We derive conditions under which the reconstruction of a target space is topologically correct via the Cech complex or the 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 Cech complex. Second, we demonstrate the homotopic equivalence of a positive \mu-reach set and its offsets. Applying these results to the restricted Cech complex and using the interleaving relations with the Cech complex (or the Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Cech complex (or the Rips complex), in terms of the \mu-reach. Our results sharpen existing results.
We derive conditions under which the reconstruction of a target space is topologically correct via the Cech 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 Cech complex. Second, we demonstrate the homotopy equivalence of a positive \mu-reach set and its offsets. Applying these results to the restricted Cech complex and using the interleaving relations with the Cech complex (or the Vietoris-Rips complex), we formulate conditions guaranteeing that the target space is homotopy equivalent to the Cech complex (or the Vietoris-Rips complex), in terms of the \mu-reach. Our results sharpen existing results.
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Rips
Vietoris-Rips
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homotopic
homotopy
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Rips
Vietoris-Rips
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Rips
Vietoris-Rips