Abstract
This article introduces a generalization of discrete Optimal Transport that includes a regularity penalty and a relaxation of the bijectivity constraint. The corresponding transport plan is solved by minimizing an energy which is a convexification of an integer optimization problem. We propose to use a proximal splitting scheme to perform the minimization on large scale imaging problems. For un-regularized relaxed transport, we show that the relaxation is tight and that the transport plan is an assignment. In the general case, the regularization prevents the solution from being an assignment, but we show that the corresponding map can be used to solve imaging problems. We show an illustrative application of this discrete regularized transport to color transfer between images. This imaging problem cannot be solved in a satisfying manner without relaxing the bijective assignment constraint because of mass variation across image color palettes. Furthermore, the regularization of the transport plan helps remove colorization artifacts due to noise amplification.
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References
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics Series. American Mathematical Society (2003)
Rubner, Y., Tomasi, C., Guibas, L.: A metric for distributions with applications to image databases. In: International Conference on Computer Vision (ICCV 1998), pp. 59–66 (1998)
Benamou, J.D., Brenier, Y.: A computational fluid mechanics solution of the monge-katorovich mass transfer problem. Numerische Mathematik 84, 375–393 (2000)
Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. International Journal of Computer Vision 60, 225–240 (2004)
Reinhard, E., Adhikhmin, M., Gooch, B., Shirley, P.: Color transfer between images. IEEE Transactions on Computer Graphics and Applications 21, 34–41 (2001)
Morovic, J., Sun, P.L.: Accurate 3d image colour histogram transformation. Pattern Recognition Letters 24, 1725–1735 (2003)
McCollum, A.J., Clocksin, W.F.: Multidimensional histogram equalization and modification. In: International Conference on Image Analysis and Processing (ICIAP 2007), pp. 659–664 (2007)
Delon, J.: Midway image equalization. Journal of Mathematical Imaging and Vision 21, 119–134 (2004)
Pitié, F., Kokaram, A.C., Dahyot, R.: Automated colour grading using colour distribution transfer. Computer Vision and Image Understanding 107, 123–137 (2007)
Papadakis, N., Provenzi, E., Caselles, V.: A variational model for histogram transfer of color images. IEEE Transactions on Image Processing 20, 1682–1695 (2011)
Rabin, J., Peyré, G.: Wasserstein regularization of imaging problem. In: IEEE International Conderence on Image Processing (ICIP 2011), pp. 1541–1544 (2011)
Louet, J., Santambrogio, F.: A sharp inequality for transport maps in via approximation. Applied Mathematics Letters 25, 648–653 (2012)
Schellewald, C., Roth, S., Schnörr, C.: Evaluation of a convex relaxation to a quadratic assignment matching approach for relational object views. Image Vision Comput 25, 1301–1314 (2007)
Tenenbaum, J.B., de Silva, V., Langford, J.C.: A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290, 2319–2323 (2000)
Elmoataz, A., Lezoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing. Trans. Img. Proc. 17, 1047–1060 (2008)
Schrijver, A.: Theory of linear and integer programming. John Wiley & Sons, Inc., New York (1986)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40, 120–145 (2011)
Duchi, J., Shalev-Shwartz, S., Singer, Y., Chandra, T.: Efficient projections onto the ℓ1-ball for learning in high dimensions. In: Proc. ICML 2008, pp. 272–279. ACM, New York (2008)
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Ferradans, S., Papadakis, N., Rabin, J., Peyré, G., Aujol, JF. (2013). Regularized Discrete Optimal Transport. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2013. Lecture Notes in Computer Science, vol 7893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38267-3_36
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DOI: https://doi.org/10.1007/978-3-642-38267-3_36
Publisher Name: Springer, Berlin, Heidelberg
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