Fuzzifying Spatial Relations - Computer Science - The University of

Fuzzifying Spatial Relations Hans W. Guesgen? Computer Science Department, University of Auckland Private Bag 92019, Auckland, New Zealand

Abstract. Reasoning about space plays an essential role in many cultures. Not only is space, like time, one of the most fundamental categories of human cognition, but also does it structure all our activities and relationships with the external world. Space serves as the basis for many metaphors, including temporal metaphors. It is inherently more complex than time, because it is multidimensional and epistemologically multiple. The way humans often deal with space in everyday situations is on a qualitative basis, allowing for imprecision in spatial descriptions when interacting with each other. Instead of using an absolute space (i.e., space viewed as a “container”, which exists independently of the objects that are located in it), it seems that they prefer a relative space, which is a construct induced by spatial relations over non-purely spatial entities. In artificial intelligence, a variety of formalisms have been developed that deal with space on the basis of relations between objects. Although most approaches provide some algorithms to reason about such relations, they usually do not make any attempt to address questions like how to handle imprecision in spatial relations or how to combine qualitative spatial relations with quantitative information. Although these questions seem to be unrelated to each other, we show in this chapter that fuzzy logic can provide an answer to both of them.

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Motivation

Researchers in artificial intelligence have been arguing successfully that human reasoning about space is of a qualitative nature in most everyday situations and that therefore computer systems should support such a form of reasoning (see, e.g., [20]). For most people, a statement like The post office is beside the city hall is more natural than a statement like The coordinates of the post office can be calculated from those of the city hall by adding the vector (−28, 10) to the latter. It is therefore not surprising that a number of publications in the area of spatial reasoning aim at some form of qualitative reasoning about space. Many of them do so by using a relational approach. In [14], for example, we introduce a form of spatial reasoning that extends Allen’s temporal logic [1] to the three dimensions of space by applying very simple methods for constructing higher-dimensional models and for reasoning about them, namely combination (i.e., building tuples of one-dimensional ?

Partly supported by the University of Auckland Research Committee through various grants (most recently through grant number XXX/9343/3414100).

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relations) and projection (i.e., extracting one-dimensional aspects from the tuples). Other approaches proceed in more or less the same way. Freksa [6] uses the same set of relations as in [1]. He shows that for an important class of problems, only a small subset of all possible combinations of spatial relations can occur. By restricting himself to sets of conceptually neighboring relations, he can restrict the complexity of the constraint satisfaction algorithms significantly. In [20], Hern´ andez introduces an extension of Allen’s approach to represent the spatial features occurring in 2D projections of 3D scenes. He suggests to establish spatial relations between objects by splitting them up into two aspects: projection and orientation. The aspect of projection describes the spatial relationship between two objects in a way similar to the one introduced in [14]. The aspect of orientation states how the objects are located relative to each other. Mukerjee and Joe’s work [22] is similar to Hern´andez’s approach. Objects of a two-dimensional world are characterized by the directions in which the objects are moving and by associating with the objects trajectories along which they are moving. In the context of spatial relations between objects, a reoccurring issue is the choice of an adequate reference frame. In [5], Frank discusses a taxonomy of frames of reference, which is inspired by cognitive models. Since we are not going into details here about choosing an adequate reference frame, we refer the reader to Frank’s publication. A common feature of the approaches sketched above (and many other approaches not referenced here) is that they represent spatial information in the form of qualitative spatial relations between objects: • The church is near the post office. • Object O1 overlaps object O2 . • The pencil is in the drawer. Most approaches provide some algorithms to reason about such relations, but they usually do not make any attempt to address the following questions: • How can imprecision in spatial relations be dealt with? • How can qualitative spatial relations be combined with quantitative information? These questions seem to be unrelated to each other, but we show in this chapter that fuzzy logic can provide an answer to both of them.

