Mathematical morphology on bipolar fuzzy sets

In many domains, it is important to be able to deal with bipolar information [5]. Positive information represents what is granted to be possible (for instance because it has already been observed or experienced), while negative information represents what is impossible (or forbidden, or surely false). This domain has recently motivated work in several directions. In particular, fuzzy and possibilistic formalisms for bipolar information have been proposed [5]. When dealing with spatial information, in image processing or for spatial reasoning applications, this bipolarity also occurs. For instance, when assessing the position of an object in space, we may have positive information expressed as a set of possible places, and negative information expressed as a set of impossible places (for instance because they are occupied by other objects). As another example, let us consider spatial relations. Human beings consider “left” and “right” as opposite relations. But this does not mean that one of them is the negation of the other one. The semantics of “opposite” captures a notion of symmetry rather than a strict complementation. In particular, there may be positions which are considered neither to the right nor to the left of some reference object, thus leaving room for some indetermination. This corresponds to the idea that the union of positive and negative information does not cover all the space. To our knowledge, bipolarity has not been much exploited in the spatial domain. The above considerations are the motivation for the present work, which aims at filling this gap by proposing formal models to manage spatial bipolar information. Additionally, imprecision has to be included, since it is an important feature of spatial information, related either to the objects themselves or to the spatial relations between them. More specifically, we consider bipolar fuzzy sets, and propose definitions of mathematical morphology operators (dilation and erosion) on these representations. To our knowledge, this is a new contribution. 2. Lattice structure and algebraic bipolar fuzzy dilations and erosions