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 { ∈ | || || ≤ } d-sphere S d x Rd 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 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 ∈ ∈ (ii) t1, t2 K t1 t2 is a face of both ∈ ⇒ ∩ Triangulation: K is a triangulation of a topological space T if T K ≈ | |

Topology Background Simplicial complex

Abstract V (K): vertex set, k-simplex: (k + 1)-subset σ V (K) ⊆ Complex K = σ σ0 σ = σ0 K { k ⊆ ⇒ ∈ }

Dey (2014) Computational Topology CCCG 14 5 / 55 Triangulation: K is a triangulation of a topological space T if T K ≈ | |

Topology Background Simplicial complex

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 { k ⊆ ⇒ ∈ } Geometric k-simplex: k + 1-point convex hull Complex K: (i) t K if t is a face of t0 K ∈ ∈ (ii) t1, t2 K t1 t2 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 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 ε-sample (Amenta-Bern-Eppstein 98)

Each x has a sample within εLfs(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 ε sample is homeomorphic to the sampled surface for ε < 0.06. −

Sampling Crust and Cocone Guarantees

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(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 < 0.06. ∈ 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 < 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. |

Dey (2014) Computational Topology CCCG 14 11 / 55 Sampling Restricted Voronoi/Delaunay

Deﬁnition

Restricted Delaunay: Del P Σ: dual of Vor P Σ | |

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. |

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. |

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. |

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 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. |

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. |

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 (ε, 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. ⇒ | 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 (ε, δ)-sampling Restricted Delaunay does not capture topology

Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ⇒ | 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 (ε, δ)-sampling Restricted Delaunay does not capture topology

Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ⇒ | 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 (ε, δ)-sampling Restricted Delaunay does not capture topology

Slivers are arbitrarily oriented [CDR05] Del P Σ Σ no matter how dense P is. ⇒ | 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. ⇒ | 6≈ Delaunay triangulation becomes harder

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 14 16 / 55 Delaunay triangulation becomes harder

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. ⇒ | 6≈

Dey (2014) Computational Topology CCCG 14 16 / 55 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. ⇒ | 6≈ Delaunay triangulation becomes harder

Dey (2014) Computational Topology CCCG 14 16 / 55 High Dimensions Reconstruction

Dey (2014) Computational Topology CCCG 14 17 / 55 High Dimensions Reconstruction

Theorem (Cheng-Dey-Ramos 2005) Given an (ε, δ)-sample P of a smooth manifold Σ Rd for appropriate ε, δ > 0, there is a weight assignment ⊂of P so that ˆ Del P Σ Σ which can be computed eﬃciently. | ≈ Theorem (Chazal-Lieutier 2006) Given an ε-noisy sample P of manifold Σ Rd , there exists ⊂ rp ρ(Σ) for each p P so that the union of balls B(p, rp) is homotopy≤ equivalent to∈ Σ.

Dey (2014) Computational Topology CCCG 14 17 / 55 Lfs vanishes, introduce µ-reach and deﬁne( ε, µ)-samples.

High Dimensions Reconstructing Compacts

Dey (2014) Computational Topology CCCG 14 18 / 55 High Dimensions Reconstructing Compacts

Lfs vanishes, introduce µ-reach and deﬁne( ε, µ)-samples.

Dey (2014) Computational Topology CCCG 14 18 / 55 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 compactK Rd 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 14 18 / 55 Mesh generation Surface and volume mesh

Dey (2014) Computational Topology CCCG 14 19 / 55 Mesh generation Surface and volume mesh

Dey (2014) Computational Topology CCCG 14 19 / 55 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)

Mesh generation Delaunay reﬁnement

Pioneered by Chew89,Ruppert92,Shewchuk98

Dey (2014) Computational Topology CCCG 14 20 / 55 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)

Mesh generation Delaunay reﬁnement

Pioneered by Chew89,Ruppert92,Shewchuk98 To mesh some domain D

Dey (2014) Computational Topology CCCG 14 20 / 55 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)

Mesh generation Delaunay reﬁnement

Pioneered by Chew89,Ruppert92,Shewchuk98 To mesh some domain D 1 Initialize points P D, compute Del P ⊂

Dey (2014) Computational Topology CCCG 14 20 / 55 3 Return Del P D | Burden is to show termination (by packing argument)

Mesh generation Delaunay reﬁnement

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 14 20 / 55 Burden is to show termination (by packing argument)

