Computational Topology in Reconstruction, Mesh Generation, and Data Analysis
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1 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
2 Outline Topological concepts: Dey (2014) Computational Topology CCCG 14 2 / 55
3 Outline Topological concepts: Topological spaces Dey (2014) Computational Topology CCCG 14 2 / 55
4 Outline Topological concepts: Topological spaces Maps Dey (2014) Computational Topology CCCG 14 2 / 55
5 Outline Topological concepts: Topological spaces Maps Complexes Dey (2014) Computational Topology CCCG 14 2 / 55
6 Outline Topological concepts: Topological spaces Maps Complexes Homology groups Dey (2014) Computational Topology CCCG 14 2 / 55
7 Outline Topological concepts: Topological spaces Maps Complexes Homology groups Applications: Dey (2014) Computational Topology CCCG 14 2 / 55
8 Outline Topological concepts: Topological spaces Maps Complexes Homology groups Applications: Manifold reconstruction Dey (2014) Computational Topology CCCG 14 2 / 55
9 Outline Topological concepts: Topological spaces Maps Complexes Homology groups Applications: Manifold reconstruction Delaunay mesh generation Dey (2014) Computational Topology CCCG 14 2 / 55
10 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
11 Topologcal spaces Topology Background A point set with open subsets closed under union and finite intersections Dey (2014) Computational Topology CCCG 14 3 / 55
12 Topologcal spaces Topology Background A point set with open subsets closed under union and finite intersections d-ball B d {x R d x 1} d-sphere S d {x R d x = 1} k-manifold: neighborhoods homeomorphic to open k-ball 2-sphere, torus, double torus are 2-manifolds Dey (2014) Computational Topology CCCG 14 3 / 55
13 Maps Topology Background Homeomorphism h : T 1 T 2 where h is continuous, bijective and has continuous inverse Isotopy : continuous deformation that maintains homeomorphism homotopy equivalence: map linked to continuous deformation only Homeomorphic No isotopy Dey (2014) Computational Topology CCCG 14 4 / 55
14 Simplicial complex Topology Background Abstract V (K): vertex set, k-simplex: (k + 1)-subset σ V (K) Complex K = {σ σ σ = σ K} Dey (2014) Computational Topology CCCG 14 5 / 55
15 Simplicial complex Topology Background Abstract V (K): vertex set, k-simplex: (k + 1)-subset σ V (K) Complex K = {σ σ σ = σ K} Geometric k-simplex: k + 1-point convex hull Complex K: (i) t K if t is a face of t K (ii) t 1, t 2 K t 1 t 2 is a face of both Dey (2014) Computational Topology CCCG 14 5 / 55
16 Simplicial complex Topology Background Abstract V (K): vertex set, k-simplex: (k + 1)-subset σ V (K) Complex K = {σ σ σ = σ K} Geometric k-simplex: k + 1-point convex hull Complex K: (i) t K if t is a face of t K (ii) t 1, t 2 K t 1 t 2 is a face of both Triangulation: K is a triangulation of a topological space T if T K Dey (2014) Computational Topology CCCG 14 5 / 55
17 Reconstruction Surface Reconstruction Dey (2014) Computational Topology CCCG 14 6 / 55
18 Sampling Sampling Sample P Σ R 3 Dey (2014) Computational Topology CCCG 14 7 / 55
19 Local Feature Size Sampling Lfs(x) is the distance to medial axis Dey (2014) Computational Topology CCCG 14 8 / 55
20 Sampling ε-sample (Amenta-Bern-Eppstein 98) Each x has a sample within εlfs(x) distance Dey (2014) Computational Topology CCCG 14 9 / 55
21 Sampling Crust and Cocone Guarantees Theorem (Crust: Amenta-Bern 1999) Any point x Σ is within O(ε)Lfs(x) distance from a point in the output. Conversely, any point of the output surface has a point x Σ within O(ε)Lfs(x) distance for ε < Dey (2014) Computational Topology CCCG / 55
22 Sampling Crust and Cocone Guarantees Theorem (Crust: Amenta-Bern 1999) Any point x Σ is within O(ε)Lfs(x) distance from a point in the output. Conversely, any point of the output surface has a point x Σ within O(ε)Lfs(x) distance for ε < Theorem (Cocone: Amenta-Choi-Dey-Leekha 2000) The output surface computed by Cocone from an ε sample is homeomorphic to the sampled surface for ε < Dey (2014) Computational Topology CCCG / 55
23 Sampling Restricted Voronoi/Delaunay Definition Restricted Voronoi: Vor P Σ : Intersection of Vor (P) with the surface/manifold Σ. Dey (2014) Computational Topology CCCG / 55
24 Sampling Restricted Voronoi/Delaunay Definition Restricted Delaunay: Del P Σ : dual of Vor P Σ Dey (2014) Computational Topology CCCG / 55
25 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. Dey (2014) Computational Topology CCCG / 55
26 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. Dey (2014) Computational Topology CCCG / 55
27 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. Theorem For a sufficiently small ε if P is an ε-sample of Σ, then (P, Σ) satisfies the closed ball property, and hence Del P Σ Σ. Dey (2014) Computational Topology CCCG / 55
28 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. Dey (2014) Computational Topology CCCG / 55
29 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. Dey (2014) Computational Topology CCCG / 55
30 Boundaries Input Variations Ambiguity in reconstruction Non-homeomorphic Restricted Delaunay [DLRW09] Non-orientabilty Dey (2014) Computational Topology CCCG / 55
![Input Variations Boundaries Ambiguity in reconstruction Non-homeomorphic Restricted Delaunay [DLRW09] Non-orientabilty Theorem (D.](/docs-images/85/91581383/images/31-0.jpg "-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.")
