Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains

ACM Transactions on Graphics

Volumn 34, Issue 4, 2015, Pages

  Solomon, Justin ^a   De Goes, Fernando ^b   Peyré, Gabriel ^c   Cuturi, Marco ^d   Butscher, Adrian ^e   Nguyen, Andy ^a   Du, Tao ^a   Guibas, Leonidas ^a  

^a STANFORD UNIVERSITY   (United States)

^b Pixar Animation Studios   (United States)

^c UNIVERSITÉ PARIS DAUPHINE   (France)

^d KYOTO UNIVERSITY   (Japan)

^e AUTODESK RESEARCH   (United States)

Author keywords

Displacement interpolation; Entropy; Optimal transportation; Wasserstein distances

Indexed keywords

CONVOLUTION; ENTROPY; GEOMETRY; IMAGE ENHANCEMENT; INTERACTIVE COMPUTER GRAPHICS; INTERPOLATION; LINEAR SYSTEMS;

DISPLACEMENT INTERPOLATION; GAUSSIAN CONVOLUTION; GEOMETRY PROCESSING; IMPROVING PERFORMANCE; ITERATIVE NUMERICAL SCHEME; OPTIMAL TRANSPORTATIONS; OPTIMIZATION PROBLEMS; WASSERSTEIN DISTANCE;

ITERATIVE METHODS;

EID: 84947219062     PISSN: 07300301     EISSN: 15577368     Source Type: Journal    
DOI: 10.1145/2766963     Document Type: Conference Paper

References (54)