Mesh generation Delaunay reﬁnement

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 14 20 / 55 Mesh generation Delaunay reﬁnement

Dey (2014) Computational Topology CCCG 14 20 / 55 It is homeomorphic to Σ Each triangle has normal aligning withinO (ε) angle to the surface normals 2 Hausdorﬀ distance between Σ and Del P Σ isO (ε ) of LFS. |

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 14 21 / 55 Each triangle has normal aligning withinO (ε) angle to the surface normals 2 Hausdorﬀ distance between Σ and Del P Σ isO (ε ) of LFS. |

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 14 21 / 55 2 Hausdorﬀ distance between Σ and Del P Σ isO (ε ) of LFS. |

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 withinO (ε) angle to the surface normals

Dey (2014) Computational Topology CCCG 14 21 / 55 Mesh generation Sampling Theorem

Dey (2014) Computational Topology CCCG 14 21 / 55 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 withinO (ε) angle to the surface normals 2 Hausdorﬀ distance between Σ and Del P Σ isO (ε ) of LFS. |

Dey (2014) Computational Topology CCCG 14 22 / 55 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 14 23 / 55 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 14 24 / 55 Mesh generation Diﬃculty

How to compute Lfs(x)? Can be approximated by computing approximatemedial axis–needs a dense sample.

Dey (2014) Computational Topology CCCG 14 25 / 55 Mesh generation Diﬃculty

How to compute Lfs(x)? Can be approximated by computing approximatemedial axis–needs a dense sample.

Dey (2014) Computational Topology CCCG 14 25 / 55 Topology guarantee is lost Require topological disks around vertices Guarantees manifolds

Mesh generation A Solution

Replace d(x, P) < εLfs(x) with d(x, P) < λ, an user parameter

Dey (2014) Computational Topology CCCG 14 26 / 55 Require topological disks around vertices Guarantees manifolds

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 14 26 / 55 Guarantees manifolds

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 14 26 / 55 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 14 26 / 55 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 14 26 / 55 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 14 27 / 55 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 14 28 / 55 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 14 29 / 55 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 withinO (λ) angle to the surface normals 2 3 Hausdorﬀ distance between Σ and Del P Σ isO (λ ) of LFS. |

Mesh generation A Meshing Theorem

Theorem Previous algorithm produces output mesh with the following guarantees:

Dey (2014) Computational Topology CCCG 14 30 / 55 1 It is related to Σ by an isotopy 2 Each triangle has normal aligning withinO (λ) angle to the surface normals 2 3 Hausdorﬀ distance between Σ and Del P Σ isO (λ ) of LFS. |

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 14 30 / 55 2 Each triangle has normal aligning withinO (λ) angle to the surface normals 2 3 Hausdorﬀ distance between Σ and Del P Σ isO (λ ) of LFS. |

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 14 30 / 55 2 3 Hausdorﬀ distance between Σ and Del P Σ isO (λ ) of LFS. |

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 2 Each triangle has normal aligning withinO (λ) angle to the surface normals

Dey (2014) Computational Topology CCCG 14 30 / 55 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 2 Each triangle has normal aligning withinO (λ) angle to the surface normals 2 3 Hausdorﬀ distance between Σ and Del P Σ isO (λ ) of LFS. |

Dey (2014) Computational Topology CCCG 14 30 / 55 Topological Data Analysis Data Analysis by Persistent Homology

Persistent homology [Edelsbrunner-Letscher-Zomorodian 00], [Zomorodian-Carlsson 02]

Dey (2014) Computational Topology CCCG 14 31 / 55 Topological Data Analysis Chain

Let be a ﬁnite simplicial complex K

Dey (2014) Computational Topology CCCG 14 32 / 55 Topological Data Analysis Chain

Let be a ﬁnite simplicial complex K

Simplicial complex

Dey (2014) Computational Topology CCCG 14 32 / 55 Topological Data Analysis Chain

Let be a ﬁnite simplicial complex K

Simplicial complex

Deﬁnition P A p-chain in is a formal sum of p-simplices: c = ai σi ; sum is K i the addition in a ring, Z, Z2, R etc.

Dey (2014) Computational Topology CCCG 14 32 / 55 Topological Data Analysis Chain

Let be a ﬁnite simplicial complex K

1-chain ab + bc + cd (ai Z2) ∈ Deﬁnition P A p-chain in is a formal sum of p-simplices: c = ai σi ; sum is K i the addition in a ring, Z, Z2, R etc.