31 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 / 55
32 Input Variations Open: Reconstructing nonsmooth surfaces Guarantee of homeomorphism is open Dey (2014) Computational Topology CCCG / 55
33 High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Dey (2014) Computational Topology CCCG / 55
34 High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Dey (2014) Computational Topology CCCG / 55
35 High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Dey (2014) Computational Topology CCCG / 55
36 High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Dey (2014) Computational Topology CCCG / 55
37 High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Use (ε, δ)-sampling Dey (2014) Computational Topology CCCG / 55
38 High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Use (ε, δ)-sampling Restricted Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. Dey (2014) Computational Topology CCCG / 55
![Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no](/docs-images/85/91581383/images/39-1.jpg "matter how dense P is.")
39 High Dimensions High Dimensional PCD Curse of dimensionality (intrinsic vs. extrinsic) Reconstruction of submanifolds brings ambiguity Use (ε, δ)-sampling Restricted Delaunay does not capture topology Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. Delaunay triangulation becomes harder Dey (2014) Computational Topology CCCG / 55
40 Reconstruction High Dimensions Theorem (Cheng-Dey-Ramos 2005) Given an (ε, δ)-sample P of a smooth manifold Σ R d for appropriate ε, δ > 0, there is a weight assignment of P so that Del ˆP Σ Σ which can be computed efficiently. Dey (2014) Computational Topology CCCG / 55
41 Reconstruction High Dimensions Theorem (Cheng-Dey-Ramos 2005) Given an (ε, δ)-sample P of a smooth manifold Σ R d for appropriate ε, δ > 0, there is a weight assignment of P so that Del ˆP Σ Σ which can be computed efficiently. Theorem (Chazal-Lieutier 2006) Given an ε-noisy sample P of manifold Σ R d, there exists r p ρ(σ) for each p P so that the union of balls B(p, r p ) is homotopy equivalent to Σ. Dey (2014) Computational Topology CCCG / 55
42 High Dimensions Reconstructing Compacts Dey (2014) Computational Topology CCCG / 55
43 High Dimensions Reconstructing Compacts Lfs vanishes, introduce µ-reach and define (ε, µ)-samples. Dey (2014) Computational Topology CCCG / 55
44 High Dimensions Reconstructing Compacts Lfs vanishes, introduce µ-reach and define (ε, µ)-samples. Theorem (Chazal-Cohen-S.-Lieutier 2006) Given an (ε, µ)-sample P of a compact K R d for appropriate ε, µ > 0, there is an α so that union of balls B(p, α) is homotopy equivalent to K η for arbitrarily small η. Dey (2014) Computational Topology CCCG / 55
45 Mesh generation Surface and volume mesh Dey (2014) Computational Topology CCCG / 55
46 Mesh generation Surface and volume mesh Dey (2014) Computational Topology CCCG / 55
47 Delaunay refinement Mesh generation Pioneered by Chew89,Ruppert92,Shewchuk98 Dey (2014) Computational Topology CCCG / 55
48 Delaunay refinement Mesh generation Pioneered by Chew89,Ruppert92,Shewchuk98 To mesh some domain D Dey (2014) Computational Topology CCCG / 55
49 Delaunay refinement Mesh generation Pioneered by Chew89,Ruppert92,Shewchuk98 To mesh some domain D 1 Initialize points P D, compute Del P Dey (2014) Computational Topology CCCG / 55
50 Delaunay refinement Mesh generation Pioneered by Chew89,Ruppert92,Shewchuk98 To mesh some domain D 1 Initialize points P D, compute Del P 2 If some condition is not satisfied, insert a point p D into P and repeat Dey (2014) Computational Topology CCCG / 55
51 Delaunay refinement Mesh generation Pioneered by Chew89,Ruppert92,Shewchuk98 To mesh some domain D 1 Initialize points P D, compute Del P 2 If some condition is not satisfied, insert a point p D into P and repeat 3 Return Del P D Dey (2014) Computational Topology CCCG / 55
52 Delaunay refinement Mesh generation Pioneered by Chew89,Ruppert92,Shewchuk98 To mesh some domain D 1 Initialize points P D, compute Del P 2 If some condition is not satisfied, insert a point p D into P and repeat 3 Return Del P D Burden is to show termination (by packing argument) Dey (2014) Computational Topology CCCG / 55
