Topological Simplification in 3-Dimensions

Transcription

1 Topological Simplification in 3-Dimensions David Letscher Saint Louis University Computational & Algorithmic Topology Sydney Letscher (SLU) Topological Simplification CATS / 33

2 Motivation Original Question? Given a finite set of points X 0 R d what can you say about the topology of the shape they represent? Letscher (SLU) Topological Simplification CATS / 33

3 Motivation Original Question? Given a finite set of points X 0 R d what can you say about the topology of the shape they represent? Discrete point set = no interesting topology... Right? Letscher (SLU) Topological Simplification CATS / 33

4 Filtrations Input X 0 R d. (Assumed to be generic) α-shape X α = x X 0 B(x, α) α-filtration More generally X α1 X α2 X αk For continuous f : R d R, define X α = {x f (x) α}. For example, f might measure density (medical imaging,...). Letscher (SLU) Topological Simplification CATS / 33

5 Topological Critical Points Definition α s where X α ɛ = Xα for all sufficiently small α. If X 0 is generic or f is a (generic) Morse function then there are finitely many critical point and at each there is a single change in topology. Letscher (SLU) Topological Simplification CATS / 33

6 Homology Suppose that X is a triangulated space. Chains C k (X ) if the vector space (with base field Z/2Z) generated by the k-simplices of X (vertices, edges, triangles,...). Boundary For a k-simplex σ, σ is the formal sum of its (k 1)-simplices. For chains, the boundary operator can be extended linearly ( ) σi = (σ i ) Letscher (SLU) Topological Simplification CATS / 33

7 Homology Cycles Z k (X ) = subspace of chains c with c = 0 Boundaries B k (X ) = spaces of boundaries of (k + 1)-chains Homology H k (X ) = Z k (X )/B k (X ) If X R 3, Rank of H 0 (X ) = number of components of X Rank of H 0 (X ) = number of independent of X Rank of H 0 (X ) = number of voids of X Letscher (SLU) Topological Simplification CATS / 33

8 Homology, examples X Rank of H 0 (X ) Rank of H 1 (X ) Rank of H 2 (X ) Letscher (SLU) Topological Simplification CATS / 33

9 Persistent Homology [L-Edelsbrunner-Zomordian, 2000] H p k (X α) = Z k (X α )/ (Z k (X α ) B k (X α+p ) Equivalently, H p k (X α) = image(h k (X α ) H k (X α+p )) H p 0 (X α) H p 1 (X α) H p 2 (X α) counts the number of components of X α that are still separate in X α+p. measures the number of (independent) loops in X α that are not filled-in in X α+p. measures the number of voids in X α that are not filled-in in X α+p. Letscher (SLU) Topological Simplification CATS / 33

10 Persistence Diagrams Every topological critical point corresponds to a birth or death of a cycle. (We assume these critical points happen at distinct times.) Birth time An α where H k (X α ) = H k (X α ɛ ) Z for all sufficiently small ɛ > 0. Letscher (SLU) Topological Simplification CATS / 33

11 Persistence Diagrams Every topological critical point corresponds to a birth or death of a cycle. (We assume these critical points happen at distinct times.) Birth time An α where H k (X α ) = H k (X α ɛ ) Z for all sufficiently small ɛ > 0. Death time If c is some cycle born at time α, then it s death time is the smallest β such that there exists a cycle c H k (X α ) such that c + c bounds a cycle in X β. Letscher (SLU) Topological Simplification CATS / 33

12 Persistence Diagrams Persistence Diagrams A persistence diagram [Cohen-Steiner-Edelsbrunner-Harer, 2005] is a plot of the birth-death pairs of the plane dim 1 dim 2 dim Letscher (SLU) Topological Simplification CATS / 33

13 Persistence Diagrams dim 1 dim 2 dim Letscher (SLU) Topological Simplification CATS / 33

14 Persistence Diagrams dim 1 dim 2 dim Letscher (SLU) Topological Simplification CATS / 33

15 Persistence Diagrams dim 1 dim 2 dim Letscher (SLU) Topological Simplification CATS / 33

16 Stability of Persistence Diagrams We will augment the persistence diagrams by adding the diagonal with infinite multiplicity. Bottleneck distance If D f and D g are two persistence diagrams the bottleneck distance d B (D f, D g ) = inf sup x φ(x) bijections φ:d f D g x D f Bottle neck distance make the space of persistence diagrams a complete separable metric space. Stability [Cohen-Steiner-Edelsbrunner-Harer, 2005] If f, g : R d R have k-dimensional persistence diagram D f and D g, respectively, then d B (D f, D g ) f g Letscher (SLU) Topological Simplification CATS / 33