Dey (2014) Computational Topology CCCG 14 32 / 55 Topological Data Analysis Boundary

Deﬁnition

A p-boundary ∂p+1c of a (p + 1)-chain c is deﬁned as the sum of boundaries of its simplices

Dey (2014) Computational Topology CCCG 14 33 / 55 Topological Data Analysis Boundary

Deﬁnition

A p-boundary ∂p+1c of a (p + 1)-chain c is deﬁned as the sum of boundaries of its simplices

Simplicial complex

Dey (2014) Computational Topology CCCG 14 33 / 55 Topological Data Analysis Boundary

Deﬁnition

A p-boundary ∂p+1c of a (p + 1)-chain c is deﬁned as the sum of boundaries of its simplices

2-chain bcd + bde (under Z2)

Dey (2014) Computational Topology CCCG 14 33 / 55 Topological Data Analysis Boundary

Deﬁnition

A p-boundary ∂p+1c 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 Z2)

Dey (2014) Computational Topology CCCG 14 33 / 55 Each p-boundary is a p-cycle: ∂p ∂p+1 = 0 ◦

Topological Data Analysis Cycles

Deﬁnition A p-cycle is a p-chain that has an empty boundary

Dey (2014) Computational Topology CCCG 14 34 / 55 Each p-boundary is a p-cycle: ∂p ∂p+1 = 0 ◦

Topological Data Analysis Cycles

Deﬁnition A p-cycle is a p-chain that has an empty boundary

Simplicial complex

Dey (2014) Computational Topology CCCG 14 34 / 55 Each p-boundary is a p-cycle: ∂p ∂p+1 = 0 ◦

Topological Data Analysis Cycles

Deﬁnition A p-cycle is a p-chain that has an empty boundary

1-cycle ab + bc + cd + de + ea (under Z2)

Dey (2014) Computational Topology CCCG 14 34 / 55 Topological Data Analysis Cycles

Deﬁnition A p-cycle is a p-chain that has an empty boundary

1-cycle ab + bc + cd + de + ea (under Z2)

Each p-boundary is a p-cycle: ∂p ∂p+1 = 0 ◦

Dey (2014) Computational Topology CCCG 14 34 / 55 Topological Data Analysis Groups

Deﬁnition

The p-chain groupC p( ) of is formed by p-chains under addition K K

Dey (2014) Computational Topology CCCG 14 35 / 55 Topological Data Analysis Groups

Deﬁnition

The p-chain groupC p( ) of is formed by p-chains under addition K K

The boundary operator ∂p induces a homomorphism

∂p :Cp( ) Cp−1( ) K → K

Dey (2014) Computational Topology CCCG 14 35 / 55 Topological Data Analysis Groups

Deﬁnition

The p-chain groupC p( ) of is formed by p-chains under addition K K

The boundary operator ∂p induces a homomorphism

∂p :Cp( ) Cp−1( ) K → K

Deﬁnition

The p-cycle groupZ p( ) of is the kernel ker ∂p K K

Dey (2014) Computational Topology CCCG 14 35 / 55 Topological Data Analysis Groups

Deﬁnition

The p-chain groupC p( ) of is formed by p-chains under addition K K

The boundary operator ∂p induces a homomorphism

∂p :Cp( ) Cp−1( ) K → K

Deﬁnition

The p-cycle groupZ p( ) of is the kernel ker ∂p K K Deﬁnition

The p-boundary groupB p( ) of is the image im ∂p+1 K K

Dey (2014) Computational Topology CCCG 14 35 / 55 Topological Data Analysis Homology

Deﬁnition The p-dimensional homology group is deﬁned as Hp( ) = Zp( )/Bp( ) K K K

Dey (2014) Computational Topology CCCG 14 36 / 55 Topological Data Analysis Homology

Definition The p-dimensional homology group is defined as Hp( ) = Zp( )/Bp( ) K K K Definition 0 0 Two p-chains c and c are homologous if c = c + ∂p+1d for some chain d

Dey (2014) Computational Topology CCCG 14 36 / 55 Topological Data Analysis Homology

Definition The p-dimensional homology group is defined as Hp( ) = Zp( )/Bp( ) K K K Definition 0 0 Two p-chains c and c are homologous if c = c + ∂p+1d for some chain d

(a) trivial (null-homologous) cycle; (b), (c) nontrivial homologous cycles

Dey (2014) Computational Topology CCCG 14 36 / 55 B(p, r) denotes an open d-ball centered at p with radius r

