Topological Inference via Meshing

Transcription

1 Topological Inference via Meshing Benoit Hudson, Gary Miller, Steve Oudot and Don Sheehy SoCG 2010

2 Mesh Generation and Persistent Homology

3 The Problem Input: Points in Euclidean space sampled from some unknown object. Output: Information about the topology of the unknown object.

4 Points, offsets, homology, and persistence.

5 Points, offsets, homology, and persistence. Input: P R d

6 Points, offsets, homology, and persistence. Input: P R d P α = ball(p, α) p P

7 Points, offsets, homology, and persistence. Input: P R d P α = ball(p, α) p P

8 Points, offsets, homology, and persistence. Input: P R d P α = ball(p, α) p P

9 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P

10 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

11 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

12 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

13 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

14 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

15 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P Compute the Homology

16 Points, offsets, homology, and persistence. Offsets Input: P R d P α = ball(p, α) p P Persistent Compute the Homology

17 Persistence Diagrams

18 Persistence Diagrams

19 Persistence Diagrams Bottleneck Distance d B = max i p i q i

20 Approximate Persistence Diagrams Bottleneck Distance d B = max i p i q i

21 Approximate Persistence Diagrams Birth and Death times differ by a constant factor. Bottleneck Distance d B = max i p i q i

22 Approximate Persistence Diagrams Birth and Death times differ by a constant factor. Bottleneck Distance d B = max i p i q i This is just the bottleneck distance of the log-scale diagrams.

23 Approximate Persistence Diagrams Birth and Death times differ by a constant factor. Bottleneck Distance d B = max i p i q i This is just the bottleneck distance of the log-scale diagrams. log a log b log a b a b < ε < ε < 1 + ε

24 We need to build a filtered simplicial complex. Associate a birth time with each simplex in complex K. At time, we have a complex K consisting of all simplices born at or before time. time

25 There are two phases, one is geometric the other is topological. Geometry Build a filtration, i.e. a filtered simplicial complex. Topology (linear algebra) Compute the persistence diagram (Run the Persistence Algorithm).

26 There are two phases, one is geometric the other is topological. Geometry Build a filtration, i.e. a filtered simplicial complex. Topology (linear algebra) Compute the persistence diagram (Run the Persistence Algorithm). We ll focus on this side.

27 There are two phases, one is geometric the other is topological. Geometry Build a filtration, i.e. a filtered simplicial complex. Topology (linear algebra) Compute the persistence diagram (Run the Persistence Algorithm). We ll focus on this side. Running time: O(N 3 ). N is the size of the complex.

28 Idea 1: Use the Delaunay Triangulation

29 Idea 1: Use the Delaunay Triangulation Good: It works, (alpha-complex filtration).

30 Idea 1: Use the Delaunay Triangulation Good: It works, (alpha-complex filtration). Bad: It can have size n O(d).

31 Idea 2: Connect up everything close.

32 Idea 2: Connect up everything close. Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at most.

33 Idea 2: Connect up everything close. Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at most. Rips Filtration: Add a k-simplex for every k+1 points that have all pairwise distances at most.

34 Idea 2: Connect up everything close. Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at most. Rips Filtration: Add a k-simplex for every k+1 points that have all pairwise distances at most. Still n d, but we can quit early.

35 Our Idea: Build a quality mesh.

36 Our Idea: Build a quality mesh.

37 Our Idea: Build a quality mesh. We can build meshes of size 2 O(d 2 ) n.

38 Meshing Counter-intuition Delaunay Refinement can take less time and space than Delaunay Triangulation.

39 Meshing Counter-intuition Delaunay Refinement can take less time and space than Delaunay Triangulation. Theorem [Hudson, Miller, Phillips, 06]: A quality mesh of a point set can be constructed in O(n log ) time, where is the spread.

40 Meshing Counter-intuition Delaunay Refinement can take less time and space than Delaunay Triangulation. Theorem [Hudson, Miller, Phillips, 06]: A quality mesh of a point set can be constructed in O(n log ) time, where is the spread. Theorem [Miller, Phillips, Sheehy, 08]: A quality mesh of a well-paced point set has size O(n).

41 The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.

42 The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.

43 The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.

44 The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.

45 The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.

46 The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.

47 The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.

48 Approximation via interleaving.

49 Approximation via interleaving. Definition: Two filtrations, {P α } and {Q α } are ε-interleaved if P α ε Q α P α+ε for all α.

50 Approximation via interleaving. Definition: Two filtrations, {P α } and {Q α } are ε-interleaved if P α ε Q α P α+ε for all α. Theorem [Chazal et al, 09]: If {P α } and {Q α } are ε-interleaved then their persistence diagrams are ε-close in the bottleneck distance.

51 The Voronoi filtration interleaves with the offset filtration.

52 The Voronoi filtration interleaves with the offset filtration.

53 The Voronoi filtration interleaves with the offset filtration.

54 The Voronoi filtration interleaves with the offset filtration.

55 The Voronoi filtration interleaves with the offset filtration. Theorem: For all α > r P, V α/ρ M P α V αρ M, where r P is minimum distance between any pair of points in P.

56 The Voronoi filtration interleaves with the offset filtration. Theorem: For all α > r P, V α/ρ M P α V αρ M, where r P is minimum distance between any pair of points in P. Finer refinement yields a tighter interleaving.

57 The Voronoi filtration interleaves with the offset filtration. Theorem: For all α > r P, V α/ρ M P α V αρ M, where r P is minimum distance between any pair of points in P. Finer refinement yields a tighter interleaving. Special case for small scales.

58 Geometric Approximation Topologically Equivalent

59 Geometric Approximation Topologically Equivalent

60 The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Work 1 n O(d) 2. Assign birth times to vertices based on distance to P (special case points very close to P).** 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm. Simple mesh filtration Over-refine the mesh Linear-Size Meshing

61 The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Work 1 n O(d) 2. Assign birth times to vertices based on distance to P (special case points very close to P).** Simple mesh filtration ρ 2 O(d2) n log 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. Over-refine the mesh 4. Feed this filtered complex to the persistence algorithm. Linear-Size Meshing

62 The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Work 1 n O(d) 2. Assign birth times to vertices based on distance to P (special case points very close to P).** Simple mesh filtration ρ =~3 2 O(d2) n log 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. Over-refine the mesh 4. Feed this filtered complex to the persistence algorithm. Linear-Size Meshing

63 The Results Approximation ratio Complex Size 1. Build a mesh M. Over-refine it. Previous Work 1 n O(d) 2. Assign birth times to vertices based on distance to P (special case points very close to P).** Simple mesh filtration ρ =~3 2 O(d2) n log 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. Over-refine the mesh 1 + ε ε O(d2) n log 4. Feed this filtered complex to the persistence algorithm. Linear-Size Meshing

64 The Results Approximation ratio Complex Size 1. Build a mesh M. Over-refine it. Use linear-size meshing. 2. Assign birth times to vertices based on distance to P (special case points very close to P).** Previous Work Simple mesh filtration 1 n O(d) ρ =~3 2 O(d2) n log 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. Over-refine the mesh 1 + ε ε O(d2) n log 4. Feed this filtered complex to the persistence algorithm. Linear-Size Meshing 1 + ε + 3θ (εθ) O(d2) n

65 Thank you.