Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips Complex

We derive conditions under which the reconstruction of a target space is topologically correct via the \v{C}ech complex or the Vietoris-Rips complex obtained from possibly noisy point cloud data. We provide two novel theoretical results. First, we describe sufficient conditions under which any non-empty intersection of finitely many Euclidean balls intersected with a positive reach set is contractible, so that the Nerve theorem applies for the restricted \v{C}ech complex. Second, we demonstrate the homotopy equivalence of a positive $\mu$-reach set and its offsets. Applying these results to the restricted \v{C}ech complex and using the interleaving relations with the \v{C}ech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the \v{C}ech complex (or the Vietoris-Rips complex), in terms of the $\mu$-reach. Our results sharpen existing results.