53 Mesh generation Sampling Theorem Theorem (Amenta-Bern 98, Cheng-D.-Edelsbrunner-Sullivan 01) If P Sigma is a discrete ε-sample of a smooth surface Σ, then for ε < 0.09, Del P Σ satisfies: Dey (2014) Computational Topology CCCG / 55
54 Mesh generation Sampling Theorem Theorem (Amenta-Bern 98, Cheng-D.-Edelsbrunner-Sullivan 01) If P Sigma is a discrete ε-sample of a smooth surface Σ, then for ε < 0.09, Del P Σ satisfies: It is homeomorphic to Σ Dey (2014) Computational Topology CCCG / 55
55 Mesh generation Sampling Theorem Theorem (Amenta-Bern 98, Cheng-D.-Edelsbrunner-Sullivan 01) If P Sigma is a discrete ε-sample of a smooth surface Σ, then for ε < 0.09, Del P Σ satisfies: It is homeomorphic to Σ Each triangle has normal aligning within O(ε) angle to the surface normals Dey (2014) Computational Topology CCCG / 55
56 Mesh generation Sampling Theorem Theorem (Amenta-Bern 98, Cheng-D.-Edelsbrunner-Sullivan 01) If P Sigma is a discrete ε-sample of a smooth surface Σ, then for ε < 0.09, Del P Σ satisfies: It is homeomorphic to Σ Each triangle has normal aligning within O(ε) angle to the surface normals Hausdorff distance between Σ and Del P Σ is O(ε 2 ) of LFS. Dey (2014) Computational Topology CCCG / 55
57 Mesh generation Sampling Theorem Modified Theorem (Boissonnat-Oudot 05) If P Σ is such that each Voronoi edge-surface intersection x lies within εlfs(x) from a sample, then for ε < 0.09, Del P Σ satisfies: It is homeomorphic to Σ Each triangle has normal aligning within O(ε) angle to the surface normals Hausdorff distance between Σ and Del P Σ is O(ε 2 ) of LFS. Dey (2014) Computational Topology CCCG / 55
58 Mesh generation Basic Delaunay Refinement 1 Initialize points P Σ, compute Del P 2 If some condition is not satisfied, insert a point c Σ into P and repeat 3 Return Del P Σ Dey (2014) Computational Topology CCCG / 55
59 Mesh generation Surface Delaunay Refinement 1 Initialize points P Σ, compute Del P 2 If some Voronoi edge intersects Σ at x with d(x, P) > εlfs(x), insert x in P and repeat 3 Return Del P Σ Dey (2014) Computational Topology CCCG / 55
60 Mesh generation Difficulty How to compute Lfs(x)? Can be approximated by computing approximatemedial axis needs a dense sample. Dey (2014) Computational Topology CCCG / 55
61 Mesh generation Difficulty How to compute Lfs(x)? Can be approximated by computing approximatemedial axis needs a dense sample. Dey (2014) Computational Topology CCCG / 55
62 A Solution Mesh generation Replace d(x, P) < εlfs(x) with d(x, P) < λ, an user parameter Dey (2014) Computational Topology CCCG / 55
63 Mesh generation A Solution Replace d(x, P) < εlfs(x) with d(x, P) < λ, an user parameter Topology guarantee is lost Dey (2014) Computational Topology CCCG / 55
64 Mesh generation A Solution Replace d(x, P) < εlfs(x) with d(x, P) < λ, an user parameter Topology guarantee is lost Require topological disks around vertices Dey (2014) Computational Topology CCCG / 55
65 Mesh generation A Solution Replace d(x, P) < εlfs(x) with d(x, P) < λ, an user parameter Topology guarantee is lost Require topological disks around vertices Guarantees manifolds Dey (2014) Computational Topology CCCG / 55
66 Mesh generation A Solution Replace d(x, P) < εlfs(x) with d(x, P) < λ, an user parameter Topology guarantee is lost Require topological disks around vertices Guarantees manifolds Dey (2014) Computational Topology CCCG / 55
67 Mesh generation A Solution 1 Initialize points P Σ, compute Del P 2 If some Voronoi edge intersects Σ at x with d(x, P) > εlfs(x), insert x in P and repeat 3 Return Del P Σ Dey (2014) Computational Topology CCCG / 55
68 Mesh generation A Solution 1 Initialize points P Σ, compute Del P 2 If some Voronoi edge intersects Σ at x with d(x, P) > λlfs(x), insert x in P and repeat 3 Return Del P Σ Dey (2014) Computational Topology CCCG / 55
69 Mesh generation A Solution 1 Initialize points P Σ, compute Del P 2 If some Voronoi edge intersects Σ at x with d(x, P) > λlfs(x), insert x in P and repeat 3 If restricted triangles around a vertex p do not form a topological disk, insert furthest x where a dual Voronoi edge of a triangle around p intersects Σ 4 Return Del P Σ Dey (2014) Computational Topology CCCG / 55