1. AGUEH, M., AND CARLIER, G. 2011. Barycenters in the Wasserstein space. SIAM J. Math. Anal. 43, 2, 904-924.
2. ASHIKHMIN, M., AND SHIRLEY, P. 2000. An anisotropic Phong BRDF model. J. of Graph. Tools 5, 2, 25-32.
3. AUBRY, M., SCHLICKEWEI, U., AND CREMERS, D. 2011. The wave kernel signature: A quantum mechanical approach to shape analysis. In Proc. ICCV Workshops, 1626-1633.
4. BENAMOU, J.-D., AND BRENIER, Y. 2000. A computational fluid mechanics solution of the Monge-Kantorovich mass transfer problem. Numerische Mathematik 84, 3, 375-393.
5. BENAMOU, J.-D., CARLIER, G., CUTURI, M., NENNA, L., AND PEYRÉ, G. 2015. Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comp., to appear.
6. BLINN, J. F. 1977. Models of light reflection for computer synthesized pictures. In Proc. SIGGRAPH, vol. 11, 192-198.
7. BONNEEL, N., VAN DE PANNE, M., PARIS, S., AND HEIDRICH, W. 2011. Displacement interpolation using Lagrangian mass transport. ACM Trans. Graph. 30, 6 (Dec.), 158:1-158:12.
8. BONNEEL, N., RABIN, J., PEYRÉ, G., AND PFISTER, H. 2014. Sliced and Radon Wasserstein barycenters of measures. J. Math. Imaging and Vision, to appear.
9. BREGMAN, L. 1967. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comp. Math. and Math. Physics 7, 3, 200-217.
10. BURKARD, R., AND ÇELA, E. 1999. Linear assignment problems and extensions. Handbook of Combinatorial Optimization: Supplement 1, 75.
11. BURKARD, R., DELL'AMICO, M., AND MARTELLO, S. 2009. Assignment Problems. SIAM.
12. CARLIER, G., OBERMAN, A., AND OUDET, E. 2014. Numerical methods for matching for teams and Wasserstein barycenters. Preprint, Ceremade.
13. COVER, T., AND THOMAS, J. 2006. Elements of Information Theory. Wiley.
14. CRANE, K., WEISCHEDEL, C., AND WARDETZKY, M. 2013. Geodesics in heat: A new approach to computing distance based on heat flow. ACM Trans. Graph. 32, 5 (Oct.), 152:1-152:11.
15. CUTURI, M., AND DOUCET, A. 2014. Fast computation of Wasserstein barycenters. In Proc. ICML, vol. 32.
16. CUTURI, M. 2013. Sinkhorn distances: Lightspeed computation of optimal transportation. In Proc. NIPS, vol. 26. 2292-2300.
17. DAVIS, T. A. 2006. Direct Methods for Sparse Linear Systems. SIAM.
18. DE GOES, F., COHEN-STEINER, D., ALLIEZ, P., AND DESBRUN, M. 2011. An optimal transport approach to robust reconstruction and simplification of 2d shapes. In Computer Graph. Forum, vol. 30, 1593-1602.
19. DE GOES, F., BREEDEN, K., OSTROMOUKHOV, V., AND DESBRUN, M. 2012. Blue noise through optimal transport. ACM Trans. Graph. 31, 6 (Nov.), 171:1-171:11.
20. DE GOES, F., MEMARI, P., MULLEN, P., AND DESBRUN, M. 2014. Weighted triangulations for geometry processing. ACM Trans. Graph. 33, 3.
21. DELON, J. 2006. Movie and video scale-time equalization application to flicker reduction. IEEE Trans. on Image Proc. 15, 1, 241-248.
22. DEMING, W. E., AND STEPHAN, F. F. 1940. On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals Math. Stat. 11, 4, 427-444.
23. DERICHE, R. 1993. Recursively implementing the Gaussian and its derivatives. Tech. Rep. RR-1893.
24. DESBRUN, M., MEYER, M., SCHRÖDER, P., AND BARR, A. H. 1999. Implicit fairing of irregular meshes using diffusion and curvature flow. In Proc. SIGGRAPH, 317-324.
25. ESCALANTE, R., AND RAYDAN, M. 2011. Alternating Projection Methods. Fundamentals of Algorithms. SIAM.
26. FERRADANS, S., PAPADAKIS, N., PEYRÉ, G., AND AUJOL, J.-F. 2014. Regularized discrete optimal transport. SIAM J. Imaging Sci. 7, 3, 1853-1882.
27. FRANKLIN, J., AND LORENZ, J. 1989. On the scaling of multidimensional matrices. Linear Algebra and its Applications 114, 717-735.
28. JOHNSON, D. B. 1977. Efficient algorithms for shortest paths in sparse networks. J. ACM 24, 1 (Jan.), 1-13.
29. JU, T., SCHAEFER, S., AND WARREN, J. 2005. Mean value coordinates for closed triangular meshes. ACM Trans. Graph. 24, 3 (July), 561-566.
30. KANTOROVICH, L. 1942. On the transfer of masses (in Russian). Doklady Akademii Nauk 37, 2, 227-229.
31. KIM, Y.-H., AND PASS, B. 2013. Multi-marginal optimal transport on Riemannian manifolds. arXiv:1303.6251.
32. KNIGHT, P. 2008. The Sinkhorn-Knopp algorithm: Convergence and applications. SIAM J. on Matrix Anal. and Applications 30, 1, 261-275.
33. LÉONARD, C. 2012. From the Schrödinger problem to the Monge-Kantorovich problem. J. Funct. Anal. 262, 4, 1879-1920.
34. LIPMAN, Y., AND DAUBECHIES, I. 2011. Conformal Wasserstein distances: Comparing surfaces in polynomial time. Adv. Math. 227, 3, 1047-1077.
35. MACNEAL, R. 1949. The Solution of Partial Differential Equations by means of Electrical Networks. PhD thesis, Caltech.
36. MALLIAVIN, P., AND STROOCK, D. W. 1996. Short time behavior of the heat kernel and its logarithmic derivatives. J. Differential Geom. 44, 3, 550-570.
37. MCCANN, R. J. 1997. A convexity principle for interacting gases. Adv. Math. 128, 1, 153-179.
38. MÉRIGOT, Q. 2011. A multiscale approach to optimal transport. Comp. Graph. Forum 30, 5, 1583-1592.
39. MOSEK APS, 2014. Mosek version 7. https://mosek.com.
40. PAPADAKIS, N., PROVENZI, E., AND CASELLES, V. 2011. A variational model for histogram transfer of color images. IEEE Trans. Image Proc. 20, 6, 1682-1695.
41. PHARR, M., AND HUMPHREYS, G. 2010. Physically Based Rendering, Second Edition: From Theory To Implementation. Morgan Kaufmann, July.
42. PITIÉ , F., KOKARAM, A. C., AND DAHYOT, R. 2007. Automated colour grading using colour distribution transfer. Comp. Vision and Image Understanding 107 (July), 123-137.
43. RUBNER, Y., TOMASI, C., AND GUIBAS, L. J. 2000. The earth mover's distance as a metric for image retrieval. Int. J. Comput. Vision 40, 2 (Nov.), 99-121.
44. SCHWARTZBURG, Y., TESTUZ, R., TAGLIASACCHI, A., AND PAULY, M. 2014. High-contrast computational caustic design. ACM Trans. Graph. 33, 4 (July), 74:1-74:11.
45. SINKHORN, R. 1964. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Math. Stat. 35, 2, 876-879.
46. SINKHORN, R. 1967. Diagonal equivalence to matrices with prescribed row and column sums. American Math. Monthly 74, 4, 402-405.
47. SOLOMON, J., NGUYEN, A., BUTSCHER, A., BEN-CHEN, M., AND GUIBAS, L. 2012. Soft maps between surfaces. Comp. Graph. Forum 31, 5 (Aug.), 1617-1626.
48. SOLOMON, J., GUIBAS, L., AND BUTSCHER, A. 2013. Dirichlet energy for analysis and synthesis of soft maps. Comp. Graph. Forum 32, 5, 197-206.
49. SOLOMON, J., RUSTAMOV, R., GUIBAS, L., AND BUTSCHER, A. 2014. Earth mover's distances on discrete surfaces. ACM Trans. Graph. 33, 4 (July), 67:1-67:12.
50. SOLOMON, J., RUSTAMOV, R., LEONIDAS, G., AND BUTSCHER, A. 2014. Wasserstein propagation for semi-supervised learning. In Proc. ICML, 306-314.
51. VARADHAN, S. R. S. 1967. On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. on Pure and Applied Math. 20, 2, 431-455.
52. VILLANI, C. 2003. Topics in Optimal Transportation. Graduate Studies in Mathematics. AMS.
53. ZHAO, X., SU, Z., GU, X. D., KAUFMAN, A., SUN, J., GAO, J., AND LUO, F. 2013. Area-preservation mapping using optimal mass transport. IEEE Trans. Vis. and Comp. Graphics 19, 12.
54. ZHU, J.-Y., LEE, Y. J., AND EFROS, A. A. 2014. AverageExplorer: Interactive exploration and alignment of visual data collections. ACM Trans. Graph. 33, 4 (July), 160:1-160:11.