A Pseudo-Metric for Weighted Point Sets
- 2.5k Downloads
We derive a pseudo-metric for weighted point sets. There are numerous situations, for example in the shape description domain, where the individual points in a feature point set have an associated attribute, a weight. A distance function that incorporates this extra information apart from the points’ position can be very useful for matching and retrieval purposes. There are two main approaches to do this. One approach is to interpret the point sets as fuzzy sets. However, a distance measure for fuzzy sets that is a metric, invariant under rigid motion and respects scaling of the underlying ground distance, does not exist. In addition, a Hausdorff-like pseudo-metric fails to differentiate between fuzzy sets with arbitrarily different maximum membership values. The other approach is the Earth Mover’s Distance. However, for sets of unequal total weights, it gives zero distance for arbitrarily different sets, and does not obey the triangle inequality. In this paper we derive a distance measure, based on weight transportation, that is invariant under rigid motion, respects scaling, and obeys the triangle inequality, so that it can be used in efficient database searching. Moreover, our pseudo-metric identifies only weight-scaled versions of the same set. We demonstrate its potential use by testing it on two different collections, one of company logos and another one of fish contours.
Keywordspseudo-metric weighted point set shape recognition indexing triangle inequality
- 1.Julio Barios, James French, Worthy Martin, Patrick Kelly, and Mike Cannon. Using the triangle inequality to reduce the number of comparisons required for similarity-based retrieval. In Proceedings of SPIE, volume 2670, Storage and Retrieval for Still Image and Video Databases IV, pages 392–403, 1996.Google Scholar
- 2.L. Boxer. On hausdorff-like metrics for fuzzy sets. Pattern Recognition Letters, 18, 1997.Google Scholar
- 3.Peter Braß. On the non-existence of hausdorff-like metrics for fuzzy sets. Technical Report (B 00-02), Freie Universität Berlin, Fachbereich Mathematik und Informatik, 2000.Google Scholar
- 4.B.B. Chaudhuri and A. Rosenfeld. On a metric distance between fuzzy sets. Pattern, Recognition Letters, 17, 1996.Google Scholar
- 5.Scott Cohen. Finding Color and Shape Patterns in Images. PhD thesis, Stanford University, Department of Computer Science, 1999.Google Scholar
- 6.Scott Cohen and Leonidas Guibas. The earth mover’s distance under transformation sets. In Proceedings of the 7th IEEE International Conference on Computer Vision, pages 173–187, September 1999.Google Scholar
- 7.SQUID database. http://www.ee.surrey.ac.uk/research/vssp/imagedb/demo.html.
- 8.Jiu-Liu Fan. Note on hausdorf-like metrics for fuzzy sets. Pattern Recognition Letters, 19, 1998.Google Scholar
- 9.Code for the Earth Movers Distence(EMD). http://vision.stanford.edn/~rubner/emd/default.htm.
- 10.David Spotts Fry. Shape Recognition using Metric on the Space of Shapes. PhD thesis, Harvard University, Division of Applied Sciences, 1993.Google Scholar
- 11.Michiel Hagerdoorn. Pattern matching using similarity measures. PhD thesis, Universiteit Utrecht, Institute of Information and Computer Science, 2000.Google Scholar
- 12.S. Hillicr and Gerald J. Lieberman. Introduction to Mathematical Programming. McGraw-Hill, 1990.Google Scholar
- 13.UMD logo database, http://documents.cfar.umd.edu/resources/database/umdlogo.html.
- 14.M. Novotni and R. Klein. A geometric approach to 3d object comparison. In International Conference on Shape Modelling and Appications, pages 167–175, 2001.Google Scholar
- 15.M. Petrou and P. Bosdogianni. Image Processing: The Fundamentals. John Wiley, 1999.Google Scholar
- 16.Fish polygon retrieval demo. http://give-lab.cs.uu.nl/matching/ptd.
- 17.Svetlozar T. Rachev and Ludger Rüschendorf. Mass Transportation Problems, volume I: Theory. Springer, 1998.Google Scholar
- 18.Svetlozar T. Rachev and Ludger Rüschendorf. Mass Transportation Problems. volume II: Applications. Springer, 1998.Google Scholar
- 19.Thomas L. Magnanti Ravindra K. Ahuja and James B. Orlin. Network Flows: Theory, Algorithms and Applications. Prentice-Hall, 1993.Google Scholar
- 21.Yossi Rubner. Perceptual Metrics for Image Database Navigation. PhD thesis, Stanford University, Department of Computer Science, 1999.Google Scholar
- 22.S.M. Smith and J. M. Brady. Susan-a new approach to low level image processing. International Journal of Computer Vsion, 23(1):45 78, May 1997.Google Scholar
- 23.Canny Edge Detection Software. http://www.computing.edu.au/~geoff/ftp.html.
- 26.SUSAN’s web page, http://www.fmrib.ox.ac.uk/~steve/susan/index.html.