17 Simplification Question Suppose f : R d R has persistence diagrams D k. For a fixed ɛ > 0 let D k be D k with all points within a distance ɛ removed. Does there exists f with f f < ɛ and the k-dimensional persistence diagrams of f equal to D k dim dim 2 dim dim 1 dim 2 dim Letscher (SLU) Topological Simplification CATS / 33

18 Simplification dim dim 2 dim dim dim 2 dim dim dim Letscher (SLU) Topological Simplification CATS / 33

19 Simplification in 2-Dimensions [Bauer-Lange-Wardetzky] If f : R 2 R is a tame Morse function then for any ɛ > 0 there exits f : R 2 R such that f f < ɛ The persistence diagram for f are the persistence diagrams for f with every point within ɛ of the boundary removed. Letscher (SLU) Topological Simplification CATS / 33

20 Simplification Counter-example in 3-Dimensions Similar simplification is not possible in R 3 using persistent homology. Letscher (SLU) Topological Simplification CATS / 33

21 Simplification Counter-example in 3-Dimensions Similar simplification is not possible in R 3 using persistent homology. t = 0 t = 1 t = 2 t = 2 + ɛ t = 3 If f is any function with the second persistence diagram then f f 1 but d B (D, D ) ɛ. Letscher (SLU) Topological Simplification CATS / 33

22 Where it Fails Letscher (SLU) Topological Simplification CATS / 33

23 Where it Fails Letscher (SLU) Topological Simplification CATS / 33

24 Homotopy Fix points x 0 X and s 0 S k. Definition π k (X, x 0 ) = {f : (S k, s 0 ) (X, x 0 )}/ where two maps are equivalent is they are homotopic through maps that map s 0 to x 0 Letscher (SLU) Topological Simplification CATS / 33

25 Homotopy Fix points x 0 X and s 0 S k. Definition π k (X, x 0 ) = {f : (S k, s 0 ) (X, x 0 )}/ where two maps are equivalent is they are homotopic through maps that map s 0 to x 0 π 0 (X, x 0 ) = connected components of X (size is the rank of H 0 (X )) Letscher (SLU) Topological Simplification CATS / 33

26 Homotopy Fix points x 0 X and s 0 S k. Definition π k (X, x 0 ) = {f : (S k, s 0 ) (X, x 0 )}/ where two maps are equivalent is they are homotopic through maps that map s 0 to x 0 π 0 (X, x 0 ) = connected components of X (size is the rank of H 0 (X )) π 1 (X, x 0 ) is a non-abelian group (for connected X ) If X is connected the Abelianization of π 1 (X, x 1 ) is H 1 (X ). Carries strictly more information than homology. Letscher (SLU) Topological Simplification CATS / 33

27 Homotopy Fix points x 0 X and s 0 S k. Definition π k (X, x 0 ) = {f : (S k, s 0 ) (X, x 0 )}/ where two maps are equivalent is they are homotopic through maps that map s 0 to x 0 π 0 (X, x 0 ) = connected components of X (size is the rank of H 0 (X )) π 1 (X, x 0 ) is a non-abelian group (for connected X ) If X is connected the Abelianization of π 1 (X, x 1 ) is H 1 (X ). Carries strictly more information than homology. For k > 1, π k (X, x 0 ) is an Abelian group (distinct from H k (X )) Letscher (SLU) Topological Simplification CATS / 33

28 Persistent Homotopy Definition π p k (X α, x 0 ) = {f : (S k, s 0 ) (X α, x 0 )}/ where two maps are equivalent is they are homotopic in X α+p through maps that fix s 0 Equivalently, π p k (X α, x 0 ) = im(π k (X α, x 0 ) π k (X α+p, x 0 )) Letscher (SLU) Topological Simplification CATS / 33

29 Persistent Homotopy Diagrams Birth times Birth time k = 0 π 0 (X α ) = π 0 (X α ɛ ) {c} k = 1 π 1 (X α ) = π 1 (X α ɛ ) Z k > 1 π k (X α ) = π k (X α ɛ ) Z Letscher (SLU) Topological Simplification CATS / 33

30 Persistent Homotopy Diagrams Birth times and death times Birth time Death time k = 0 π 0 (X α ) = π 0 (X α ɛ ) {c} First β where the component c merges with a component existing before time α k = 1 π 1 (X α ) = π 1 (X α ɛ ) Z First β where there exists γ π 1 (X α ) π 1 (X α ɛ ) where γ maps to trivial element in π 1 (X β ) k > 1 π k (X α ) = π k (X α ɛ ) Z First β where there exists γ π k (X α ) π k (X α ɛ ) where γ maps to trivial element in π k (X β ) Letscher (SLU) Topological Simplification CATS / 33

