49017 10.1007/BF02392620 (Pubitemid 126345872)
14. C. Gurwitz 1990 Weighted median algorithms for L1 approximation BIT Numer. Math. 30 2 301 310 1039669 0704.65044 10.1007/BF02017350
15. T. Hurtut Y. Gousseau F. Schmitt 2008 Adaptive image retrieval based on the spatial organization of colors Comput. Vis. Image Underst. 112 2 101 113 10.1016/j.cviu.2007.12.006
16. S. Haker L. Zhu A. Tannenbaum S. Angenent 2004 Optimal mass transport for registration and warping Int. J. Comput. Vis. 60 3 225 240 10.1023/B:VISI. 0000036836.66311.97
17. Indyk, P., Thaper, N.: Fast image retrieval via embeddings. In: 3rd International Workshop on Statistical and Computational Theories of Vision, Nice, France (2003)
18. A.C. Jalba M.H.F. Wilkinson J.B.T.M. Roerdink 2006 Shape representation and recognition through morphological curvature scale spaces IEEE Trans. Image Process. 15 2 331 341 10.1109/TIP.2005.860606 (Pubitemid 43126158)
19. L. Kantorovich 1942 On the transfer of masses Dokl. Akad. Nauk 37 2 227 229 (in Russian)
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) (Pubitemid 40673448)
21. H. Ling K. Okada 2007 An efficient Earth Mover's distance algorithm for robust histogram comparison IEEE Trans. Pattern Anal. Mach. Intell. 29 5 840 853 10.1109/TPAMI.2007.1058 (Pubitemid 46534780)
22. D.G. Lowe 2004 Distinctive image features from scale-invariant keypoints Int. J. Comput. Vis. 60 2 91 110 10.1023/B:VISI.0000029664.99615.94
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)
24. R.J. McCann 1995 Existence and uniqueness of monotone measure-preserving maps Duke Math. J. 80 2 309 323 1369395 0873.28009 10.1215/S0012-7094-95-08013-2
25. McCann, R.J.: Exact solutions to the transportation problem on the line. In: Proceedings: Mathematical, Physical and Engineering Sciences, pp. 1341-1380 (1999)
26. F. Mokhtarian 1997 Silhouette-based occluded object recognition through curvature scale space Mach. Vis. Appl. 10 3 87 97 10.1007/s001380050062 (Pubitemid 127569524)
27. Monge, G.: Mémoire sur la théorie des déblais et des remblais. In: Histoire de l'Académie Royale des Sciences (1781)
28. J. Morovic P.L. Sun 2003 Accurate 3d image colour histogram transformation Pattern Recognit. Lett. 24 11 1725 1735 10.1016/S0167-8655(02) 00328-8 (Pubitemid 36386786)
29. Orlin, J.: A faster strongly polynomial minimum cost flow algorithm. In: STOC (1988)
30. Pele, O.: Source code for EMD. http://www.cs.huji.ac.il/~ofirpele/ FastEMD/code/
31. T2: http://www.cs.huji.ac.il/ ~ofirpele/SiftDist/code/
32. Pitié, F., Kokaram, A., Dahyot, R.: Automated colour grading using colour distribution transfer. Comput. Vision Image Underst., February 2007
33. Pele, O., Werman, M.: A linear time histogram metric for improved sift matching. In: ECCV08 (2008)
34. Pele, O., Werman, M.: Fast and robust earth mover's distances. In: ICCV (2009)
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)
36. Rabin, J., Delon, J., Gousseau, Y.: A statistical approach to the matching of local features. SIAM J. Imaging Sci. (2009)
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)
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)
39. M.A. Ruzon C. Tomasi 2001 Edge, junction, and corner detection using color distributions IEEE Trans. Pattern Anal. Mach. Intell. 23 11 1281 1295 10.1109/34.969118 (Pubitemid 33137963)
40. Y. Rubner C. Tomasi L.J. Guibas 2000 The Earth Mover's distance as a metric for image retrieval Int. J. Comput. Vis. 40 2 99 121 1012.68705 10.1023/A:1026543900054 (Pubitemid 32136368)
41. Rubner, Y.: Source code for EMD. http://robotics.stanford.edu/~rubner/
42. Shirdhonkar, S., Jacobs, D.W.: Approximate earth mover's distance in linear time. In: CVPR08, pp. 1-8 (2008)
43. H.C. Shen A.K.C. Wong 1983 Generalized texture representation and metric Comput. Vis. Graph. Image Process. 23 2 187 206 1332570 10.1016/0734-189X(83) 90112-3
44. Villani, C.: Topics in Optimal Transportation. Am. Math. Soc., Providence (2003)
45. Villani, C.: Optimal Transport: Old and New. Springer, Berlin (2008)
46. M. Werman S. Peleg R. Melter T.Y. Kong 1986 Bipartite graph matching for points on a line or a circle J. Algorithms 7 2 277 284 840776 0605.68059 10.1016/0196-6774(86)90009-X
47. M. Werman S. Peleg A. Rosenfeld 1985 A distance metric for multidimensional histograms Comput. Vis. Graph. Image Process. 32 3 328 336 10.1016/0734-189X(85)90055-6
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) (Pubitemid 47113508)
49. L. Zhu Y. Yang S. Haker A. Tannenbaum 2007 An image morphing technique based on optimal mass preserving mapping IEEE Trans. Image Process. 16 6 1481 1495 2465225 10.1109/TIP.2007.896637 (Pubitemid 46804334)