Cubical singular simplex model for 3D objects and fast computation of homology groups

Abstract

This paper proposes a new simplex model for 3D objects: a cubical singular simplex model instead of the traditional triangularization simplex model. An extended Kohonen mapping is then presented as an efficient learning rule of the model. Based on this model, one can derive a pyramid structure for multi-resolution hierarchy, which is not possible for triangularization simplex. Besides, a fast algorithm is shown to calculate the homology groups of the objects as topological invariants from the proposed model.