TOPOLOGY OF POINT CLOUD DATA

Transcription

1 TOPOLOGY OF POINT CLOUD DATA CS 468 Lecture 8 11/12/2 Afra Zomorodian CS 468 Lecture 8 - Page 1

2 PROJECTS Writeups: Introduction to Knot Theory (Giovanni De Santi) Mesh Processing with Morse Theory (Niloy Mitra) Presentations: November 27th: * Surface Flattening (Jie Gao) * Simplicial Sets (Patrick Perry) * Complexity of Knot Problems (Krishnaram Kenthapadi) December 4th * Discrete Morse Theory(?) (Yichi Gu) * Irreducible Triangulations (Jon McAlister) * Homotopy in the Plane (Rachel Kolodny) Afra Zomorodian CS 468 Lecture 8 - Page 2

3 OVERVIEW Points Complexes Cěch Rips Alpha Filtrations Persistence Afra Zomorodian CS 468 Lecture 8 - Page 3

4 POINTS m samples M = {m 1, m 2,..., m m } from a manifold M Samples are embedded, but intrinsic topology is lost Error: acquisition device noise and approximation Afra Zomorodian CS 468 Lecture 8 - Page 4

5 POINT CLOUD DATA (a) Surface (b) Molecule (c) Universe Afra Zomorodian CS 468 Lecture 8 - Page 5

6 ɛ-balls ɛ-ball: B ɛ (x) = {y d(x, y) < ɛ}. Open sets and topology Manifold is M = m i M B ɛ(m i ) Afra Zomorodian CS 468 Lecture 8 - Page 6

7 CĚCH COMPLEX C ɛ (M) = { conv T T M, m i T B ɛ(m i ) }. m k=0 ( m k ) = 2 m+1 1 C ɛ (M) M Afra Zomorodian CS 468 Lecture 8 - Page 7

8 RIPS COMPLEX R ɛ (M) = {conv T T M, d(m i, m j ) < ɛ, m i, m j T }. Still O (( m k )) for the kth skeleton Need (k + 1)st skeleton for computing H k Afra Zomorodian CS 468 Lecture 8 - Page 8

9 ALPHA COMPLEX V (m i ) = {x R 3 d(x, m i ) d(x, m j ) m j M} ˆV (m i ) = B ɛ (m i ) V (m i ) { A ɛ = conv T T M, ˆV } m i T (m i ) A ɛ (M) M, A ɛ D, the Delaunay complex O(n log n + n d/2 ) Afra Zomorodian CS 468 Lecture 8 - Page 9

10 ALPHA COMPLEX Extendible to points with weights van der Waals model of molecules Afra Zomorodian CS 468 Lecture 8 - Page 10

11 FILTRATIONS a b a b a b a b a b a b d c d c d c d c d c 0 a, b 1 c, d, ab, bc 2 cd, ad 3 ac 4 abc 5 acd Complexes C ɛ, R ɛ, A ɛ, compute homology! Which ɛ? Vary and get a filtration! A filtration of a complex K is = K 0 K 1... K m = K. Afra Zomorodian CS 468 Lecture 8 - Page 11

12 BUNNY 34,834 points, 1,026,111 complexes Afra Zomorodian CS 468 Lecture 8 - Page 12

13 GRAMICIDIN A 312 atoms, 8,591 complexes Afra Zomorodian CS 468 Lecture 8 - Page 13

14 APPROACH Input: point cloud Procedure: Put ɛ-balls around points Compute complex K ɛ Compute homology of complex Varying ɛ gives us a filtration Incremental algorithm gives homology of filtration (demo) Afra Zomorodian CS 468 Lecture 8 - Page 14

15 HOMOLOGY OF A FILTRATION K l is a filtration Z l k = Z k (K l ) and B l k = B k (K l ) are the kth cycle and boundary group of K l, respectively. The kth homology group of K l is H l k = Z l k/b l k. The kth Betti number β l k of Kl is the rank of H l k. Afra Zomorodian CS 468 Lecture 8 - Page 15

16 PROBLEM Space β { }. Filtration l Signature (Tunnels) Features Noise: spawned by noise, representation, etc. Afra Zomorodian CS 468 Lecture 8 - Page 16

17 PERSISTENCE K l be a filtration. The p-persistent kth homology group of K l is H l,p k = Z l k/(b l+p k Z l k), The p-persistent kth Betti number β l,p k Well-defined of Kl is the rank of H l,p k. η l,p k im η l,p k : Hl k H l+p k, Hl,p k. This lecture: Z 2 homology Afra Zomorodian CS 468 Lecture 8 - Page 17

18 LIFETIMES Let z be a non-bounding k-cycle, created when σ enters complex at time i That is, β k ++ at time i z creates a class of homologous cycles [z] [z] is merged with the boundary class at time j when τ enters (β k ) τ destroys z and the cycle class [z]. The persistence of z, and its homology class [z], is j i 1. σ is the creator (positive) and τ is the destroyer (negative) of [z]. If a cycle class does not have a destroyer, its persistence is. Afra Zomorodian CS 468 Lecture 8 - Page 18

19 LIFETIME REGIONS H l,p k = Zl k / (B l+p k Z l k) Basis element z + B k lives during [i, j) z B l k for l j Therefore, z B l+p k for l + p < j. p 0 l i Afra Zomorodian CS 468 Lecture 8 - Page 19

20 TRIANGLE l > i l+p < j p > 0 (i, 0) (j, 0) index (l) (i, j i) (i, 0) (j, 0) (i, j i) persistence (p) p 0 l i l < j Afra Zomorodian CS 468 Lecture 8 - Page 20

21 TRIANGLES [ [ [ [ ) [ [ ) [ ) [ ) [ ) ) ) ) [ index persistence Afra Zomorodian CS 468 Lecture 8 - Page 21

22 GRAPH OF log(β l,p 1 + 1) log 2 (β 1 l,p +1) p l Afra Zomorodian CS 468 Lecture 8 - Page 22

23 CMY COLOR SPACE Green Yellow (minus blue) (minus red) Cyan Black Red Blue Magenta (minus green) Afra Zomorodian CS 468 Lecture 8 - Page 23

24 TOPOLOGY MAPS Afra Zomorodian CS 468 Lecture 8 - Page 24

25 ALGORITHM a b a b a b a b a b a b d c d c d c d c d c 0 a, b 1 c, d, ab, bc 2 cd, ad 3 ac 4 abc 5 acd a b c d ab bc cd ad ac abc acd b a c b d c ad ac Compute k σ i Eliminate negative simplices in chain Look for youngest cycle creator and store Afra Zomorodian CS 468 Lecture 8 - Page 25

26 DISCUSSION We can compute: Cycles (components, cycles, voids) Bounding manifolds (Demo) Points can be anything samples from high dimensional manifolds: configuration spaces for robots (PRM), time-variant data, etc. samples of tangent complex for data-set M S 2 Need fast d-dim complex builder Afra Zomorodian CS 468 Lecture 8 - Page 26