Local Conditions for Triangulating Submanifolds of Euclidean Space

Abstract

We consider the following setting: suppose that we are given a manifold M in R d with positive reach. Moreover assume that we have an embedded simplical complex A without boundary, whose vertex set lies on the manifold, is sufficiently dense and such that all simplices in A have sufficient quality. We prove that if, locally, interiors of the projection of the simplices onto the tangent space do not intersect, then A is a triangulation of the manifold, that is, they are homeomorphic.

Flat Delaunay Complexes for Homeomorphic Manifold Reconstruction

“…= ηε 20 , ε = 21 20 ε, and δ = 2δ. There are positive constants c 1 , c 2 , c 3 , and c 4 that depend only upon η, C ste , and d such that if ε R < c 1 then, given a point set P such that M ⊆ P ⊕ε , P ⊆ M ⊕δ , and separation(P ) > ηε, the point set P obtained after resetting each of its points satisfies M ⊆ (P ) ⊕ε , P ⊆ M ⊕δ , and separation(P ) > 9 10 ηε. Moreover, whenever we apply the Moser-Tardos Algorithm with Heigh min = c 2 ρ R 1 2 ρ and Prot min = c 3 ρ R 1 2 ρ, the algorithm terminates with expected time O( P ) and returns a point set P that satisfies:…”

“…First, while the criterion of star consistency in TDC is simple and elegant, the proof of homeomorphism for TDC, once this consistency is assumed, prove to be rather involved, requiring in particular the use of a lemma by Whitney about the projection of oriented PL pseudo-manifolds; see [5,Lemma 5.14], [9] and [19,Lemma 15a, Appendix II]. Our construction defines instead what we call prestars everywhere in space, not merely at the points of the data set and, for each d-simplex σ in the FDC, requires these prestars to agree at every pair of points in conv σ and not merely at the vertices of σ.…”

Delaunay-like Triangulation of Smooth Orientable Submanifolds by L1-Norm Minimization

“…We thus have: 21 20 ε 0 , and δ = 2δ 0 . There are positive constants c 1 , c 2 , c 3 , and c 4 that depend only upon η 0 , C ste , and d such that if ε 0 R < c 1 then, given a point set P 0 such that M ⊆ (P 0 ) ⊕ε 0 , P 0 ⊆ M ⊕δ 0 , and separation(P 0 ) > η 0 ε 0 , the point set P obtained after resetting each of its points satisfies M ⊆ P ⊕ε , P ⊆ M ⊕δ , and separation(P ) > 9 10 η 0 ε 0 . Moreover, whenever we apply the Moser-Tardos Algorithm with Heigh min = c 2 ρ R…”

The Topological Correctness of PL Approximations of Isomanifolds

“…The techniques used in this paper are also different from many of the standard tools and do not rely on Delaunay triangulations [32,33], the closed ball property [15,34,35], Whitney's lemma [36] or collapses [37]. The current paper mainly relies on the non-smooth implicit function theorem [38] with some Morse theory.…”

“…By the Leibniz rule and the triangle inequality)≤ (Γ B max (γ φ + γ ψ ) + 1)2D 2 α B max + 8d Dα B max T + γ φ Γ B max 2D 2 α max (byLemma 28,(35),(36), and Lemma 9) + 4d Dα B max T because φ(y) ∈ [0, 1], ( Proposition 10, and )α max ≤ α B max = (Γ B max (2γ φ + γ ψ ) + 1)2D 2 α B max + Transversality with respect to τ for Step 2) If both the regularity condition (57) and the transversality condition (58) hold, then, inside each σ × [0, 1], the gradient of τ on F −1 L,2, (0) is smooth and does not vanish. Both conditions hold if D = O(1/d 2 ).…”