Characterization of WOSF by Equivalent Classes through Support Vector Machine

It is known that a weighted order statistic filter (WOSF) generates a linearly separable Boolean function and 2 different WOSF may generate the same Boolean function. Therefore a natural question “how to characterize WOSF which correspond the same Boolean function” arises. In this paper, we propose a different representation of WOSF induced from SVM. Also, we construct equivalent classes for WOSF based on the maximal margin classification of SVM. Two types of equivalent classes are proposed. The first one is called BF equivalent class. The parameters representing the hyperplane are adopted as the representative of the class, which is unique. The second class is the global equivalent class which is derived by additional sign change and permutation on the components of the BF class representatives. Therefore we can efficiently characterize all of the WOSF through only few representatives of equivalent classes and save computation cost when searching for various WOSF. Finally, we provide 3 formulas to directly generate the corresponding outputs of each WOSF.