Abstract
In this paper, we consider the problem (denoted as EMDRT) of minimizing the earth mover’s distance between two sets of weighted points A and B in a fixed dimensional ℝd space under rigid transformation. EMDRT is an important problem in both theory and applications and has received considerable attentions in recent years. In this paper, we present the first FPTAS algorithm for EMDRT. Our algorithm runs roughly in O((nm)d + 2(lognm)2d) time which matches the order of magnitude of the degree of a lower bound for any PTAS of this problem, where n and m are the sizes of A and B, respectively. Our result is based on several new techniques, such as the Sequential Orthogonal Decomposition (SOD) and Optimum Guided Base (OGB). Our technique can also be extended to several related problems, such as the alignment problem, and achieves FPTAS for each of them.
This research was supported in part by NSF under grant IIS-1115220.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alt, H., Guibas, L.: Discrete geometric shapes: matching, interpolation, and approximation. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 121–153. Elsevier, Amsterdam (1999)
Andoni, A., Indyk, P., Krauthgamer, R.: Earth mover distance over high-dimensional spaces. In: SODA, pp. 343–352 (2008)
Andoni, A., Do Ba, K., Indyk, P., Woodruff, D.P.: Efficient Sketches for Earth-Mover Distance, with Applications. In: FOCS, pp. 324–330 (2009)
Benkert, M., Gudmundsson, J., Merrick, D., Wolle, T.: Approximate one-to-one point pattern matching. J. Discrete Algorithms 15, 1–15 (2012)
Cohen, S.: Finding Color and Shape Patterns in Images. PhD thesis, Stanford University, Department of Compute Science (1999)
Cabello, S., Giannopoulos, P., Knauer, C.: On the parameterized complexity of d-dimensional point set pattern matching. Inf. Process. Lett. 105(2), 73–77 (2008)
Cabello, S., Giannopoulos, P., Knauer, C., Rote, G.: Matching Point Sets with Respect to the Earth Mover’s Distance. Computational Geometry: Theory and Applications 39(2), 118–133 (2008)
Cardoze, D.E., Schulman, L.J.: Pattern Matching for Spatial Point Sets. In: FOCS, pp. 156–165 (1998)
Gavrilov, M., Indyk, P., Motwani, R., Venkatasubramanian, S.: Combinatorial and Experimental Methods for Approximatie Point Pattern Matching. Algorithmica 38(1), 59–90 (2004)
Goodrich, M.T., Mitchell, J.S.B., Orletsky, M.W.: Approximate Geometric Pattern Matching Under Rigid Motions. IEEE PAMI 21(4), 371–379 (1999)
Klein, O., Veltkamp, R.C.: Approximation Algorithms for Computing the Earth Mover’s Distance Under Transformations. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1019–1028. Springer, Heidelberg (2005)
Rubner, Y., Tomasi, C., Guibas, L.J.: The Earth Movers Distance as a Metric for Image Retrieval. Int. J. of Comp. Vision 40(2), 99–121 (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ding, H., Xu, J. (2013). FPTAS for Minimizing Earth Mover’s Distance under Rigid Transformations. In: Bodlaender, H.L., Italiano, G.F. (eds) Algorithms – ESA 2013. ESA 2013. Lecture Notes in Computer Science, vol 8125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40450-4_34
Download citation
DOI: https://doi.org/10.1007/978-3-642-40450-4_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40449-8
Online ISBN: 978-3-642-40450-4
eBook Packages: Computer ScienceComputer Science (R0)