<title>Computing part hierarchies of 3D object shape from metric and nonmetric surface representations</title>

Abstract

Hierarchical representation of three dimensional (3D) object shape has been based on different levels of resolution. This paper introduces a representational hierarchy that is based on the connectedness and neighborliness of object shape expressed through topologies on the bounding surface with increasing strength. The topology at the object part level is weaker than at the level of simply connected elliptic, parabolic, plane and hyperbolic regions and the strongest topology is given by the classical topology for smooth surfaces. This provides a unified view on the representation of three-dimensional object shape for recognition. The open sets have a natural interpretation in the context of object recognition and relate to different types of recognition processes. More elaborate descriptions are naturally obtained by the introduction of additional structure, such as affine and metric. Qualitative shape features are defined at each level of the hierarchy their usefulness and limitation for shape discrimination is discussed. The possibility of deriving the topologies from ordinal structure is considered and examples of object description presented.