70 A Meshing Theorem Mesh generation Theorem Previous algorithm produces output mesh with the following guarantees: Dey (2014) Computational Topology CCCG / 55
71 Mesh generation A Meshing Theorem Theorem Previous algorithm produces output mesh with the following guarantees: 1 Output mesh is always a 2-manifold 2 If λ is sufficiently small, the output mesh satisfies topological and geometric guarantees: Dey (2014) Computational Topology CCCG / 55
72 Mesh generation A Meshing Theorem Theorem Previous algorithm produces output mesh with the following guarantees: 1 Output mesh is always a 2-manifold 2 If λ is sufficiently small, the output mesh satisfies topological and geometric guarantees: 1 It is related to Σ by an isotopy Dey (2014) Computational Topology CCCG / 55
73 A Meshing Theorem Mesh generation Theorem Previous algorithm produces output mesh with the following guarantees: 1 Output mesh is always a 2-manifold 2 If λ is sufficiently small, the output mesh satisfies topological and geometric guarantees: 1 It is related to Σ by an isotopy 2 Each triangle has normal aligning within O(λ) angle to the surface normals Dey (2014) Computational Topology CCCG / 55
74 A Meshing Theorem Mesh generation Theorem Previous algorithm produces output mesh with the following guarantees: 1 Output mesh is always a 2-manifold 2 If λ is sufficiently small, the output mesh satisfies topological and geometric guarantees: 1 It is related to Σ by an isotopy 2 Each triangle has normal aligning within O(λ) angle to the surface normals 3 Hausdorff distance between Σ and Del P Σ is O(λ 2 ) of LFS. Dey (2014) Computational Topology CCCG / 55
75 Topological Data Analysis Data Analysis by Persistent Homology Persistent homology [Edelsbrunner-Letscher-Zomorodian 00], [Zomorodian-Carlsson 02] Dey (2014) Computational Topology CCCG / 55
76 Chain Topological Data Analysis Let K be a finite simplicial complex Dey (2014) Computational Topology CCCG / 55
77 Chain Topological Data Analysis Let K be a finite simplicial complex Simplicial complex Dey (2014) Computational Topology CCCG / 55
78 Chain Topological Data Analysis Let K be a finite simplicial complex Simplicial complex Definition A p-chain in K is a formal sum of p-simplices: c = i the addition in a ring, Z, Z 2, R etc. a i σ i ; sum is Dey (2014) Computational Topology CCCG / 55
79 Chain Topological Data Analysis Let K be a finite simplicial complex 1-chain ab + bc + cd (a i Z 2 ) Definition A p-chain in K is a formal sum of p-simplices: c = i the addition in a ring, Z, Z 2, R etc. a i σ i ; sum is Dey (2014) Computational Topology CCCG / 55
80 Boundary Topological Data Analysis Definition A p-boundary p+1 c of a (p + 1)-chain c is defined as the sum of boundaries of its simplices Dey (2014) Computational Topology CCCG / 55
81 Boundary Topological Data Analysis Definition A p-boundary p+1 c of a (p + 1)-chain c is defined as the sum of boundaries of its simplices Simplicial complex Dey (2014) Computational Topology CCCG / 55
82 Boundary Topological Data Analysis Definition A p-boundary p+1 c of a (p + 1)-chain c is defined as the sum of boundaries of its simplices 2-chain bcd + bde (under Z 2 ) Dey (2014) Computational Topology CCCG / 55
83 Boundary Topological Data Analysis Definition A p-boundary p+1 c of a (p + 1)-chain c is defined as the sum of boundaries of its simplices 1-boundary bc +cd +db+bd +de +eb = bc +cd +de +eb = 2 (bcd +bde) (under Z 2 ) Dey (2014) Computational Topology CCCG / 55
84 Cycles Topological Data Analysis Definition A p-cycle is a p-chain that has an empty boundary Dey (2014) Computational Topology CCCG / 55
85 Cycles Topological Data Analysis Definition A p-cycle is a p-chain that has an empty boundary Simplicial complex Dey (2014) Computational Topology CCCG / 55
86 Cycles Topological Data Analysis Definition A p-cycle is a p-chain that has an empty boundary 1-cycle ab + bc + cd + de + ea (under Z 2 ) Dey (2014) Computational Topology CCCG / 55
87 Cycles Topological Data Analysis Definition A p-cycle is a p-chain that has an empty boundary 1-cycle ab + bc + cd + de + ea (under Z 2 ) Each p-boundary is a p-cycle: p p+1 = 0 Dey (2014) Computational Topology CCCG / 55
88 Groups Topological Data Analysis Definition The p-chain group C p (K) of K is formed by p-chains under addition Dey (2014) Computational Topology CCCG / 55
