Space-Efficient Approximation Scheme for Circular Earth Mover Distance

The Earth Mover Distance (EMD) between point sets A and B is the minimum cost of a bipartite matching between A and B. EMD is an important measure for estimating similarities between objects with quantifiable features and has important applications in several areas including computer vision. The streaming complexity of approximating EMD between point sets in a two-dimensional discretized grid is an important open problem proposed in [8,9]. We study the problem of approximating EMD in the streaming model, when points lie on a discretized circle. Computing the EMD in this setting has applications to computer vision [13] and can be seen as a special case of computing EMD on a discretized grid. We achieve a (1 ±e) approximation for EMD in O (e-3) space, for every 0<e<1. To our knowledge, this is the first streaming algorithm for a natural and widely applied EMD model that matches the space bound asked in [9].

[1]  Trevor Darrell,et al.  Fast contour matching using approximate earth mover's distance , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[2]  Julien Rabin,et al.  Transportation Distances on the Circle , 2009, Journal of Mathematical Imaging and Vision.

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

[4]  Julien Rabin,et al.  Circular Earth Mover’s Distance for the comparison of local features , 2008, 2008 19th International Conference on Pattern Recognition.

[5]  Patrick Pérez,et al.  Robust tracking with motion estimation and local Kernel-based color modeling , 2007, Image Vis. Comput..

[6]  S. Muthukrishnan,et al.  Data streams: algorithms and applications , 2005, SODA '03.

[7]  Leonidas J. Guibas,et al.  A metric for distributions with applications to image databases , 1998, Sixth International Conference on Computer Vision (IEEE Cat. No.98CH36271).

[8]  Michael Werman,et al.  Bipartite Graph Matching for Points on a Line or a Circle , 1986, J. Algorithms.

[9]  C. Villani.  Topics in Optimal Transportation , 2003 .

[10]  Matthijs C. Dorst.  Distinctive Image Features from Scale-Invariant Keypoints , 2011 .

[11]  G. Burton.  TOPICS IN OPTIMAL TRANSPORTATION (Graduate Studies in Mathematics 58) By CÉDRIC VILLANI: 370 pp., US$59.00, ISBN 0-8218-3312-X (American Mathematical Society, Providence, RI, 2003) , 2004 .

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

[13]  J. Ian Munro,et al.  Selection and sorting with limited storage , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[14]  Julien Rabin,et al.  A Statistical Approach to the Matching of Local Features , 2009, SIAM J. Imaging Sci..

[15]  Trevor Darrell,et al.  Efficient image matching with distributions of local invariant features , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[16]  David P. Woodruff,et al.  On the exact space complexity of sketching and streaming small norms , 2010, SODA '10.

[17]  David P. Woodruff,et al.  Open Problems in Data Streams, Property Testing, and Related Topics , 2011 .

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

[19]  Carlos Cabrelli,et al.  A Linear Time Algorithm for a Matching Problem on the Circle , 1998, Inf. Process. Lett..

[20]  T. S. Jayram,et al.  OPEN PROBLEMS IN DATA STREAMS AND RELATED TOPICS IITK WORKSHOP ON ALGORITHMS FOR DATA STREAMS ’06 , 2007 .

[21]  Noga Alon,et al.  The Space Complexity of Approximating the Frequency Moments , 1999 .

[22]  Gideon Schechtman,et al.  Planar Earthmover is not in L_1 , 2005, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).