2 2.1

Imprecision in Spatial Relations Conceptual Neighborhoods

Allen [1] introduced a temporal logic based on a set of thirteen atomic temporal relations between time intervals (see Figure 1), together with an algorithm

Fuzzifying Spatial Relations

Relation

O2

O1 mO2 O2 miO1

O1 sO2 O2 siO1

O1 dO2 O2 diO1

O1 fO2 O2 fiO1

O1 =O2

Interpretation

O1

O1 O1

O1 oO2 O2 oiO1

Illustration

O1 O2

O1 O2

O1

O1

O1 meets O2 O2 met by O1

O1 overlaps O2 O2 overlapped by O1

O1 during O2 O2 contains O1

O2

O1

O1

O1 before O2 O2 after O1

O1 starts O2 O2 started by O1

O2

O2

3

O1 finishes O2 O2 finished by O1

O1 equals O2

O2

Fig. 1. Allen’s thirteen atomic relations.

to reason about networks of such relations. As indicated in the introductory section of this chapter, Allen’s logic can be adopted for spatial reasoning by interpreting the thirteen Allen relations as spatial relations between objects. By applying Allen’s algorithm to these spatial relations, we obtain an instrument to reason about space. The basis of Allen’s algorithm is a composition table, which determines the possible relations between two objects like O1 and O3 given the relations between O1 and another object O2 as well as the relation between O2 and O3 . The composition table is shown in Figure 2. If, for example, O1 is enclosing O2 (O1 diO2 ) and O2 is overlapping O3 (O2 oO3 ), then it can be concluded

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Hans W. Guesgen
oi, mi oi, mi > >

Fig. 2. Allen’s composition table including = as arranged in [7]. The entry at row r1 and column r2 in the table denotes the possible relations between O1 and O3 , assuming that O1 r1 O2 and O2 r2 O3 .

Fuzzifying Spatial Relations

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that the relation between O1 and O3 is an element of the set {o, fi, di}. Such a set is called a non-atomic Allen relation. To adopt the composition table for imprecise reasoning, we explore the notion of conceptional neighbors [7]. Assume that two objects O1 and O2 , for example, are in relation m, then by moving or deforming the objects slightly we can change this relation to < or o. Therefore, < and o are conceptual neighbors of m. The relation f, for example, is not a conceptual neighbor of m, as f cannot be obtained directly from m by deforming or moving objects. Freksa [7] distinguishes three different types of conceptual neighbors (A-, B-, and C-neighbors), depending on how the objects can be changed. For example, if the objects can be moved but not deformed, s is not a conceptual neighbor of =. However, if we allow for deformation, s and = are neighbors. To visualize the notion of conceptual neighbors, Freksa rearranged the Allen relations in such a way that conceptual neighbors are close in the topological sense if they are close in the conceptual sense. The result is shown in Figure 3 for the case of allowing movement of objects but no deformation. In the n

Fig. 3. Allen’s thirteen atomic relations arranged according to their conceptual neighborhood. Only movement of objects is considered, no deformation.

following, we restrict our discussions to this case. 2.2

Fuzzification of Allen Relations

The notion of conceptual neighbors can be used to introduce imprecision into reasoning about spatial relations [15]. The first step in that directions involves the introduction of characteristic functions to denote atomic relations: µr : A −→ {0, 1} The domain of µr is the set of atomic Allen relations, i.e.: A = {}

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The function yields a value of 1 if and only if the argument is equal to the atomic relation denoted by the characteristic function:  1, if r0 = r 0 µr (r ) = 0, else The next step towards the introduction of imprecision is to transform the atomic Allen relations into fuzzy sets. For that purpose, we represent each atomic relation as a set of pairs, each pair consisting of an element of A and the value of the characteristic function of the relation applied to that element. For example, if two object O1 and O2 are adjacent to each other, i.e., O1 mO2 , we use the characteristic function of the relation m to convert this statement into the following: O1 {(r, µm (r)) | r ∈ A}O2 = O1 {(m, 1), (