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

The problem considered below is that of determining information about the topology of a subset X ⊂ R n 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 ˇ Cech homology. Computer implementations involve constructions from discrete geometry such as alpha shapes and the minimal spanning tree.

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