FPTAS for Minimizing Earth Mover's Distance under Rigid Transformations

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 ℝ d 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.

[1]  L. Guibas,et al.  Finding color and shape patterns in images , 1999 .

[2]  Günter Rote,et al.  Matching point sets with respect to the Earth mover's distance , 2005, EuroCG.

[3]  Joachim Gudmundsson,et al.  Approximate one-to-one point pattern matching , 2012, J. Discrete Algorithms.

[4]  David P. Woodruff,et al.  Efficient Sketches for Earth-Mover Distance, with Applications , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[5]  Alexandr Andoni,et al.  Earth mover distance over high-dimensional spaces , 2008, SODA '08.

[6]  Leonidas J. Guibas,et al.  Discrete Geometric Shapes: Matching, Interpolation, and Approximation , 2000, Handbook of Computational Geometry.

[7]  Leonidas J. Guibas,et al.  The Earth Mover's Distance as a Metric for Image Retrieval , 2000, International Journal of Computer Vision.

[8]  David E. Cardoze,et al.  Pattern matching for spatial point sets , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[9]  Remco C. Veltkamp,et al.  Approximation Algorithms for Computing the Earth Mover's Distance Under Transformations , 2005, ISAAC.

[10]  Joseph S. B. Mitchell,et al.  Approximate Geometric Pattern Matching Under Rigid Motions , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Piotr Indyk,et al.  Combinatorial and Experimental Methods for Approximate Point Pattern Matching , 2003, Algorithmica.

[12]  Sergio Cabello,et al.  On the parameterized complexity of d-dimensional point set pattern matching , 2006, Inf. Process. Lett..

[13]  J. Sack,et al.  Handbook of computational geometry , 2000 .