A new combinatorial model for boundary representations

Abstract

Boundary representations contain explicitly adjacency information for the so-called topological entities of the surface that bounds a manifold object. They have been shown to be useful for scene recognition (analysis) and stepwise construction of the models (synthesis). However, the current underlying combinatorial models and the operations that guarantee closure cover only environments that satisfy the minimal-topology partitioning requirement. Trivial partitions, such as a surface composed of a unique closed face ( e.g., sphere) or a surface composed of two simply-connected faces and one multiply-connected face ( e.g., cylinder), do not belong to the environments. In this paper we present a new approach to a combinatorial model which supports any kind of partition of a closed surface. It is achieved by introducing new geometrical interpretations to the topological entities. The proposed combinatorial scheme can be schematically represented by a face-based graph model. The Euler formula is adequately extended to express valid relationships between the number of topological entities, the Euler characteristic and the geometrical interpretations of the topological entities. Based on this formula, a set of transformation operators is suggested to build and change consistently the proposed model.