31 How Can the Diagrams Differ in R 3 Persistent homology vs persistent homotopy 0-cycles 1-cycles 2-cycles Letscher (SLU) Topological Simplification CATS / 33

32 How Can the Diagrams Differ in R 3 Persistent homology vs persistent homotopy 0-cycles 1-cycles 2-cycles Unchanged Letscher (SLU) Topological Simplification CATS / 33

33 How Can the Diagrams Differ in R 3 Persistent homology vs persistent homotopy 0-cycles 1-cycles 2-cycles Unchanged Letscher (SLU) Topological Simplification CATS / 33

34 How Can the Diagrams Differ in R 3 Persistent homology vs persistent homotopy 0-cycles 1-cycles 2-cycles Unchanged Moved up Letscher (SLU) Topological Simplification CATS / 33

35 How Can the Diagrams Differ in R 3 Persistent homology vs persistent homotopy 0-cycles 1-cycles 2-cycles Unchanged Moved up Letscher (SLU) Topological Simplification CATS / 33

36 How Can the Diagrams Differ in R 3 Persistent homology vs persistent homotopy 0-cycles 1-cycles 2-cycles Unchanged Moved up Moved Right Letscher (SLU) Topological Simplification CATS / 33

37 Simplification in 3-Dimensions Theorem If M is either a closed, irreducible 3-manifold or R 3 and f : M R is a tame Morse function then for any ɛ > 0 there exits f : M R such that f f < ɛ The persistence diagram for f are the persistence diagrams for f with every point within ɛ of the boundary removed. Letscher (SLU) Topological Simplification CATS / 33

38 Classical 3-Manifold Results Loop Theorem Suppose that M is a compact 3-manifold and B M a compact surface. For any normal subgroup N of π 1 (B) that does not contain the kernel of π 1 (B) π 1 (M) there exists an embedding f : (D, S 1 ) (M, B) such that f (S 1 ) represents a loop in π 1 (B) N. Dehn s Lemma Let M be a compact 3-manifold and f : (D, S 1 ) (M, M) be an embedding on some collar neighborhood A of D. Then there is an embedding g : D M that agrees with f on A. Sphere Theorem Suppose M is a compact 3-manifold with non-trivial π 2 (M). Then there exists an essential 2-sided embedded sphere or projective plane. Letscher (SLU) Topological Simplification CATS / 33

39 3-Manifold Algorithms Issue Non-injective π 1 ( M) π 1 (M) Theory Existence of embedded compressing disk (Loop theorem) Non-trivial π 2 (M) Existence of embedded essential sphere (Sphere theorem) Letscher (SLU) Topological Simplification CATS / 33

40 3-Manifold Algorithms Issue Theory Algorithm Non-injective Existence of embedded compressing Normal surfaces π 1 ( M) π 1 (M) disk (Loop theorem) Non-trivial Existence of embedded essential Normal and almost π 2 (M) sphere (Sphere theorem) normal surfaces Letscher (SLU) Topological Simplification CATS / 33

41 Simplified Case Assumption Two spaces in a filtration X 0 X 1 S 3 Letscher (SLU) Topological Simplification CATS / 33

42 Simplified Case Assumption Two spaces in a filtration X 0 X 1 S 3 What could happen? New components form New 1-handles added forming loops A void forms Letscher (SLU) Topological Simplification CATS / 33

43 Simplified Case Assumption Two spaces in a filtration X 0 X 1 S 3 What could happen? New components form New 1-handles added forming loops A void forms Components merge 2-handles added closing loops Void filled Letscher (SLU) Topological Simplification CATS / 33

44 Simplified Case Goal Find Y such that X 0 Y X 1 π k (Y ) = im(π k (X 0 ) π k (X 1 )) for all k 1 π k (X 0 ) π k (Y ) π k (X 1 ) 1 is exact Letscher (SLU) Topological Simplification CATS / 33

45 Simplified Case Goal Find Y such that X 0 Y X 1 π k (Y ) = im(π k (X 0 ) π k (X 1 )) for all k 1 π k (X 0 ) π k (Y ) π k (X 1 ) 1 is exact Letscher (SLU) Topological Simplification CATS / 33

46 Simplified Case Goal Find Y such that X 0 Y X 1 π k (Y ) = im(π k (X 0 ) π k (X 1 )) for all k 1 π k (X 0 ) π k (Y ) π k (X 1 ) 1 is exact Letscher (SLU) Topological Simplification CATS / 33

47 Existence of Y Non-trivial kernel π 1 (X 0 ) π 1 (X 1 ) Loop theorem implies there exists an embedded disk D with D is an essential curve in X 0 D X 1 X 0 Add a small neighborhood of D to Y. Note: we might need to add several disks. Letscher (SLU) Topological Simplification CATS / 33

