Reasoning about Shape using the Tangential Axis Transform or the Shape’s “Grain”

Abstract

Several related areas of inquiry are concerned with developing qualitative characterisations of shape. Database query and support for natural language processing are two target problems. A general capability to reason about shapes would also be useful in a wide variety of contexts. In this paper, the use of a modified version of the medial axis transform (MAT), also called the skeleton, is investigated. The MAT is known to have several desirable properties for a general shape descriptor. However, it is also known to be sensitive to small perturbations at the boundary of an arbitrary shape, limiting its general utility. The tangential axis transform (TAT) is introduced to overcome these difficulties. The TAT leads to a definition of the “grain” of a region within the shape, characterised by a constant orientation. The locus of points which form the skeleton is then reinterpreted as the fracturing of the grain orientation within the shape. The concept of grain orientation leads further to the development of a modified version of the traditional skeleton, called here the angular skeleton. The angular skeleton retains the advantages of the radial skeleton, but does not exhibit the same kind of sensitivity to boundary deviations that was found in the standard (or radial) skeleton. As a result, it can be used as a general purpose shape quantifier. In addition, the notion of grain allows for a path-based query capability and hence supports spatial reasoning dealing with shape in a wide variety of situations. Finally, the concept of grain and the angular skeleton allow a simple procedure to be defined for trimming the angular skeleton and hence generalising the shape. It is shown how the latter allows for the characterisation of the spatial content of open class lexical elements such as adjectives, verbs, nouns, etc.