Topological Data Analysis Complexes

Let P Rd be a point set ⊂

Dey (2014) Computational Topology CCCG 14 37 / 55 Topological Data Analysis Complexes

Let P Rd be a point set ⊂ B(p, r) denotes an open d-ball centered at p with radius r

Dey (2014) Computational Topology CCCG 14 37 / 55 Topological Data Analysis Complexes

Let P Rd be a point set ⊂ B(p, r) denotes an open d-ball centered at p with radius r

Deﬁnition The Cechˇ complex r (P) is a simplicial complex where a simplex r C σ (P) iﬀ Vert(σ) P and p∈Vert(σ)B(p, r/2) = 0 ∈ C ⊆ ∩ 6

Dey (2014) Computational Topology CCCG 14 37 / 55 Topological Data Analysis Complexes

Let P Rd be a point set ⊂ B(p, r) denotes an open d-ball centered at p with radius r

Dey (2014) Computational Topology CCCG 14 37 / 55 Topological Data Analysis Complexes

Let P Rd be a point set ⊂ B(p, r) denotes an open d-ball centered at p with radius r

Definition The Cechˇ complex r (P) is a simplicial complex where a simplex r C σ (P) iff Vert(σ) P and p∈Vert(σ)B(p, r/2) = 0 ∈ C ⊆ ∩ 6 Definition The Rips complex r (P) is a simplicial complex where a simplex σ r (P) iff Vert(Rσ) are within pairwise Euclidean distance of r ∈ R Proposition For any finite set P Rd and any r 0, r (P) r (P) 2r (P) ⊂ ≥ C ⊆ R ⊆ C Dey (2014) Computational Topology CCCG 14 37 / 55 Topological Data Analysis Point set P

Dey (2014) Computational Topology CCCG 14 38 / 55 Topological Data Analysis Balls B(p, r/2) for p P ∈

Dey (2014) Computational Topology CCCG 14 39 / 55 Topological Data Analysis Cechˇ complex r (P) C

Dey (2014) Computational Topology CCCG 14 40 / 55 Topological Data Analysis Rips complex r (P) R

Dey (2014) Computational Topology CCCG 14 41 / 55 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 14 42 / 55 Topological Data Analysis Persistent Homology

−1 f : T R; Ta = f ( , a], the sublevel→ set −∞

Ta Tb for a b provides inclusion ⊆ ≤ map ι : Ta Tb → Induced map ι∗ :Hp(Ta) Hp(Tb) giving the sequence →

0 Hp(Ta1 ) Hp(Ta2 ) Hp(Tan ) Hp(T) → → → · · · → →

Persistent homology classes: Image of ij fp :Hp(Tai ) Hp(Taj ) → Dey (2014) Computational Topology CCCG 14 43 / 55 Union of balls with its nerve Cechˇ complex

Topological Data Analysis Continuous to Discrete

Replace T with a simplicial complex K := K(T)

Dey (2014) Computational Topology CCCG 14 44 / 55 Topological Data Analysis Continuous to Discrete

Replace T with a simplicial complex K := K(T) Union of balls with its nerve Cechˇ complex

Dey (2014) Computational Topology CCCG 14 44 / 55 Topological Data Analysis Continuous to Discrete

Replace T with a simplicial complex K := K(T) Union of balls with its nerve Cechˇ complex

Dey (2014) Computational Topology CCCG 14 44 / 55 Topological Data Analysis Continuous to Discrete

Replace T with a simplicial complex K := K(T) Union of balls with its nerve Cechˇ (b) (a) complex

Dey (2014) Computational Topology CCCG 14 44 / 55 Topological Data Analysis Continuous to Discrete

Replace T with a simplicial complex K := K(T) Union of balls with its nerve Cechˇ (b) (a) complex Evolution of sublevel sets becomes Filtration:

Dey (2014) Computational Topology CCCG 14 44 / 55 Topological Data Analysis Continuous to Discrete

Replace T with a simplicial complex K := K(T) Union of balls with its nerve Cechˇ (b) (a) complex Evolution of sublevel sets becomes Filtration:

Dey (2014) Computational Topology CCCG 14 44 / 55 Topological Data Analysis Bar Codes

birth-death and bar codes

Dey (2014) Computational Topology CCCG 14 45 / 55 incorporate the diagonal in the diagram Dgm p(f )

