I Pattern Recognition Letters ELSEVIER Pattern Recognition Letters 19 ( 1998) 247-254 Minkowski decomposition of convex polygons into their symmetric and asymmetric parts Alexander Tuzikov a, Henk J.A.M. Heijmans b,• ' lnstilllte <!f' Engineering Cybernetics, Academy <!f' Sciences of' Republic Belarus, Surganorn 6, 220012 Minsk, Belarus h CWJ, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands Received 23 September 1996; revised 7 January 1998 Abstract This paper discusses Minkowski decomposition of convex polygons into their symmetric and totally asymmetric parts. Two different types of symmetries are considered: finite-order rotations and line reflections. The approach is based on the representation of convex polygons through their perimetric measure. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Convex polygon: Minkowski addition; Perimetric measure: Rotation; Reflection; Symmetry; (Strongly) cyclic transformation; Polygon decomposition 1. Introduction In this paper, the following problem will be ad- dressed: given a compact, convex set P s;;; 2 , find a decomposition of the form P = Ps e P, , ( 1 ) where Ps is symmetric in a sense to be specified, and where P, is totally asymmetric (i.e., P, does not contain any symmetric parts). Here e denotes Minkowski addition. Matheron and Serra (1988), who considered this problem for the case of central sym- metry, used a perimetric representation to obtain such decompositions. In the work of Jourlin and Laget (1988) and Schneider (1989) one can find • Corresponding author. related material concerning the Minkowski decom- position of convex sets. In this paper we show how the approach by Matheron and Serra ( 1988) can also be used to deal with rotation as well as {line) reflection symmetry. It turns out, however, that these two cases are essen- tially different. In the rotation-symmetric case, the perimetric measure of the symmetric part equals the minimum of the corresponding rotations of the peri- metric measure of the original shape. In the reflec- tion-symmetric case, such a result does not hold, but we are able to present an algorithm which finds the symmetric part with largest area. Although our argu- ments apply to arbitrary compact, convex sets, we shall restrict ourselves to convex polygons in order to obtain efficient algorithms. Minkowski addition is one of the basic operations in mathematical morphology (e.g. (Heijmans, 1994; Serra, 1982)), where it is used to define dilation. 0167-8655/98/$19.00 © 1998 Elsevier Science B.Y. All rights reserved. PI/ SOl67-8655(98)00009-9