TOPOLOGY OF POINT CLOUD DATA

Transcription

1 TOPOLOGY OF POINT CLOUD DATA CS 468 Lecture 8 3/3/4 Afra Zomorodian CS 468 Lecture 8 - Page 1

2 OVERVIEW Points Complexes Cěch Rips Alpha Filtrations Persistence Afra Zomorodian CS 468 Lecture 8 - Page 2

3 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 3

4 POINT CLOUD DATA (a) Surface (b) Molecule (c) Universe Afra Zomorodian CS 468 Lecture 8 - Page 4

5 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 5

6 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 6

7 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 7

8 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 8

9 ALPHA COMPLEX Extendible to points with weights van der Waals model of molecules Afra Zomorodian CS 468 Lecture 8 - Page 9

10 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 10

11 BUNNY 34,834 points, 1,026,111 complexes Afra Zomorodian CS 468 Lecture 8 - Page 11

12 GRAMICIDIN A Afra Zomorodian CS 468 Lecture 8 - Page 12

13 312 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 GRADED F[t]-MODULE 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 Degrees of homogeneous elements: a b c d ab bc cd ad ac abc acd M 1 = 1 ab bc cd ad ac d 0 0 t t 0 c 0 1 t 0 t 2 b t t a t 0 0 t 2 t 3 deg ê i + deg M k (i, j) = deg e j Afra Zomorodian CS 468 Lecture 8 - Page 25

26 COLUMN ECHELON FORM M 1 = cd bc ab z 1 z 2 d t c t b 0 t t 0 0 a 0 0 t 0 0 Only column operations of type (1, 3) z 1 = ad cd t bc t ab z 2 = ac t 2 bc t 2 ab {z 1, z 2 } form homogeneous basis for Z 1 rank M k = rank B k 1 is number of pivots Afra Zomorodian CS 468 Lecture 8 - Page 26

27 ECHELON FORM LEMMA M 1 = cd bc ab z 1 z 2 d t c t b 0 t t 0 0 a 0 0 t 0 0 (Lemma) The pivots in column-echelon form are the same as the diagonal elements in normal form. Moreover, the degree of the basis elements on pivot rows is the same in both forms. If only interested in degree of basis elements, read them off the column echelon form. Afra Zomorodian CS 468 Lecture 8 - Page 27

28 BASIS CHANGE LEMMA j i j M k i = 0 M k+1 m k 1 x mk m k xmk+1 Represent k+1 in terms of the basis computed for Z k k k+1 =, M k M k+1 = 0 (Lemma) To represent k+1 relative to the standard basis for C k+1 and the basis computed for Z k, simply delete rows in M k+1 that correspond to pivot columns in M k. Afra Zomorodian CS 468 Lecture 8 - Page 28

29 PROOF j i j M k i = 0 M k+1 m k 1 x mk m k xmk+1 Replace column i by (column i) + q(column j) to eliminate element in pivot row j replacing column basis element e i by e i + qe j in M k replacing row j with (row j) q(row i) in M k+1 But row j is eventually zero and row i is not changed. QED Afra Zomorodian CS 468 Lecture 8 - Page 29

30 ALGORITHM No need for row operations Free columns correspond to positive simplices Pivot columns correspond to negative simplices No need for matrix representation Sparse matrix computation of Betti numbers based on persistence Afra Zomorodian CS 468 Lecture 8 - Page 30

31 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 31