89 Groups Topological Data Analysis Definition The p-chain group C p (K) of K is formed by p-chains under addition The boundary operator p induces a homomorphism p : C p (K) C p 1 (K) Dey (2014) Computational Topology CCCG / 55
90 Groups Topological Data Analysis Definition The p-chain group C p (K) of K is formed by p-chains under addition The boundary operator p induces a homomorphism p : C p (K) C p 1 (K) Definition The p-cycle group Z p (K) of K is the kernel ker p Dey (2014) Computational Topology CCCG / 55
91 Groups Topological Data Analysis Definition The p-chain group C p (K) of K is formed by p-chains under addition The boundary operator p induces a homomorphism p : C p (K) C p 1 (K) Definition The p-cycle group Z p (K) of K is the kernel ker p Definition The p-boundary group B p (K) of K is the image im p+1 Dey (2014) Computational Topology CCCG / 55
92 Homology Topological Data Analysis Definition The p-dimensional homology group is defined as H p (K) = Z p (K)/B p (K) Dey (2014) Computational Topology CCCG / 55
93 Homology Topological Data Analysis Definition The p-dimensional homology group is defined as H p (K) = Z p (K)/B p (K) Definition Two p-chains c and c are homologous if c = c + p+1 d for some chain d Dey (2014) Computational Topology CCCG / 55
94 Homology Topological Data Analysis Definition The p-dimensional homology group is defined as H p (K) = Z p (K)/B p (K) Definition Two p-chains c and c are homologous if c = c + p+1 d for some chain d (a) trivial (null-homologous) cycle; (b), (c) nontrivial homologous cycles Dey (2014) Computational Topology CCCG / 55
95 Complexes Topological Data Analysis Let P R d be a point set Dey (2014) Computational Topology CCCG / 55
96 Topological Data Analysis Complexes Let P R d be a point set B(p, r) denotes an open d-ball centered at p with radius r Dey (2014) Computational Topology CCCG / 55
97 Topological Data Analysis Complexes Let P R d be a point set B(p, r) denotes an open d-ball centered at p with radius r Definition The Čech complex Cr (P) is a simplicial complex where a simplex σ C r (P) iff Vert(σ) P and p Vert(σ) B(p, r/2) 0 Dey (2014) Computational Topology CCCG / 55
98 Topological Data Analysis Complexes Let P R d be a point set B(p, r) denotes an open d-ball centered at p with radius r Definition The Čech complex Cr (P) is a simplicial complex where a simplex σ C r (P) iff Vert(σ) P and p Vert(σ) B(p, r/2) 0 Definition The Rips complex R r (P) is a simplicial complex where a simplex σ R r (P) iff Vert(σ) are within pairwise Euclidean distance of r Dey (2014) Computational Topology CCCG / 55
99 Topological Data Analysis Complexes Let P R d be a point set B(p, r) denotes an open d-ball centered at p with radius r Definition The Čech complex C r (P) is a simplicial complex where a simplex σ C r (P) iff Vert(σ) P and p Vert(σ) B(p, r/2) 0 Definition The Rips complex R r (P) is a simplicial complex where a simplex σ R r (P) iff Vert(σ) are within pairwise Euclidean distance of r Proposition For any finite set P R d and any r 0, C r (P) R r (P) C 2r (P) Dey (2014) Computational Topology CCCG / 55
100 Point set P Topological Data Analysis Dey (2014) Computational Topology CCCG / 55
101 Topological Data Analysis Balls B(p, r/2) for p P Dey (2014) Computational Topology CCCG / 55
102 Topological Data Analysis Čech complex C r (P) Dey (2014) Computational Topology CCCG / 55
103 Topological Data Analysis Rips complex R r (P) Dey (2014) Computational Topology CCCG / 55
104 Topological Data Analysis Topological persistence r(x) = d(x, P): distance to point cloud P Sublevel sets r 1 [0, a] are union of balls Evolution of the sublevel sets with increasing a left hole persists longer Persistent homology quantizes this idea Dey (2014) Computational Topology CCCG / 55
105 Topological Data Analysis Persistent Homology f : T R; T a = f 1 (, a], the sublevel set T a T b for a b provides inclusion map ι : T a T b Induced map ι : H p (T a ) H p (T b ) giving the sequence 0 H p (T a1 ) H p (T a2 ) H p (T an ) H p (T) Persistent homology classes: Image of f ij p : H p (T ai ) H p (T aj ) Dey (2014) Computational Topology CCCG / 55
