Matching Point Sets with Respect to the Earth Mover's Distance

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 + e)-approximation algorithms for the minimum Euclidean EMD between A and B under translations and rigid motions.

[1]  Arie Tamir,et al.  Algebraic optimization: The Fermat-Weber location problem , 1990, Math. Program..

[2]  David Mumford,et al.  Mathematical theories of shape: do they model perception? , 1991, Optics & Photonics.

[3]  James B. Orlin.  A Faster Strongly Polynomial Minimum Cost Flow Algorithm , 1993, Oper. Res..

[4]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[5]  S. Rao Kosaraju,et al.  Faster algorithms for some geometric graph problems in higher dimensions , 1993, SODA '93.

[6]  Günter Rote,et al.  Matching Shapes with a Reference Point , 1997, Int. J. Comput. Geom. Appl..

[7]  Leonidas J. Guibas,et al.  The Earth Mover's Distance under transformation sets , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[8]  Pankaj K. Agarwal,et al.  Approximation algorithms for bipartite and non-bipartite matching in the plane , 1999, SODA '99.

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

[10]  Michiel H. M. Smid,et al.  Dynamic algorithms for geometric spanners of small diameter: Randomized solutions , 1999, Comput. Geom..

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

[12]  Remco C. Veltkamp,et al.  State of the Art in Shape Matching , 2001, Principles of Visual Information Retrieval.

[13]  Remco C. Veltkamp,et al.  A Survey of Content-Based Image Retrieval Systems , 2002 .

[14]  Remco C. Veltkamp,et al.  A Pseudo-Metric for Weighted Point Sets , 2002, ECCV.

[15]  Zvi Drezner,et al.  The Weber Problem , 2002 .

[16]  Remco C. Veltkamp,et al.  Using transportation distances for measuring melodic similarity , 2003, ISMIR.

[17]  P. Bose,et al.  Fast approximations for sums of distances, clustering and the Fermat-Weber problem , 2003, Comput. Geom..

[18]  Trevor Darrell,et al.  Fast contour matching using approximate earth mover's distance , 2004, CVPR 2004.

[19]  Kai Li,et al.  Image similarity search with compact data structures , 2004, CIKM '04.

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

[21]  Pravin M. Vaidya,et al.  Using geometry to solve the transportation problem in the plane , 2005, Algorithmica.

[22]  Remco C. Veltkamp,et al.  Approximation algorithms for the Earth mover's distance under transformations using reference points , 2005, EuroCG.

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