Computational Topology in Reconstruction, Mesh Generation, and Data Analysis Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Dey (2014) Computational Topology CCCG 14 1 / 55 Manifold reconstruction Delaunay mesh generation Topological data analysis Topological spaces Maps Complexes Homology groups Applications: Outline Topological concepts: Dey (2014) Computational Topology CCCG 14 2 / 55 Manifold reconstruction Delaunay mesh generation Topological data analysis Maps Complexes Homology groups Applications: Outline Topological concepts: Topological spaces Dey (2014) Computational Topology CCCG 14 2 / 55 Manifold reconstruction Delaunay mesh generation Topological data analysis Complexes Homology groups Applications: Outline Topological concepts: Topological spaces Maps Dey (2014) Computational Topology CCCG 14 2 / 55 Manifold reconstruction Delaunay mesh generation Topological data analysis Homology groups Applications: Outline Topological concepts: Topological spaces Maps Complexes Dey (2014) Computational Topology CCCG 14 2 / 55 Manifold reconstruction Delaunay mesh generation Topological data analysis Applications: Outline Topological concepts: Topological spaces Maps Complexes Homology groups Dey (2014) Computational Topology CCCG 14 2 / 55 Manifold reconstruction Delaunay mesh generation Topological data analysis Outline Topological concepts: Topological spaces Maps Complexes Homology groups Applications: Dey (2014) Computational Topology CCCG 14 2 / 55 Delaunay mesh generation Topological data analysis Outline Topological concepts: Topological spaces Maps Complexes Homology groups Applications: Manifold reconstruction Dey (2014) Computational Topology CCCG 14 2 / 55 Topological data analysis Outline Topological concepts: Topological spaces Maps Complexes Homology groups Applications: Manifold reconstruction Delaunay mesh generation Dey (2014) Computational Topology CCCG 14 2 / 55 Outline Topological concepts: Topological spaces Maps Complexes Homology groups Applications: Manifold reconstruction Delaunay mesh generation Topological data analysis Dey (2014) Computational Topology CCCG 14 2 / 55 Topology Background Topologcal spaces A point set with open subsets closed under union and finite intersections Dey (2014) Computational Topology CCCG 14 3 / 55 Topology Background Topologcal spaces A point set with open subsets closed under union and finite intersections d-ball Bd x Rd x 1 f 2 j jj jj ≤ g d-sphere S d x Rd x = 1 f 2 j jj jj g k-manifold: neighborhoods `homeomorphic' to open k-ball 2-sphere, torus, double torus are 2-manifolds Dey (2014) Computational Topology CCCG 14 3 / 55 Topology Background Maps Homeomorphic Homeomorphism h : T1 T2 where h is continuous, bijective! and has continuous inverse Isotopy : continuous deformation that maintains homeomorphism homotopy equivalence: map linked to No isotopy continuous deformation only Dey (2014) Computational Topology CCCG 14 4 / 55 Geometric k-simplex: k + 1-point convex hull Complex K: (i) t K if t is a face of t0 K 2 2 (ii) t1; t2 K t1 t2 is a face of both 2 ) \ Triangulation: K is a triangulation of a topological space T if T K ≈ j j Topology Background Simplicial complex Abstract V (K): vertex set, k-simplex: (k + 1)-subset σ V (K) ⊆ Complex K = σ σ0 σ = σ0 K f k ⊆ ) 2 g Dey (2014) Computational Topology CCCG 14 5 / 55 Triangulation: K is a triangulation of a topological space T if T K ≈ j j Topology Background Simplicial complex Abstract V (K): vertex set, k-simplex: (k + 1)-subset σ V (K) ⊆ Complex K = σ σ0 σ = σ0 K f k ⊆ ) 2 g Geometric k-simplex: k + 1-point convex hull Complex K: (i) t K if t is a face of t0 K 2 2 (ii) t1; t2 K t1 t2 is a face of both 2 ) \ Dey (2014) Computational Topology CCCG 14 5 / 55 Topology Background Simplicial complex Abstract V (K): vertex set, k-simplex: (k + 1)-subset σ V (K) ⊆ Complex K = σ σ0 σ = σ0 K f k ⊆ ) 2 g Geometric k-simplex: k + 1-point convex hull Complex K: (i) t K if t is a face of t0 K 2 2 (ii) t1; t2 K t1 t2 is a face of both 2 ) \ Triangulation: K is a triangulation of a topological space T if T K ≈ j j Dey (2014) Computational Topology CCCG 14 5 / 55 Reconstruction Surface Reconstruction Dey (2014) Computational Topology CCCG 14 6 / 55 Sampling Sampling Sample P Σ R3 ⊂ ⊂ Dey (2014) Computational Topology CCCG 14 7 / 55 Sampling Local Feature Size Lfs(x) is the distance to medial axis Dey (2014) Computational Topology CCCG 14 8 / 55 Sampling "-sample (Amenta-Bern-Eppstein 98) Each x has a sample within "Lfs(x) distance Dey (2014) Computational Topology CCCG 14 9 / 55 Theorem (Cocone: Amenta-Choi-Dey-Leekha 2000) The output surface computed by Cocone from an " sample is homeomorphic to the sampled surface for " < 0:06. − Sampling Crust and Cocone Guarantees Theorem (Crust: Amenta-Bern 1999) Any point x Σ is within O(")Lfs(x) distance from a point in the output. Conversely,2 any point of the output surface has a point x Σ within O(")Lfs(x) distance for " < 0:06. 