Topological algorithms mapping point cloud data
Gunnar Carlsson, Yi Ding · 2009
It has been a challenging problem to associate complex point clouds with simple prototypes while preserving global topological structures. In this work, we present a functorial approach mapping topological invariants in all dimensions. We produce maps in the form of chain transformations that could be perturbed to incorporate local geometry. This leads to a difficult optimization problem, which we solve using heuristic algorithms. We begin by introducing the mathematical background including Hom complexes and the homotopy classification theorem. We then proceed to discuss how to compute chain maps that induce the desired homology and how to optimize them within homotopy classes. We conclude by discussing applications in point cloud data analysis.