48 Existence of Y Non-trivial kernel π 1 (X 0 ) π 1 (X 1 ) Loop theorem implies there exists an embedded disk D with D is an essential curve in X 0 D X 1 X 0 Add a small neighborhood of D to Y. Note: we might need to add several disks. Non-trivial kernel π 2 (X 0 ) π 2 (X 1 ) Sphere theorem implies there is a essential sphere in X 0 bounding a ball B X 1. Add B to Y. Letscher (SLU) Topological Simplification CATS / 33

49 Existence of Y Non-trivial kernel π 1 (X 0 ) π 1 (X 1 ) Loop theorem implies there exists an embedded disk D with D is an essential curve in X 0 D X 1 X 0 Add a small neighborhood of D to Y. Note: we might need to add several disks. Non-trivial kernel π 2 (X 0 ) π 2 (X 1 ) Sphere theorem implies there is a essential sphere in X 0 bounding a ball B X 1. Add B to Y. Non-trivial kernel π 0 (X 0 ) π 0 (X 1 ) Connect components of X 0 by shortest paths in X 1 and add small neighborhoods of the paths to Y. Letscher (SLU) Topological Simplification CATS / 33

50 General Case X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

51 General Case X 1 0 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

52 General Case X 1 0 X 1 1 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

53 General Case X 1 0 X 1 1 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

54 General Case X 1 0 X 1 1 X 1 2 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

55 General Case X 1 0 X 1 1 X 1 2 X 1 3 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

56 General Case X 2 0 X 2 1 X 2 2 X 1 0 X 1 1 X 1 2 X 1 3 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

57 General Case X 3 0 X 3 1 X 3 2 X 2 0 X 2 1 X 2 2 X 1 0 X 1 1 X 1 2 X 1 3 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

58 General Case X 3 0 X 3 1 X 3 2 X 2 0 X 2 1 X 2 2 X 1 0 X 1 1 X 1 2 X 1 3 X 0 X 1 X 2 X 3 Letscher (SLU) Topological Simplification CATS / 33

59 General Case X 3 0 X 3 1 X 3 2 X 2 0 X 2 1 X 2 2 X 1 0 X 1 1 X 1 2 X 1 3 X 0 X 1 X 2 X 3 Note: Upward diagonal arrows are π k -surjective for all k Downward diagonal arrows are π k -injective for all k Letscher (SLU) Topological Simplification CATS / 33

60 Is it Practical? Letscher (SLU) Topological Simplification CATS / 33

61 Is it Practical? Yes Letscher (SLU) Topological Simplification CATS / 33

62 Is it Practical? Yes and No Letscher (SLU) Topological Simplification CATS / 33

63 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams agree. Letscher (SLU) Topological Simplification CATS / 33

64 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Letscher (SLU) Topological Simplification CATS / 33

65 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Algorithm A simple modification of the O(n 3 ) standard persistence algorithm [ELZ] computes the following: Paired birth/death times Letscher (SLU) Topological Simplification CATS / 33

66 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Algorithm A simple modification of the O(n 3 ) standard persistence algorithm [ELZ] computes the following: Paired birth/death times Generators for the homology groups at each birth that are minimal in a particular sense Letscher (SLU) Topological Simplification CATS / 33

67 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Algorithm A simple modification of the O(n 3 ) standard persistence algorithm [ELZ] computes the following: Paired birth/death times Generators for the homology groups at each birth that are minimal in a particular sense Boundary complexes for each birth that are minimal in a particular sense. Letscher (SLU) Topological Simplification CATS / 33

68 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Guarantees? O(n 3 ) worst case runtime (appears quadratic in practice) Letscher (SLU) Topological Simplification CATS / 33

69 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Guarantees? O(n 3 ) worst case runtime (appears quadratic in practice) If we re luckly all of the death complexes are disks and we re done. Letscher (SLU) Topological Simplification CATS / 33

70 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Guarantees? O(n 3 ) worst case runtime (appears quadratic in practice) If we re luckly all of the death complexes are disks and we re done. Hope we re were lucky. Letscher (SLU) Topological Simplification CATS / 33

71 Is it Practical? Yes and No In practice it appears that persistent homology and persistent homotopy diagrams mostly agree. Guarantees? O(n 3 ) worst case runtime (appears quadratic in practice) If we re luckly all of the death complexes are disks and we re done. Hope we re were lucky. If we re unlucky resort to normal surface theory for the (hopefully few) where the death complexes are not disks (Regina). Letscher (SLU) Topological Simplification CATS / 33

72 Thanks Questions? Letscher (SLU) Topological Simplification CATS / 33