2 Dey (2014) Computational Topology CCCG 14 10 / 55 Sampling Crust and Cocone Guarantees Theorem (Crust: Amenta-Bern 1999) Any point x Σ is within O(")Lfs(x) distance from a point in the output. Conversely,2 any point of the output surface has a point x Σ within O(")Lfs(x) distance for " < 0:06. 2 Theorem (Cocone: Amenta-Choi-Dey-Leekha 2000) The output surface computed by Cocone from an " sample is homeomorphic to the sampled surface for " < 0:06. − Dey (2014) Computational Topology CCCG 14 10 / 55 Sampling Restricted Voronoi/Delaunay Definition Restricted Voronoi: Vor P Σ: Intersection of Vor (P) with the surface/manifold Σ. j Dey (2014) Computational Topology CCCG 14 11 / 55 Sampling Restricted Voronoi/Delaunay Definition Restricted Delaunay: Del P Σ: dual of Vor P Σ j j Dey (2014) Computational Topology CCCG 14 12 / 55 Sampling Topology Closed Ball property (Edelsbrunner, Shah 94) If restricted Voronoi cell is a closed ball in each dimension, then Del P Σ is homeomorphic to Σ. j Dey (2014) Computational Topology CCCG 14 13 / 55 Sampling Topology Closed Ball property (Edelsbrunner, Shah 94) If restricted Voronoi cell is a closed ball in each dimension, then Del P Σ is homeomorphic to Σ. j Dey (2014) Computational Topology CCCG 14 13 / 55 Sampling Topology Closed Ball property (Edelsbrunner, Shah 94) If restricted Voronoi cell is a closed ball in each dimension, then Del P Σ is homeomorphic to Σ. j Theorem For a sufficiently small " if P is an "-sample of Σ, then (P, Σ) satisfies the closed ball property, and hence Del P Σ Σ. j ≈ Dey (2014) Computational Topology CCCG 14 13 / 55 Sampling Topology Closed Ball property (Edelsbrunner, Shah 94) If restricted Voronoi cell is a closed ball in each dimension, then Del P Σ is homeomorphic to Σ. j Dey (2014) Computational Topology CCCG 14 13 / 55 Sampling Topology Closed Ball property (Edelsbrunner, Shah 94) If restricted Voronoi cell is a closed ball in each dimension, then Del P Σ is homeomorphic to Σ. j Dey (2014) Computational Topology CCCG 14 13 / 55 Theorem (D.-Li-Ramos-Wenger 2009) Given a sufficiently dense sample of a smooth compact surface Σ with boundary one can compute a Delaunay mesh isotopic to Σ. Input Variations Boundaries Ambiguity in reconstruction Non-homeomorphic Restricted Delaunay [DLRW09] Non-orientabilty Dey (2014) Computational Topology CCCG 14 14 / 55 Input Variations Boundaries Ambiguity in reconstruction Non-homeomorphic Restricted Delaunay [DLRW09] Non-orientabilty Theorem (D.-Li-Ramos-Wenger 2009) Given a sufficiently dense sample of a smooth compact surface Σ with boundary one can compute a Delaunay mesh isotopic to Σ. Dey (2014) Computational Topology CCCG 14 14 / 55 Input Variations Open: Reconstructing nonsmooth surfaces Guarantee of homeomorphism is open Dey (2014) Computational Topology CCCG 14 15 / 55 Use ("; δ)-sampling Reconstruction of submanifolds brings ambiguity Restricted Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ) j 6≈ Delaunay triangulation becomes harder High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Dey (2014) Computational Topology CCCG 14 16 / 55 Use ("; δ)-sampling Restricted Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ) j 6≈ Delaunay triangulation becomes harder High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Dey (2014) Computational Topology CCCG 14 16 / 55 Use ("; δ)-sampling Restricted Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ) j 6≈ Delaunay triangulation becomes harder High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Dey (2014) Computational Topology CCCG 14 16 / 55 Use ("; δ)-sampling Restricted Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ) j 6≈ Delaunay triangulation becomes harder High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Dey (2014) Computational Topology CCCG 14 16 / 55 Restricted Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ) j 6≈ Delaunay triangulation becomes harder High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds