Persistent Homology Computation Using Combinatorial Map Simplification

Transcription

1 Persistent Homology Computation Using Combinatorial Map Simplification Guillaume Damiand 1, Rocio Gonzalez-Diaz 2 1 CNRS, Université de Lyon, LIRIS, UMR5205, F France 2 Universidad de Sevilla, Dpto. de Matemática Aplicada I, S-41012, Spain January 24-25, 2019

2 Motivations Framework Cellular model Describe any orientable mesh: cells, incidence and adjacency relations in this work, we use 2-maps Goal Compute persistent homology of objects described by 2-maps good complexity How 1 Dispatch faces in clusters 2 Simplify each cluster, while preserving homology 3 Compute persistence: AT-model Control filtration: a new parameter δ Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 2 / 23

3 Outline 1 Preliminary Notions 2 Computing Persistence 3 Experiments 4 Conclusion

4 1. Preliminary Notions 1 Preliminary Notions 2 Computing Persistence 3 Experiments 4 Conclusion

5 2-maps: represent cellular 2D meshes 2-map: M = (D, β 1, β 2 ) 1 D is a finite set of darts; 2 β 1 is a partial permuation on D; 3 β 2 is a partial involution on D. f 5 e 7 v 3 e 6 f 1 v 2 e 1 v 1 f 4 e 4 e e 3 2 f 2 e 5 f 3 A mesh: 5 faces, 14 edges, 12 vertices; Corresponding 2-map e 1 is inner; e 6 is dangling; e 7 is isolated Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 3 / 23

6 What is Homology? Homology Topological invariant Characterizes holes in all dimensions Hk = homology generator in dimension k group of k-cycles quotient group of (k + 1)-boundaries In 2D H0: one vertex per connected component Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 4 / 23

7 What is Homology? Homology Topological invariant Characterizes holes in all dimensions Hk = homology generator in dimension k group of k-cycles quotient group of (k + 1)-boundaries In 2D H1: one loop of edges around each tunnel Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 4 / 23

8 What is Homology? Homology Topological invariant Characterizes holes in all dimensions Hk = homology generator in dimension k group of k-cycles quotient group of (k + 1)-boundaries In 2D H2: one set of faces around each cavity Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 4 / 23

9 Persistent Homology Pertistent Homology Captures the topological changes in a growing sequence of meshes, called filtration Lower-start filtration Well-known filtration h: real-valued function on vertices order vertices h(v 1 ) h(v 2 ) h(v n ) lower-star of vertex v: cells incident to v s.t. vertices of such cells v have h(v ) h(v) M i : lower-star of all vertices v s.t. h(v) h(v i ) Lower-star filtration: sequence of nested meshes: = M 0 M 1 M 2 M n 1 M n = M Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 5 / 23

10 Persistence Diagram Pertistence Homology classes may appear (be born) and disappear (die) Homology class γ is born at M i and dies at M j : h(v j ) h(v i ): persistence of γ. Persistence Diagram for each homology class: plot point (birth time, death time) Betti0 Betti1 Death Birth Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 6 / 23

11 AT-models AT-model Allows to compute persistent homology [GR2005,GIJP2011]: Input: an ordering of cells: {σ 1,..., σ m } σ a k-cell: (σ) is the set of (k 1)-cells in its boundary f : C(M) C(H) maps each k-cell to a sum of surviving cells: satisfying if a, b are two homologous k-cycles then f k (a) = f k (b) Step i of the algorithm if f (σ i ) = 0: a new homology class is born if f (σ i ) 0: a homology class dies remove from H the youngest element of f (σ i ) i.e. σ j s.t. j = max{ index(µ) : µ f (σ i ) } Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 7 / 23

12 2-map Simplification Algorithm to simplify a 2-map preserves its homology [DPF2006,DG2016] (1) foreach inner edge e do Remove e; (2) foreach edge e do while e is dangling do e one edge adjacent to e; Remove e; e e ; (3) foreach edge e do if e is not a loop then Contract e; Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 8 / 23

13 2-map Simplification (1) foreach inner edge e do Remove e; (2) foreach edge e do while e is dangling do e one edge adjacent to e; Remove e; e e ; (3) foreach edge e do if e is not a loop then Contract e; Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 9 / 23

14 2-map Simplification (1) foreach inner edge e do Remove e; (2) foreach edge e do while e is dangling do e one edge adjacent to e; Remove e; e e ; (3) foreach edge e do if e is not a loop then Contract e; Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 10 / 23

15 2-map Simplification (1) foreach inner edge e do Remove e; (2) foreach edge e do while e is dangling do e one edge adjacent to e; Remove e; e e ; (3) foreach edge e do if e is not a loop then Contract e; Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 11 / 23

16 2. Computing Persistence 1 Preliminary Notions 2 Computing Persistence 2-map Simplification Compute Filtration Compute Persistent Homology 3 Experiments 4 Conclusion

17 Computing Persistence Our method: based on 3 steps 1 Simplify the 2-map according to a parameter δ; 2 Compute the lower-star filtration of the simplified mesh; 3 Compute persistent homology of the given filtration. Goal of step (1) Decrease the number of faces in each filtration level Improve computation Filter small persistent homology events Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 12 / 23

18 2-map Simplification (1.1) 2-map Simplification Dispatch faces in clusters: according to a parameter δ order vertices of the mesh by h(v), non-decreasing way face C: h(c) maximum value of its vertices each cluster: set of faces: smallest unasigned face C plus all unasigned faces at distance" δ i.e. path of adjacent unasigned faces from C to C, length δ Example of clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 13 / 23

19 2-map Simplification (1.1) 2-map Simplification Dispatch faces in clusters: according to a parameter δ order vertices of the mesh by h(v), non-decreasing way face C: h(c) maximum value of its vertices each cluster: set of faces: smallest unasigned face C plus all unasigned faces at distance" δ i.e. path of adjacent unasigned faces from C to C, length δ Example of clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 13 / 23

20 2-map Simplification (1.1) 2-map Simplification Dispatch faces in clusters: according to a parameter δ order vertices of the mesh by h(v), non-decreasing way face C: h(c) maximum value of its vertices each cluster: set of faces: smallest unasigned face C plus all unasigned faces at distance" δ i.e. path of adjacent unasigned faces from C to C, length δ Example of clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 13 / 23

21 2-map Simplification (1.1) 2-map Simplification Dispatch faces in clusters: according to a parameter δ order vertices of the mesh by h(v), non-decreasing way face C: h(c) maximum value of its vertices each cluster: set of faces: smallest unasigned face C plus all unasigned faces at distance" δ i.e. path of adjacent unasigned faces from C to C, length δ Example of clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 13 / 23

22 2-map Simplification (1.1) 2-map Simplification Dispatch faces in clusters: according to a parameter δ order vertices of the mesh by h(v), non-decreasing way face C: h(c) maximum value of its vertices each cluster: set of faces: smallest unasigned face C plus all unasigned faces at distance" δ i.e. path of adjacent unasigned faces from C to C, length δ Example of clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 13 / 23

23 2-map Simplification (1.1) 2-map Simplification Dispatch faces in clusters: according to a parameter δ order vertices of the mesh by h(v), non-decreasing way face C: h(c) maximum value of its vertices each cluster: set of faces: smallest unasigned face C plus all unasigned faces at distance" δ i.e. path of adjacent unasigned faces from C to C, length δ Example of clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 13 / 23

24 2-map Simplification (1.1) 2-map Simplification Dispatch faces in clusters: according to a parameter δ order vertices of the mesh by h(v), non-decreasing way face C: h(c) maximum value of its vertices each cluster: set of faces: smallest unasigned face C plus all unasigned faces at distance" δ i.e. path of adjacent unasigned faces from C to C, length δ Example of clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 13 / 23

25 2-map Simplification (1.2) 2-map Simplification Simplify cluster while preserving homology Use previous simplification algorithm, BUT: remove only inner edges between two faces of the same cluster do not use the contraction step Example of simplification inside clusters: δ = Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 14 / 23

26 Compute Filtration (2) Compute Filtration Compute lower star filtration On the simplified mesh Attention Increase δ decrease number of cells δ > 0 persistent homology computed pertistent homology of the original mesh δ allows to remove small persistent homology events Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 15 / 23

27 Compute Persistent Homology (3) Compute Persistent Homology In this work: AT-model On the simplified mesh order cells according to the filtration store points of persistence diagram Compute bootleneck distances Measures similarity between persistence diagrams A and B, different δ shortest distance d s.t. perfect matching between points of A and B s.t. distance between any couple of matched points d. Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 16 / 23

28 3. Experiments 1 Preliminary Notions 2 Computing Persistence 3 Experiments 4 Conclusion

29 Experiments Tests Software based on CGAL Combinatorial Maps Intel R i CPU, GHz, 32 Go RAM 6 meshes #0-cells #1-cells #2-cells #H0 #H1 #H2 Blade 882,954 2,648,082 1,765, DrumDancer 1,335,436 4,006,302 2,670, Neptune 2,003,932 6,011,808 4,007, HappyBuddha 543,652 1,631,574 1,087, Iphigenia 351,750 1,055, , ThaiStatue 4,999,996 15,000,000 10,000, Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 17 / 23

30 Impact of δ Zoom in on the trident for the Neptune mesh: δ = 0, 1, 4 and 32 Number of cells of the simplified combinatorial map 4,194,304 0-cells 1-cells 2-cells 1,048,576 Number of cells 262,144 65,536 16, Average values for the six meshes Delta Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 18 / 23

31 Evolution of Finite Persistence Events Number of finite persistence events 16,384 Betti 0 Bettti Number of events Delta Average values for the six meshes log 2 scale. Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 19 / 23

32 Evolution of Computation Times Computation Times of the Whole Process 8 7 Filtration Simplification AT-model 6 Time (sec) Delta δ Blade DrumDancer Neptune HappyBuddha Iphigenia Statuette Average Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 20 / 23

33 Bottleneck Distance: 0-Dimensional Persistence Events Distances between events for δ = 0 and different δ δ Blade DrumDancer Neptune HappyBuddha Iphigenia Statuette Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 21 / 23

34 Bottleneck Distance: 1-Dimensional Persistence Events Distances between events for δ = 0 and different δ δ Blade DrumDancer Neptune HappyBuddha Iphigenia Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 22 / 23

35 4. Conclusion 1 Preliminary Notions 2 Computing Persistence 3 Experiments 4 Conclusion

36 Conclusion / Future works Conclusion Algorithm for computing persistent homology on 2D mesh Filtrations defined by parameter δ Flexibility: speed-up/precision Filter small events Future works Define other filtrations e.g. use face heights Theoretical results: filtration is stable w.r.t. δ Non orientable n-gmaps package in CGAL Parallel version Simplification already parallel need some work for AT-model or use existing parallel method (PHAT, DIPHA,... ) Extension in higher dimensions Persistent Homology Computation Using Combinatorial Map Simplification G. Damiand, R. Gonzalez-Diaz 23 / 23

37 Persistent Homology Computation Using Combinatorial Map Simplification Guillaume Damiand 1, Rocio Gonzalez-Diaz 2 1 CNRS, Université de Lyon, LIRIS, UMR5205, F France 2 Universidad de Sevilla, Dpto. de Matemática Aplicada I, S-41012, Spain January 24-25, 2019