Pattern Recognition, Vol. 28, No. 4, pp. 511-579, 1995 Elsevier Science Ltd Copyright 0 1995 Pattern Recognition Society Printed in Great Britain. All rights reserved 0031-3203195 $9.50 + .oo 0031-3203(94)00121-9 NEAR-OPTIMAL MST-BASED SHAPE DESCRIPTION USING GENETIC ALGORITHM SVEN LONCARIC* and ATAM P. DHAWAN Department of Electrical and Computer Engineering, University of Cincinnati, Cincinnati, OH 45221-0030, U.S.A. (Received 26 October 1993; in revisedform 21 July 1994; receivedfor publication 14 September 1994) Abstract-A new method for the selection of the optimal structuring element for shape description and matching based on the morphological signature transform (MST) is presented in this paper. For a given class of shapes the optimal structuring element for MST method is selected by means of a genetic algorithm. The optimization criteria is formulated to enable a robust shape matching. Experiments have been performed on a class of model shapes. The proposed optimal shape description method is applied to the problem ofshape matching which evolves in many object recognition applications. Here, an unknown object is matched to a set of known objects in order to classify it into one of finite number of classes. Experimental results are presented and discussed. Shape description Mathematical morphology Genetic algorithms Shape matching Multiscale representation Multiresolution pyramid 1. INTRODUCTION Mathematical morphology has been successfully used for various image processing tasks.(‘-3) It is useful for processing of shape because morphological operations directly affect the shape. A new approach for morphological shape represen- tation was presented. (4-7) It is based on the morpho- logical signature transform (MST). The method uses a structuring element to compute the MST of objects to be described but does not solve the problem of the selection of the structuring element. In this paper, we present a new method for selection of optimal structuring elements for use in the MST- based method for shape representation and shape matching. The proposed method uses the optimal structuring element for multiresolution morphological operations. The optimal SE is selected by means of genetic algorithm. The optimality criteria is defined to enable best discrimination of shapes in a certain class. In other words, the structuring element provides the best shape matching capability for shapes in a given class. Experiments were performed where the proposed method for optimal structuring elements was applied to a certain class of test shapes. The optimal structuring element was determined by means of genetic algorithm and used for shape description using the MST-based algorithm. The proposed shape description method is * Author to whom correspondence should be addressed at: Faculty of Electrical Engineering, University of Zagreb, 41000 Zagreb, Croatia. applied to the problem of shape matching which evolves in many object recognition applications. Here, an unknown object must be matched to a set of known objects in order to classify it into one of a finite number of possible classes. In practical applications, the un- known object may be corrupted by noise. 1.1. Morphological signature transform A brief description of the MST(4,5) is given below. For generality, let W be a set which contains shapes to be described. Some common examples for W are: W = RZ (planar shapes) or W = R3 (three dimensional shapes), where R denotes the set of real numbers. The multiplication of a set by a real number is defined as rX= (_j {rx) xcx where r E R, and X c W. Another notation that we use in this paper is: S” = S @S @ . . . @S, where there are n summands in the Minkowski addition on the right side of the equation. Note that generally S” # nS, but in the special case when S is a convex set it holds that S” = nS. The MST of shape X with respect to structuring element S is defined as: X,(r, n) = M(rX, Sn) where r E R, and n E M is a binary morphological opera- tor, i.e. M:P x P + P, where P is the set of all subsets (i.e. the partitive set) of W. In other words M takes two sets from P as arguments and produces a third set as result. Some common examples for binary operator M operator are morphological erosion, dilation, opening, and closing. 571