Estimation of Euler Characteristic from Point Data

Determination of the geometry and topology of a 3-dimensional body is an important problem appearing in Computer Aided Design, medical tomography, crystallography, molecular biology, etc. The methods used to address this problem depend on the form of input data. Let S be a finite set of sample points from a 3-dimensional body K ⊂ R. In applications one may need to estimate certain integral properties of K, like Euler characteristic, surface area, volume, etc. Sometimes a reconstruction of K can be attempted and many reconstruction methods have been suggested – e.g., methods based on Fourier analysis, methods based on weighted Delaunay or Voronoi complexes, such as e.g. the α-shapes method of Edelsbrunner [Ede99], etc. However, if the number of sample points is very large, a piecewise-linear reconstruction is not always possible – for example, at present it is impossible to use any methods based on Delaunay complexes when the number of sample points is about 2 or larger. On the other hand, even if a certain reconstruction method is applicable, naturally appears the question of verification of topological properties of a reconstruction. Most reconstruction methods yield either a simplicial complex or, more generally, a semialgebraic, or even semianalytic set. How well does this new body represent the shape of the original point set S? While S itself has very simple topology, being a finite set, we may think of S as a uniform, Poisson, or lattice sampling from some 3-dimensional body with a well-defined shape, whose intrinsic topology we seek to determine.