106 Topological Data Analysis Continuous to Discrete Replace T with a simplicial complex K := K(T) Dey (2014) Computational Topology CCCG / 55
107 Topological Data Analysis Continuous to Discrete Replace T with a simplicial complex K := K(T) Union of balls with its nerve Čech complex Dey (2014) Computational Topology CCCG / 55
108 Topological Data Analysis Continuous to Discrete Replace T with a simplicial complex K := K(T) Union of balls with its nerve Čech complex Dey (2014) Computational Topology CCCG / 55
109 Topological Data Analysis Continuous to Discrete Replace T with a simplicial complex K := K(T) Union of balls with its nerve Čech complex (a) (b) (c) (d) Dey (2014) Computational Topology CCCG / 55
110 Topological Data Analysis Continuous to Discrete Replace T with a simplicial complex K := K(T) Union of balls with its nerve Čech complex Evolution of sublevel sets becomes Filtration: (a) (b) = K 0 K 1 K n = K 0 H p (K 1 ) H p (K n ) = H p (K). (c) (d) Dey (2014) Computational Topology CCCG / 55
111 Topological Data Analysis Continuous to Discrete Replace T with a simplicial complex K := K(T) Union of balls with its nerve Čech complex Evolution of sublevel sets becomes Filtration: (a) (b) = K 0 K 1 K n = K 0 H p (K 1 ) H p (K n ) = H p (K). Birth and Death of homology classes (c) (d) Dey (2014) Computational Topology CCCG / 55
112 Bar Codes Topological Data Analysis birth-death and bar codes Dey (2014) Computational Topology CCCG / 55
113 Topological Data Analysis Persistence Diagram a bar [a, b] is represented as a point in the plane Dey (2014) Computational Topology CCCG / 55
114 Topological Data Analysis Persistence Diagram a bar [a, b] is represented as a point in the plane incorporate the diagonal in the diagram Dgm p (f ) Dey (2014) Computational Topology CCCG / 55
115 Topological Data Analysis Persistence Diagram a bar [a, b] is represented as a point in the plane incorporate the diagonal in the diagram Dgm p (f ) Dey (2014) Computational Topology CCCG / 55
116 Topological Data Analysis Persistence Diagram a bar [a, b] is represented as a point in the plane incorporate the diagonal in the diagram Dgm p (f ) Death Birth Dey (2014) Computational Topology CCCG / 55
117 Topological Data Analysis Stability of Persistence Diagram Bottleneck distance (C all bijections) d B (Dgm p (f ), Dgm p (g)) := inf sup x c(x) c C x Dgm p (f ) Theorem (Cohen-Steiner,Edelsbrunner,Harer 06) d B (Dgm p (f ), Dgm p (g)) f g Dey (2014) Computational Topology CCCG / 55
118 Back to Point Data Topological Data Analysis d T be the distance function from the space T. Dey (2014) Computational Topology CCCG / 55
119 Topological Data Analysis Back to Point Data d T be the distance function from the space T. d P be the distance function from sample P Dey (2014) Computational Topology CCCG / 55
120 Topological Data Analysis Back to Point Data Death d T be the distance function from the space T. d P be the distance function from sample P Birth Dey (2014) Computational Topology CCCG / 55
121 Topological Data Analysis Back to Point Data Death d T be the distance function from the space T. d P be the distance function from sample P d T d P d(t, P) = ε Birth Dey (2014) Computational Topology CCCG / 55
122 Topological Data Analysis Back to Point Data Death d T be the distance function from the space T. d P be the distance function from sample P d T d P d(t, P) = ε d B (Dgm p (d T ), Dgm (d P )) ε Birth Dey (2014) Computational Topology CCCG / 55
123 Topological Data Analysis Back to Point Data Death d T be the distance function from the space T. d P be the distance function from sample P d T d P d(t, P) = ε d B (Dgm p (d T ), Dgm (d P )) ε Birth Dey (2014) Computational Topology CCCG / 55
124 Topological Data Analysis Issues: Choice of Complexes Čech complexes are difficult to compute Dey (2014) Computational Topology CCCG / 55
125 Topological Data Analysis Issues: Choice of Complexes Čech complexes are difficult to compute Most literature considers Rips complexes, but they are Huge Dey (2014) Computational Topology CCCG / 55
126 Topological Data Analysis Issues: Choice of Complexes Čech complexes are difficult to compute Most literature considers Rips complexes, but they are Huge Sparsified Rips complex [Sheehy 12] Dey (2014) Computational Topology CCCG / 55
