Topological approximation by small simplicial complexes

Given a point-cloud dataset sampled from an underlying geometric space X, it is often desirable to build a simplicial complex S approximating the geometric or topological structure of X. For example, recent techniques in automatic feature location depend on the ability to estimate topological invariants of X. These calculations can be prohibitively expensive if the number of cells in the approximating complex S is large. Unfortunately, most existing simplicial approximation algorithms either give too many cells, or involve calculations which are tractable or valid only in low dimensional Euclidean geometry. In this paper we introduce the combinatorial Delaunay triangulation, a simplicial complex construction which can be efficiently computed in arbitrary metric spaces, and which gives reliable topological approximations using comparatively few cells.