Computational Topology for Point Data: Betti Numbers of α-Shapes

The problem considered belowis that of determining information about the topology of a subset X ⊂ ℝⁿ given only a finite point approximation to X. The basic approach is to compute topological properties — such as the number of components and number of holes — at a sequence of resolutions, and then to extrapolate. Theoretical foundations for taking this limit come from the inverse limit systems of shape theory and Čech homology. Computer implementations involve constructions from discrete geometry such as alpha shapes and the minimal spanning tree.