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Journal of Mathematical Imaging and Vision

, Volume 7, Issue 1, pp 53–68 | Cite as

Convex Set Symmetry Measurement via Minkowski Addition

  • A.V. Tuzikov
  • G.L. Margolin
  • A.I. Grenov
Article

Abstract

We introduce and investigate measures of rotation and reflectionsymmetries for compact convex sets. The appropriate symmetrization transformations are used to transform original sets into symmetrical ones. Symmetry measures are defined as the ratio of volumes (Lebesgue measure) of the original set and the corresponding symmetrical set.

For the case of rotation symmetry we use as a symmetrizationtransformation a generalization of the Minkowski symmetric set (a difference body) for a cyclic group of rotations.

For the case of reflection symmetry we investigate Blaschke symmetrization. Given a convex set and a hyperplane E in \(\mathbb{R}^n\)we get a set symmetrical with respect to this hyperplane. Analyzing all hyperplanes containing the coordinate center one gets the measure of reflection symmetry. We discuss the lower bounds of this symmetry measure and also a derived symmetry measure.

In the two dimensional case a perimetric measure representation of convexsets is applied for convex sets symmetrization as well as for the symmetry measure calculation. The perimetric measure allows also to perform a decomposition of a compact convex set into Minkowski sum of two sets. The first one is rotationally symmetrical and the second one is completely asymmetrical in the sense that it does not allow such a decomposition.

We discuss a problem of the fast computation of symmetrization transformations. Minkowski addition of two sets is reduced to the convolution of their characteristic functions. Therefore, in the case of the discrete plane \(\mathbb{Z}^2\), Fast Fourier Transformation can be applied for the fast computation of symmetrization transformations.

rotation symmetry reflection symmetry convex sets symmetrization transformation symmetry measure Minkowski addition 

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Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • A.V. Tuzikov
    • 1
  • G.L. Margolin
    • 1
  • A.I. Grenov
    • 1
  1. 1.Academy of Sciences of Republic BelarusInstitute of Engineering CyberneticsMinskBelarus

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