ABSTRACT
Drawing spatial relationship between entities that possess several attributes is a challenging task. Entities and their attributes in this section respectively denote sets represented on Z2 and their shape properties. Let X1 and X2 be two compact disjoint spatial objects of which the intersection yields an empty set, and X be the third spatial input of which the degree of containedness in the “between” space determined between X1 and X2 is addressed as one of the important parts of this study. What degree of object Y is contained in the “between” space? To answer this seemingly trivial question of GIS relevance, one should have (1) an approach to automatically derive the “between” space of X1 and X2 and (2) an approach to quantify the degree of containedness. In Chaudhuri (1990), authors investigated the latter aspect. In this study (Chaudhuri 1990), using fuzzy set theoretic concepts, (1) computation of
geometrical and topological properties of planar sets and (2) spatial relationships between sets are addressed. Besides several geometric and topological properties, one of the important spatial relationships considered in this study (Chaudhuri 1990) includes “degree of betweenness.” A basic prerequisite to study the set X’s degree of betweenness in between the sets X1 and X2 (e.g., Figure 13.1a) is to automatically determine the “between” space β(X1, X2) of sets X1 and X2. Here, “between” denotes spatial relation. Studying the spatial relations “between,” such sets come under the topic of spatial reasoning. To some extent, initial impetus to such problem was given in Aiello and van Bentha (2002) and Larvor (2004), where coliniariti between points
or center of spheres was considered, without taking the shape of the objects into account. Definitions of the degree of adjacency of two regions in the plane, and the degree of surroundedness of one region by another, are proposed by Rosenfeld and Klette (1985). Of late, this study is further extended in a series of papers (Bloch 1999, Serra 1982, Beucher and Meyer 1992, Bloch and Ralescu 2003, Bloch et al. 2006). Recursive dilations with certain logical operations are considered in these studies in determining the “between” space of sets. However, sets with concavities on both visible and non-visible sides pose limitations to deal with the approach proposed in these studies. Besides, authors dealt with by means of mixed approaches such as fuzzy morphology (Bloch et al. 2006).
