The Semantics of Relations in 2D Space Using Representative Points: Spatial Indexes

the number of spatial relations that we can represent as a function of the representative points per object), the notion of image equivalence with respect to a modelling space, as well as information equivalence of relation based representations. 13 coordinates of the cells that store the representative points for the corresponding intersections): Thus the representation formalism is equivalent to Egenhofer's 4-intersection matrices with respect to topological relations. Notice that the array S M TD4 preserves more information than the set of direction relations of M D4 plus the topological relations of M T. That is, the array cannot be represented using, for instance, the array of S M D4 and a set of intersection matrices, since it also contains the relative positions of intersections with respect to the direction representative points. This property allows for the definition of relations that belong to M TD4 , but not to M T or M D4. For example, we can define a relation west_touch as: Objects a and b in Figure 1 satisfy the relation west_touch since there exists a point in the leftmost part of their boundaries where the two objects touch. In the array of Figure 11 this point corresponds to the cell (1,3). When the positions of the intersections in the array are not important, further compression of the array is possible. For instance we can move the contents of column 2 to another column and remove the column from the index array. Eventually we can generate an array with the dimensions of the array used for the direction relations (i.e., array of Figure 9b). The paper describes spatial indexes, a 2D array structure which can be used for the representation of a set of direction and topological relations, called the modelling space. Spatial indexes preserve the spatial relations of the modelling space and discard visual information and irrelevant spatial relations. Although we assumed distinct modelling spaces M D and M T in this paper, we do not argue that direction relations are completely independent from topological relations. For instance, the partitions denoted with * (number 5 in Figure 4a and numbers 13, 29, 53 and 69 in Figure 7) incorporate topological in addition to direction information. Furthermore, it is not allowed to have certain direction relations in conjunction with some topological relations among the same objects. For example, according to the previous definitions it is not permitted to have …