Abstract
A significant class of variational models in connection with matching general data structures and comparison of metric measure spaces, lead to computationally intensive dense linear assignment and mass transportation problems. To accelerate the computation we present an extension of the auction algorithm that exploits the regularity of the otherwise arbitrary cost function. The algorithm only takes into account a sparse subset of possible assignment pairs while still guaranteeing global optimality of the solution. These subsets are determined by a multiscale approach together with a hierarchical consistency check in order to solve problems at successively finer scales. While the theoretical worst-case complexity is limited, the average-case complexity observed for a variety of realistic experimental scenarios yields a significant gain in computation time that increases with the problem size.
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References
Bertsekas, D.P.: A distributed algorithm for the assignment problem. Tech. rep., Lab. for Information and Decision Systems Report, MIT (May 1979)
Bertsekas, D.P.: The auction algorithm: A distributed relaxation method for the assignment problem. Annals of Operations Research 14, 105–123 (1988)
Bertsekas, D.P., Eckstein, J.: Dual coordinate step methods for linear network flow problems. Mathematical Programming, Series B 42, 203–243 (1988)
Bertsekas, D., Castanon, D.: The auction algorithm for the transportation problem. Annals of Operations Research 20, 67–96 (1989)
Carlier, G., Galichon, A., Santambrogio, F.: From Knothe’s transport to Brenier’s map and a continuation method for optimal transport. SIAM J. Math. Anal. 41, 2554–2576 (2010)
Haker, S., Zhu, L., Tannenbaum, A., Angenent, S.: Optimal mass transport for registration and warping. Int. J. Comput. Vision 60, 225–240 (2004)
Kuhn, H.W.: The hungarian method for the assignment problem. Naval Research Logistics 2, 83–97 (1955)
Mémoli, F.: Gromov-Wasserstein distances and the metric approach to object matching. Found. Comp. Math. 11, 417–487 (2011)
Pele, O., Werman, W.: Fast and Robust Earth Mover’s Distances. In: Proc. Int. Conf. Comp. Vision, ICCV (2009)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, vol. 24. Springer (2003)
Shirdhonkar, S., Jacobs, D.W.: Approximate earth mover’s distance in linear time. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008 (2008)
Villani, C.: Optimal Transport: Old and New. Springer (2009)
Wang, W., Slepčev, D., Basu, S., Ozolek, J.A., Rohde, G.K.: A linear optimal transportation framework for quantifying and visualizing variations in sets of images. International Journal of Computer Vision pp. 1–16 (2012)
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Schmitzer, B., Schnörr, C. (2013). A Hierarchical Approach to 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_38
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DOI: https://doi.org/10.1007/978-3-642-38267-3_38
Publisher Name: Springer, Berlin, Heidelberg
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