Death

Birth

Topological Data Analysis Persistence Diagram

a bar [a, b] is represented as a point in the plane

Dey (2014) Computational Topology CCCG 14 46 / 55 Death

Birth

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 14 46 / 55 Death

Birth

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 14 46 / 55 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 14 46 / 55 Topological Data Analysis Stability of Persistence Diagram

Bottleneck distance (C all bijections)

dB (Dgm p(f ), Dgm p(g)) := inf sup x c(x) c∈C x∈Dgm p(f ) k − k

Theorem (Cohen-Steiner,Edelsbrunner,Harer 06)

dB (Dgm (f ), Dgm (g)) f g ∞ p p ≤ k − k Dey (2014) Computational Topology CCCG 14 47 / 55 dP be the distance function from sample P

Topological Data Analysis Back to Point Data

dT be the distance function from the space T.

Dey (2014) Computational Topology CCCG 14 48 / 55 Topological Data Analysis Back to Point Data

dT be the distance function from the space T. dP be the distance function from sample P

Dey (2014) Computational Topology CCCG 14 48 / 55 Topological Data Analysis Back to Point Data

Death

Birth dT be the distance function from the space T. dP be the distance function from sample P

Dey (2014) Computational Topology CCCG 14 48 / 55 Topological Data Analysis Back to Point Data

Death

Birth dT be the distance function from the space T. dP be the distance function from sample P d dP ∞ d(T, P) = ε k T − k ≤

Dey (2014) Computational Topology CCCG 14 48 / 55 Topological Data Analysis Back to Point Data

Death

Birth dT be the distance function from the space T. dP be the distance function from sample P d dP ∞ d(T, P) = ε k T − k ≤ dB (Dgm (d ), Dgm (dP )) ε p T ≤

Dey (2014) Computational Topology CCCG 14 48 / 55 Topological Data Analysis Back to Point Data

Death

Birth dT be the distance function from the space T. dP be the distance function from sample P d dP ∞ d(T, P) = ε k T − k ≤ dB (Dgm (d ), Dgm (dP )) ε p T ≤

Dey (2014) Computational Topology CCCG 14 48 / 55 Most literature considers Rips complexes, but they are Huge Sparsiﬁed Rips complex [Sheehy 12] Graph Induced Complex (GIC) [D.-Fang-Wang 13]

Topological Data Analysis Issues: Choice of Complexes

Cechˇ complexes are diﬃcult to compute

Dey (2014) Computational Topology CCCG 14 49 / 55 Sparsiﬁed Rips complex [Sheehy 12] Graph Induced Complex (GIC) [D.-Fang-Wang 13]

Topological Data Analysis Issues: Choice of Complexes

Cechˇ complexes are diﬃcult to compute Most literature considers Rips complexes, but they are Huge

Dey (2014) Computational Topology CCCG 14 49 / 55 Graph Induced Complex (GIC) [D.-Fang-Wang 13]

r(P ) R

Topological Data Analysis Issues: Choice of Complexes

Cechˇ complexes are diﬃcult to compute Most literature considers Rips complexes, but they are Huge Sparsiﬁed Rips complex [Sheehy 12]

Dey (2014) Computational Topology CCCG 14 49 / 55 Topological Data Analysis Issues: Choice of Complexes

Cechˇ 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(P ) R

Dey (2014) Computational Topology CCCG 14 49 / 55 Topological Data Analysis Issues: Choice of Complexes

Cechˇ 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(P ) Q R

Dey (2014) Computational Topology CCCG 14 49 / 55 Topological Data Analysis Issues: Choice of Complexes

Cechˇ 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 (P ) r0(Q) R R

Dey (2014) Computational Topology CCCG 14 49 / 55 Topological Data Analysis Issues: Choice of Complexes

Cechˇ 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 (P ) r0(Q) r(P,Q) R R G

Dey (2014) Computational Topology CCCG 14 49 / 55 f1 f2 fn 0 K K1 K2 Kn = K −→ −→ ··· −→

Eﬃcient algorithm for Zigzag simplicial maps?

Simplicial maps instead of inclusions [D.-Fan-Wang 14]

Topological Data Analysis Issues: Filtration Maps

Zigzag inclusions[Carlsson-de Silva-Morozov 09]

K1 K2 K3 ... Kn ⊆ ⊆ ⊆ ⊆

Dey (2014) Computational Topology CCCG 14 50 / 55 Eﬃcient algorithm for Zigzag simplicial maps?

