Abstract
The Earth Mover’s Distance (EMD) between two weighted point sets (point distributions) is a distance measure commonly used in computer vision for color-based image retrieval and shape matching. It measures the minimum amount of work needed to transform one set into the other one by weight transportation.
We study the following shape matching problem: Given two weighted point sets A and B in the plane, compute a rigid motion of A that minimizes its Earth Mover’s Distance to B. No algorithm is known that computes an exact solution to this problem. We present simple FPTAS and polynomial-time (2 + ε)-approximation algorithms for the minimum Euclidean EMD between A and B under translations and rigid motions.
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References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)
Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Comp. Geom., pp. 121–153. Elsevier Science Publishers B.V, North-Holland (1999)
Atkinson, D.S., Vaidya, P.M.: Using geometry to solve the transportation problem in the plane. Algorithmica 13, 442–461 (1995)
Bose, P., Maheshwari, A., Morin, P.: Fast approximations for sums of distances clustering and the Fermat-Weber problem. Comp. Geom. Theory & Appl. 24, 135–146 (2003)
Callahan, P.B., Kosaraju, S.R.: Faster algorithms for some geometric graph problems in higher dimensions. In: Proc. of the 4th ACM-SIAM SODA, pp. 291–300 (1993)
Chandrasekaran, R., Tamir, A.: Algebraic optimization: The Fermat-Weber location problem. Math. Programming 46(2), 219–224 (1990)
Cohen, S., Guibas, L.: The Earth Mover’s Distance under transformation sets. In: Proc. of the 7th IEEE ICCV, pp. 173–187 (1999)
Giannopoulos, P., Veltkamp, R.C.: A pseudo-metric for weighted point sets. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 715–731. Springer, Heidelberg (2002)
Grauman, K., Darell, T.: Fast contour matching using approximate Earth Mover’s Distance. In: Proc. of the IEEE CVPR, pp. 220–227 (2004)
Indyk, P., Thaper, N.: Fast image retrieval via embeddings. In: 3rd Int. Workshop on Statistical and Computational Theories of Vision (2003)
Klein, O., Veltkamp, R.C.: Approximation algorithms for the Earth Mover’s Distance under transformations using reference points. Technical Report UU-CS-2005-003, IICS, Utrecht University, The Netherlands (2005)
Lv, Q., Charikar, M., Li, K.: Image similarity search with compact data structures. In: Proc. of the 13th ACM CIKM, pp. 208–217 (2004)
Mumford, D.: Mathematical theories of shape: Do they model perception? In: SPIE Geometric Methods in Comp. Vision, vol. 1570, pp. 2–10 (1991)
Orlin, J.B.: A faster strongly polynomial minimum cost flow algorithm. Operations Research 41(2), 338–350 (1993)
Rubner, Y., Tomasi, C., Guibas, L.J.: The Earth Mover’s Distance as a metric for image retrieval. Int. Journal of Computer Vision 40(2), 99–121 (2000)
Typke, R., Giannopoulos, P., Veltkamp, R.C., Wiering, F., van Oostrum, R.: Using transportation distances for measuring melodic similarity. In: Proc of 4th Int. Symp. on Music Inf. Retrieval (ISMIR), pp. 107–114 (2003)
Varadarajan, K.R., Agarwal, P.K.: Approximation algorithms for bipartite and non-bipartite matching in the plane. In: Proc. of the 10th ACM-SIAM SODA 1999, pp. 805–814 (1999)
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Cabello, S., Giannopoulos, P., Knauer, C., Rote, G. (2005). Matching Point Sets with Respect to the Earth Mover’s Distance. In: Brodal, G.S., Leonardi, S. (eds) Algorithms – ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11561071_47
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DOI: https://doi.org/10.1007/11561071_47
Publisher Name: Springer, Berlin, Heidelberg
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