Abstract
This report deals with similarity measures for convex shapes whose definition is based on Minkowski addition and the Brunn-Minkowski inequality. All measures considered here are invariant under translations. In addition, they may be invariant under rotations, multiplications, reflections, or affine transformations. Restricting oneselves to the class of convex polygons, it is possible to develop efficient algorithms for the computation of such similarity measures. These algorithms use a special representation of convex polygons known as the perimetric measure. Such representations are unique for convex sets and linear with respect to Minkowski addition. Although the paper deals exclusively with the 2-dimensional case, many of the results carry over almost immediately to higher-dimensional spaces.
A. Tuzikov was supported by the Netherlands Organization for Scientific Research (NWO) through a visitor's grant.
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References
E. L. Allgower and P. H. Schmidt. Computing volumes of polyhedra. Mathematics of Computation, 46:171–174, 1986.
E. M. Arkin, L. P. Chew, D. P. Huttenlocher, K. Kedem, and J. S. B. Mitchell. An efficiently computable metric for comparing polygonal shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13:209–216, 1991.
D. Ballard and C. Brown. Computer Vision. Prentice-Hall, Englewood Cliffs, 1982.
P. K. Ghosh. A unified computational framework for Minkowski operations. Computers and Graphics, 17:357–378, 1993.
H. Goldstein. Classical Mechanics. Addison-Wesley, Reading, MA, 1950.
B. Griinbaum. Convex Polytopes. Interscience Publishers, 1967.
H. J. A. M. Heijmans and A. Tuzikov. Similarity and symmetry measures for convex sets based on Minkowski addition. Research report BS-119610, CWI, 1996.
J. Hong and X. Tan. Recognize the similarity between shapes under afine transformation. In Proc. Second Int. Conf. Computer Vision, pages 489–493. IEEE Comput. Soc. Press, 1988.
B. K. P. Horn. Robot Vision. MIT Press, Cambridge, 1986.
G. Matheron and J. Serra. Convexity and symmetry: Part 2. In J. Serra, editor, Image Analysis and Mathematical Morphology, vol. 2: Theoretical Advances, pages 359–375, London, 1988. Academic Press.
A. Rosenfeld and A. C. Kak. Digital Picture Processing. Academic Press, Orlando, 2nd edition, 1982.
R. Schneider. Convex Bodies. The Brunn-Minkowski Theory. Cambridge University Press, Cambridge, 1993.
P. J. van Otterloo. A Contour-Oriented Approach to Digital Shape Analysis. PhD thesis, Delft University of Technology, Delft, The Netherlands, 1988.
M. Werman and D. Weinshall. Similarity and affine invariant distances between 2D point sets. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17:810–814, 1995.
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© 1997 Springer-Verlag Berlin Heidelberg
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Tuzikov, A., Heijmans, H.J.A.M. (1997). Comparing convex shapes using Minkowski addition. In: Sommer, G., Daniilidis, K., Pauli, J. (eds) Computer Analysis of Images and Patterns. CAIP 1997. Lecture Notes in Computer Science, vol 1296. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63460-6_110
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DOI: https://doi.org/10.1007/3-540-63460-6_110
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