# Convex Set Symmetry Measurement via Minkowski Addition

- 126 Downloads
- 7 Citations

## 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.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Bleau, J. Guise, and A.R. LeBlanc, “A new set of fast algorithms for mathematical morphology. 1. idempotent geodesic transforms,”
*CVGIP: Image Understanding*, Vol. 56, No.2, pp. 178–209, 1992.Google Scholar - 2.S.J. Crabtree, L.-P. Yaun, and R. Ehrlich, “A fast and accurate erosion-dilation method suitable for microcomputers,”, Vol. 53, No.3, pp. 283–290, 1991.Google Scholar
- 3.B.A. de Valcourt, “Measures of axial symmetry for ovals,”, Vol. 72, No.2, pp. 289–290, 1966.Google Scholar
- 4.V.V. Dobronravov, N.N. Nikitin, and A.L. Dvornikov,
*The Course of Theoretical Mechanics*, Moscow, in Russian, 1974.Google Scholar - 5.B. Grünbaum, “Measures of symmetry for convex sets,” in
*Proc. Sympos. Pure Math.*, Providence, USA, Vol. 7, pp. 233–270, 1963.Google Scholar - 6.H. Hadwiger,
*Vorlesungen über Inhalt, Oberfläche und Isoperimetrie*, Springer: Berlin, 1957.Google Scholar - 7.A. Imiya and T. Nakamura, “Morphological operations via convolution,” in
*Proceedings of the 6th Scandinavian Conference on Image Analysis*, Oulu, Finland, 1989, Vol. II, pp. 878–881.Google Scholar - 8.K. Leichtweiß,
*Konvexe Mengen*, VEB Deutscher Verlag der Wissenschaften: Berlin, 1980.Google Scholar - 9.G.L. Margolin and A.V. Tuzikov, “On a rotational symmetry of convex sets,” in
*Aspects of Visual Form Processing*, C. Arcelli, L. Cordella, and G. Sanniti di Baja (Eds.), World Scientific, pp. 354–363, 1994.Google Scholar - 10.G.L. Margolin, A.V. Tuzikov, and A.I. Grenov, “Reflection symmetry measure for convex sets,” in
*Proceedings of the First IEEE International Conference on Image Processing*, Austin, Texas, USA, Nov. 13–16, 1994, Vol. 1, pp. 691–695.Google Scholar - 11.G. Matheron and J. Serra, “Convexity and symmetry: Part 2,” in
*Image Analysis and Mathematical Morphology, Vol. 2: Theoretical Advances*, J. Serra, (Ed.), Academic Press: London, pp. 359–375, 1988.Google Scholar - 12.J.E. Mazille, “Mathematical morphology and convolutions,”
*Journal of Microscopy*, Vol. 156, No.1, pp. 3–13, 1988.Google Scholar - 13.H. Minkowski, “Volumen and Oberfläche,”
*Math. Ann.*, Vol. 57, pp. 447–495, 1903.Google Scholar - 14.I. Rangnemalm, “Fast erosion and dilation by contour processing and thresholding of distance maps,”
*Pattern Recognition Letters*, Vol. 13, pp. 161–166, 1992.Google Scholar - 15.A.V. Tuzikov, G.L. Margolin, and A.I. Grenov, “Convex polygon symmetrization and decomposition via perimetric measure,” in
*SPIE Proceedings*, 1994, Vol. 2180, pp. 239–247.Google Scholar - 16.R. van den Boomgaard,
*Mathematical Morphology: Extensions towards Computer Vision*, Ph.D. thesis, University of Amsterdam, The Netherlands, 1992.Google Scholar - 17.R. van den Boomgaard and R. van Balen, “Methods for fast morphological image transforms using bitmapped binary images,”
*Computer Vision, Graphics and Image Processing*, Vol. 54, No. 3, pp. 252–258, 1992.Google Scholar - 18.Luc Vincent, “Morphological transformations of binary images with arbitrary structuring elements,”
*Signal Processing*, Vol. 22, No.1, pp. 3–23, 1991.Google Scholar