Simplicial maps instead of inclusions [D.-Fan-Wang 14]

f1 f2 fn 0 K K1 K2 Kn = K −→ −→ ··· −→

Topological Data Analysis Issues: Filtration Maps

Zigzag inclusions[Carlsson-de Silva-Morozov 09]

K1 K2 K3 ... Kn ⊆ ⊇ ⊇ ⊆

Dey (2014) Computational Topology CCCG 14 50 / 55 Eﬃcient algorithm for Zigzag simplicial maps?

Topological Data Analysis Issues: Filtration Maps

Zigzag inclusions[Carlsson-de Silva-Morozov 09]

K1 K2 K3 ... Kn ⊆ ⊇ ⊇ ⊆ Simplicial maps instead of inclusions [D.-Fan-Wang 14]

f1 f2 fn 0 K K1 K2 Kn = K −→ −→ ··· −→

Dey (2014) Computational Topology CCCG 14 50 / 55 Topological Data Analysis Issues: Filtration Maps

Zigzag inclusions[Carlsson-de Silva-Morozov 09]

K1 K2 K3 ... Kn ⊆ ⊇ ⊇ ⊆ Simplicial maps instead of inclusions [D.-Fan-Wang 14]

f1 f2 fn 0 K K1 K2 Kn = K −→ −→ ··· −→

Eﬃcient algorithm for Zigzag simplicial maps?

Dey (2014) Computational Topology CCCG 14 50 / 55 Reeb graphs [C-MEHNP 03, D.-Wang 11]

Interleaving [Chazal-Oudot 08], [CCGGO08] Scalar ﬁeld data analysis [CGOS 09] Computing topological structures

Further Problems Further Issues/Problems

What about stabilty when domains aren’t same.

Dey (2014) Computational Topology CCCG 14 51 / 55 Reeb graphs [C-MEHNP 03, D.-Wang 11]

Scalar ﬁeld data analysis [CGOS 09] Computing topological structures

Further Problems Further Issues/Problems

What about stabilty when domains aren’t same. Interleaving [Chazal-Oudot 08], [CCGGO08]

Dey (2014) Computational Topology CCCG 14 51 / 55 Reeb graphs [C-MEHNP 03, D.-Wang 11]

Computing topological structures

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 14 51 / 55 Reeb graphs [C-MEHNP 03, D.-Wang 11]

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 14 51 / 55 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 14 51 / 55 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 14 51 / 55 Further Problems Further Issues/Problems

Dey (2014) Computational Topology CCCG 14 51 / 55 First solution for surfaces: Erickson-Whittlesey [SODA05] General problem NP-hard: Chen-Freedman [SODA10]

H1 basis for simplicial complexes: D.-Sun-Wang [SoCG10]

Further Problems Optimal Homology Basis Problem

Compute an optimal set of cycles forming a basis

Dey (2014) Computational Topology CCCG 14 52 / 55 First solution for surfaces: Erickson-Whittlesey [SODA05] General problem NP-hard: Chen-Freedman [SODA10]

H1 basis for simplicial complexes: D.-Sun-Wang [SoCG10]

Further Problems Optimal Homology Basis Problem

Compute an optimal set of cycles forming a basis

Dey (2014) Computational Topology CCCG 14 52 / 55 First solution for surfaces: Erickson-Whittlesey [SODA05] General problem NP-hard: Chen-Freedman [SODA10]

H1 basis for simplicial complexes: D.-Sun-Wang [SoCG10]

Further Problems Optimal Homology Basis Problem

Compute an optimal set of cycles forming a basis

Dey (2014) Computational Topology CCCG 14 52 / 55 General problem NP-hard: Chen-Freedman [SODA10]

H1 basis for simplicial complexes: D.-Sun-Wang [SoCG10]

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 14 52 / 55 H1 basis for simplicial complexes: D.-Sun-Wang [SoCG10]

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 14 52 / 55 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]

H1 basis for simplicial complexes: D.-Sun-Wang [SoCG10]

Dey (2014) Computational Topology CCCG 14 52 / 55 Surfaces: Colin de Verdi`ere-Lazarus [DCG05], Colin de Verdi`ere-Erickson[SODA06], Chambers-Erickson-Nayyeri [SoCG09] General problem NP-hard: Chen-Freedman [SODA10] Special cases: Dey-Hirani-Krishnamoorthy [STOC10]

Further Problems Optimal Localization