A General Theory of Boundary-based Qualitative Representation of Two-dimensional Shape

Shape is an important spatial attribute that features in much of our everyday reasoning and intelligent action. Consequently, representations of shape are important in the domain of Artificial Intelligence (AI). For the kind of commonsense reasoning that AI is interested in, it is necessary to represent shape in qualitative terms, using higher-level symbolic representations in preference to (or in addition to) low-level numerical data. Schemes for representing shape are most commonly categorised as either region-based or boundary-based. In this thesis, we focus our attention on the boundary-based approach to the qualitative representation of two-dimensional shape. Existing boundary-based schemes exhibit a number of common features. They all use high-level descriptors which are ultimately analysable in terms of qualitative curvature variation. The central claim of this thesis is that such an analysis provides an adequate theoretical underpinning for the boundary-based approach, leading to a theory which provides a unifying account of, and generalises, existing boundary-based schemes. Our analysis results in an unbounded hierarchy of atomic tokens, each of which corresponds to a particular qualitative composition of tangent bearing and zero or more of its derivatives. A shape has a token-string description at each level of the hierarchy and each level provides a conceptual granularity that is finer than the previous level and coarser than the next. Strings of atomic tokens give rise to complex tokens capable of representing localised curve features of greater abstraction. A local-feature scheme is defined as a set of complex tokens that supports token-string description. Given a local-feature scheme, a token-ordering graph can be constructed that visually encodes its token-ordering constraints. The atomic and complex tokens, local-feature schemes, and token-ordering graphs constitute a general theory of the boundary-based approach, which we call qualitative boundary theory (QBT). We apply QBT by showing that each of the existing boundary-based schemes is definable as a local-feature scheme.