Overlap Index, Overlap Functions and Migrativity

In this work we study overlap degrees expressed in terms of overlap functions. We present the basic properties that from our point of view must satisfy these overlap functions. We study a construction method, we analyze which t-norms are also overlap functions and we prove that if we apply particular aggregations to such functions we recover the overlap index between fuzzy sets as defined by Dubois, and the consistency index of Zadeh. We also consider some properties that can be required to overlap functions, as k-Lipschitzianity or migrativity . Keywords— Overlap degree, Overlap function, Overlap index, t-norm, Migrativity. 1 Overlap function. Definition and properties Zadeh’s fuzzy sets theory has been very useful for solving problems which are described by imprecise models and with a large amount of noise. In particular, this theory is very appropriate to study the problem of identifying the objects in an image (see [18, 26]). To separate the object from the background in an image, the first thing to do is to represent the object by means of a fuzzy set and the background by means of another one. The success of the separation method lies on the correct choice of those fuzzy sets, which do not need to be disjoint in the sense of Ruspini [27] (see [1, 2, 17]). To build these sets it is necessary to know the exact property that characterizes the pixels belonging to the object (background). This property determines the expression of the membership function associated to the fuzzy set representing the object (background)(see [8, 9]). Usually, this function is not precisely known. There are some pixels for which the expert is sure they belong to the object or the background, but there are other pixels for which the expert hesitates. It is for these last ones that the value of the membership function is not accurately known. So, suppose that for a given an image, we ask an expert to assign to each pixel of intensity q the following values: μB(q), representing the membership of the pixel to the background. μO(q), representing the membership of the pixel to the object. In Fig.1 we show the two membership functions provided by the expert to represent an image in an L gray-level scale. We can deduce that, for intensities less than qi, the expert is Figure 1: Overlap between two functions sure that the pixels do not belong to the object. For intensities greater than qj he is sure that the pixels do not belong to the background. However, for intensities between qi and qj the expert is not sure about the membership of the pixels, with intensity qk corresponding to the maximum lack of knowledge. From this analysis we deduce that the overlap degree between the two functions can be understood as a representation of the lack of knowledge of the expert when he has to settle if a given pixel belongs to the background or to the object. So we can define the overlap degree between μB(q) and μO(q) by means of an overlap function GS : [0, 1] × [0, 1] → [0, 1] such that: (GS0) GS depends only on μB(q) and μO(q). (GS1) GS is symmetric. The overlap degree does not depend on the order we consider the membership degrees. (GS2) GS(μB(q), μO(q)) = 0 if and only if μB(q) = 0 or μO(q) = 0, (i.e., min(μB(q), μO(q)) = 0) . (GS3) GS(μB(q), μO(q)) = 1 if and only if μB(q) = 1 and μO(q) = 1, (i.e., min(μB(q), μO(q)) = 1). (GS4) If the membership degrees increase, so does the overlap degree. (GS5) Continuity. The overlap degree between the two memberships associated to a given pixel must not react chaotically ISBN: 978-989-95079-6-8 IFSA-EUSFLAT 2009