Qualitative shape representation

Boundaries are closely related to shape and it is in the context of shape representation that the problem of undetermined boundaries appears in spatial reasoning. Shape indeterminacy may occur at different levels of abstraction, namely at the topological and the geometrical level. It shows that there is quite a gap in expressiveness between the topological and the geometrical approaches to shape representation. Ordering information, e.g. the fact that a region is convex, falls into this representational gap and is therefore difficult to handle with standard spatial reasoning techniques. It is shown how a representational scheme for positional ordering information that was originally developed for qualitative robot navigation can be adapted to the problem of shape representation. The ordering information approach to qualitative shape description proves to be useful when the indeterminacy arises from a lack of shape information and also handles to a certain extent recognition indeterminacy which is due to competing interpretations of the spatial information. Introduction We will discuss problems arising with undetermined boundaries from a spatial reasoning perspective. Spatial reasoning is a subfield of AI research on knowledge representation which studies formalisms for encoding spatial information (Kak, 1988, Chen, 1990, McDermott, 1992). A major concern in this field is the comparison of different representational formalisms in terms of expressive power, another the analysis of the computational costs of the underlying inference mechanisms. Such questions seem similar to those raised by algorithmic disciplines outside AI which also deal with spatial information, e.g. computational geometry. Although there is a number of techniques that one will find in both areas there is a clear difference in emphasis: the algorithmic disciplines consider well-posed problems trying to find optimal solutions for them, whereas spatial reasoning seeks to draw inferences even from incomplete spatial information, that is from ill-posed problems. Suboptimal results may well be acceptable in spatial reasoning if the efficiency of the inference process can be increased that way. For example, a strategy solving a spatial constraint satisfaction problem could be considered useful if it works for some frequently encountered cases though failing to produce a solution for every solvable problem. The approach to shape representation that is advocated for in this paper applies the idea of qualitative abstraction which is implicitly used in many spatial reasoning formalisms to a special kind of spatial information, namely ordering information. The first section of this paper is devoted to indeterminacy of spatial information in general. In the two following sections the procedure of qualitative abstraction and the concept of ordering information are described. The last section then develops the ordering information approach to qualitative shape description. In Frank, A. (in press). Spatial conceptual models for geographic objects with undeterminded boundaries, London: Taylor & Francis