127 Topological Data Analysis Issues: Choice of Complexes Čech complexes are difficult to compute Most literature considers Rips complexes, but they are Huge Sparsified Rips complex [Sheehy 12] Graph Induced Complex (GIC) [D.-Fang-Wang 13] R r (P ) Dey (2014) Computational Topology CCCG / 55
128 Topological Data Analysis Issues: Choice of Complexes Čech complexes are difficult to compute Most literature considers Rips complexes, but they are Huge Sparsified Rips complex [Sheehy 12] Graph Induced Complex (GIC) [D.-Fang-Wang 13] R r (P ) Q Dey (2014) Computational Topology CCCG / 55
129 Topological Data Analysis Issues: Choice of Complexes Čech complexes are difficult to compute Most literature considers Rips complexes, but they are Huge Sparsified Rips complex [Sheehy 12] Graph Induced Complex (GIC) [D.-Fang-Wang 13] R r (P ) R r (Q) Dey (2014) Computational Topology CCCG / 55
130 Topological Data Analysis Issues: Choice of Complexes Čech complexes are difficult to compute Most literature considers Rips complexes, but they are Huge Sparsified Rips complex [Sheehy 12] Graph Induced Complex (GIC) [D.-Fang-Wang 13] R r (P ) R r (Q) G r (P, Q) Dey (2014) Computational Topology CCCG / 55
131 Topological Data Analysis Issues: Filtration Maps Zigzag inclusions[carlsson-de Silva-Morozov 09] K 1 K 2 K 3... K n Dey (2014) Computational Topology CCCG / 55
132 Topological Data Analysis Issues: Filtration Maps Zigzag inclusions[carlsson-de Silva-Morozov 09] K 1 K 2 K 3... K n Dey (2014) Computational Topology CCCG / 55
133 Topological Data Analysis Issues: Filtration Maps Zigzag inclusions[carlsson-de Silva-Morozov 09] K 1 K 2 K 3... K n Simplicial maps instead of inclusions [D.-Fan-Wang 14] K f 1 f K 2 f 1 K2 n Kn = K Dey (2014) Computational Topology CCCG / 55
134 Topological Data Analysis Issues: Filtration Maps Zigzag inclusions[carlsson-de Silva-Morozov 09] K 1 K 2 K 3... K n Simplicial maps instead of inclusions [D.-Fan-Wang 14] K f 1 f K 2 f 1 K2 n Kn = K Efficient algorithm for Zigzag simplicial maps? Dey (2014) Computational Topology CCCG / 55
135 Further Problems Further Issues/Problems What about stabilty when domains aren t same. Dey (2014) Computational Topology CCCG / 55
136 Further Problems Further Issues/Problems What about stabilty when domains aren t same. Interleaving [Chazal-Oudot 08], [CCGGO08] Dey (2014) Computational Topology CCCG / 55
137 Further Problems Further Issues/Problems What about stabilty when domains aren t same. Interleaving [Chazal-Oudot 08], [CCGGO08] Scalar field data analysis [CGOS 09] Dey (2014) Computational Topology CCCG / 55
138 Further Problems Further Issues/Problems What about stabilty when domains aren t same. Interleaving [Chazal-Oudot 08], [CCGGO08] Scalar field data analysis [CGOS 09] Computing topological structures Dey (2014) Computational Topology CCCG / 55
139 Further Problems Further Issues/Problems What about stabilty when domains aren t same. Interleaving [Chazal-Oudot 08], [CCGGO08] Scalar field data analysis [CGOS 09] Computing topological structures Reeb graphs [C-MEHNP 03, D.-Wang 11] Dey (2014) Computational Topology CCCG / 55
140 Further Problems Further Issues/Problems What about stabilty when domains aren t same. Interleaving [Chazal-Oudot 08], [CCGGO08] Scalar field data analysis [CGOS 09] Computing topological structures Reeb graphs [C-MEHNP 03, D.-Wang 11] Homology cycles Dey (2014) Computational Topology CCCG / 55
141 Further Problems Further Issues/Problems What about stabilty when domains aren t same. Interleaving [Chazal-Oudot 08], [CCGGO08] Scalar field data analysis [CGOS 09] Computing topological structures Reeb graphs [C-MEHNP 03, D.-Wang 11] Homology cycles Dey (2014) Computational Topology CCCG / 55
142 Further Problems Optimal Homology Basis Problem Compute an optimal set of cycles forming a basis Dey (2014) Computational Topology CCCG / 55
143 Further Problems Optimal Homology Basis Problem Compute an optimal set of cycles forming a basis Dey (2014) Computational Topology CCCG / 55
144 Further Problems Optimal Homology Basis Problem Compute an optimal set of cycles forming a basis Dey (2014) Computational Topology CCCG / 55
145 Further Problems Optimal Homology Basis Problem Compute an optimal set of cycles forming a basis First solution for surfaces: Erickson-Whittlesey [SODA05] Dey (2014) Computational Topology CCCG / 55
146 Further Problems Optimal Homology Basis Problem Compute an optimal set of cycles forming a basis First solution for surfaces: Erickson-Whittlesey [SODA05] General problem NP-hard: Chen-Freedman [SODA10] Dey (2014) Computational Topology CCCG / 55
![Further Problems Optimal Homology Basis Problem Compute an optimal set of cycles forming a basis First solution for surfaces: Erickson-Whittlesey [SODA05]](/docs-images/85/91581383/images/147-0.jpg "General problem NP-hard: Chen-Freedman [SODA10] H 1 basis for simplicial complexes: D.-Sun-Wang [SoCG10] Dey (2014) Computational Topology CCCG 14 52 / 55")
147 Further Problems Optimal Homology Basis Problem Compute an optimal set of cycles forming a basis First solution for surfaces: Erickson-Whittlesey [SODA05] General problem NP-hard: Chen-Freedman [SODA10] H 1 basis for simplicial complexes: D.-Sun-Wang [SoCG10] Dey (2014) Computational Topology CCCG / 55
148 Optimal Localization Further Problems Compute an optimal cycle in a given class. Dey (2014) Computational Topology CCCG / 55
149 Optimal Localization Further Problems Compute an optimal cycle in a given class. Dey (2014) Computational Topology CCCG / 55
150 Optimal Localization Further Problems Compute an optimal cycle in a given class. Dey (2014) Computational Topology CCCG / 55
151 Optimal Localization Further Problems Compute an optimal cycle in a given class. Dey (2014) Computational Topology CCCG / 55
152 Optimal Localization Further Problems Compute an optimal cycle in a given class. Surfaces: Colin de Verdière-Lazarus [DCG05], Colin de Verdière-Erickson [SODA06], Chambers-Erickson-Nayyeri [SoCG09] Dey (2014) Computational Topology CCCG / 55
153 Optimal Localization Further Problems Compute an optimal cycle in a given class. Surfaces: Colin de Verdière-Lazarus [DCG05], Colin de Verdière-Erickson [SODA06], Chambers-Erickson-Nayyeri [SoCG09] General problem NP-hard: Chen-Freedman [SODA10] Dey (2014) Computational Topology CCCG / 55
154 Optimal Localization Further Problems Compute an optimal cycle in a given class. Surfaces: Colin de Verdière-Lazarus [DCG05], Colin de Verdière-Erickson [SODA06], Chambers-Erickson-Nayyeri [SoCG09] General problem NP-hard: Chen-Freedman [SODA10] Special cases: Dey-Hirani-Krishnamoorthy [STOC10] Dey (2014) Computational Topology CCCG / 55
155 Conclusions Further Problems Reconstructions : Dey (2014) Computational Topology CCCG / 55
156 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open Dey (2014) Computational Topology CCCG / 55
157 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open high dimensions still not satisfactory Dey (2014) Computational Topology CCCG / 55
158 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open high dimensions still not satisfactory Mesh Generation : Dey (2014) Computational Topology CCCG / 55
159 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open high dimensions still not satisfactory Mesh Generation : piecewise smooth surfaces, complexes Dey (2014) Computational Topology CCCG / 55
160 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open high dimensions still not satisfactory Mesh Generation : piecewise smooth surfaces, complexes Dey (2014) Computational Topology CCCG / 55
161 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open high dimensions still not satisfactory Mesh Generation : piecewise smooth surfaces, complexes Data Analysis : Dey (2014) Computational Topology CCCG / 55
162 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open high dimensions still not satisfactory Mesh Generation : piecewise smooth surfaces, complexes Data Analysis : functions on spaces Dey (2014) Computational Topology CCCG / 55
163 Conclusions Further Problems Reconstructions : non-smooth surfaces remain open high dimensions still not satisfactory Mesh Generation : piecewise smooth surfaces, complexes Data Analysis : functions on spaces connecting to data mining, machine learning. Dey (2014) Computational Topology CCCG / 55
164 Thank Thank You Dey (2014) Computational Topology CCCG / 55
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