Reconstruction of Filament Structure

Transcription

1 Reconstruction of Filament Structure Ruqi HUANG INRIA-Geometrica Joint work with Frédéric CHAZAL and Jian SUN 27/10/2014

2 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches Reeb Graph Geometrical Guarantee α Reeb Graph Topological Guarantee

3 We are interested in the dataset with underlying linear structure. Example Earthquake faults:

4 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches Reeb Graph Geometrical Guarantee α Reeb Graph Topological Guarantee

5 ɛ correspondence Let (X, d X ) and (Y, d Y ) be two metric spaces, an ɛ correspondence between them is a subset C X Y such that: for any x X, there exists y Y such that (x, y) C; for any y Y, there exists x X such that (x, y) C; for any (x, y), (x, y ) C, d X (x, x ) d Y (y, y ) ɛ.

6 ɛ correspondence Let (X, d X ) and (Y, d Y ) be two metric spaces, an ɛ correspondence between them is a subset C X Y such that: for any x X, there exists y Y such that (x, y) C; for any y Y, there exists x X such that (x, y) C; for any (x, y), (x, y ) C, d X (x, x ) d Y (y, y ) ɛ. Gromov-Hausdorff Distance d GH (X, Y ) = 1 2 inf {ɛ : there exists an ɛ correspondence between X and Y }

7 Problem Statement Input A compact path metric space (X, d X ), which is close to some unknown metric graph (G, d G ), meaning d GH (X, G ) < ɛ.

8 Problem Statement Input A compact path metric space (X, d X ), which is close to some unknown metric graph (G, d G ), meaning d GH (X, G ) < ɛ. Output A metric graph (G, d G ) as an approximation of (G, d G ).

9 Problem Statement Input A compact path metric space (X, d X ), which is close to some unknown metric graph (G, d G ), meaning d GH (X, G ) < ɛ. Output A metric graph (G, d G ) as an approximation of (G, d G ).

10 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches Reeb Graph Geometrical Guarantee α Reeb Graph Topological Guarantee

11 Definition Let (X, d X ) be a compact path metric. We set r X as a root point, let d(x) = d X (r, x) the distance function to r in X.

12 Definition Let (X, d X ) be a compact path metric. We set r X as a root point, let d(x) = d X (r, x) the distance function to r in X. Equivalence x y IFF d(x) = d(y) and x, y are in the same connected component of d 1 (d(x)).

13 Definition Let (X, d X ) be a compact path metric. We set r X as a root point, let d(x) = d X (r, x) the distance function to r in X. Equivalence x y IFF d(x) = d(y) and x, y are in the same connected component of d 1 (d(x)). Reeb Graph G = X \

14 v 10 v 9 Illustration v 8 v 7 v 3 v 2 r v 1 v 4 v 11 v 6 v5

15 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches Reeb Graph Geometrical Guarantee α Reeb Graph Topological Guarantee

16 Properties Let π : X G be π(x) = [x]. Then we consider correspondence (x, π(x)) X G.

17 Properties Let π : X G be π(x) = [x]. Then we consider correspondence (x, π(x)) X G. 1-Lipschitz: d X (x, y) d G (π(x), π(y));

18 Properties Let π : X G be π(x) = [x]. Then we consider correspondence (x, π(x)) X G. 1-Lipschitz: d X (x, y) d G (π(x), π(y)); d X (x, y) d G (π(x), π(y)) + 2(β 1 (G) + 1)M;

19 Properties Let π : X G be π(x) = [x]. Then we consider correspondence (x, π(x)) X G. 1-Lipschitz: d X (x, y) d G (π(x), π(y)); d X (x, y) d G (π(x), π(y)) + 2(β 1 (G) + 1)M; bounding the diameter of level-set, i.e. M = sup x X diam(d 1 (d(x))).

20 Bounding M Since X is close to some underlying graph G, there is an upper bound for the level-set.

21 Bounding M Since X is close to some underlying graph G, there is an upper bound for the level-set. Theorem For any l > α, the diameter of any connected component L of d 1 [l α, l + α] satisfies: diam(l) 4(2 + N E,G (4(α + 2ɛ)))(α + 2ɛ) + ɛ N E,G (δ) is the number of edges of G with length small than δ.

22 Bounding M Since X is close to some underlying graph G, there is an upper bound for the level-set. Theorem For any l > α, the diameter of any connected component L of d 1 [l α, l + α] satisfies: diam(l) 4(2 + N E,G (4(α + 2ɛ)))(α + 2ɛ) + ɛ N E,G (δ) is the number of edges of G with length small than δ. Let α = 0, M (8N E,G (8ɛ) + 17)ɛ

23 Geometrical guarantee To conclude, we proved that: d GH (X, G) < (β 1 (G) + 1)(17 + 8N E,G (8ɛ))ɛ

24 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches Reeb Graph Geometrical Guarantee α Reeb Graph Topological Guarantee

25 α Reeb Graph We modify the equivalence relation to construct the α Reeb graph. Cover the range of d(x) with the intervals {I i } of length at most 2α, for example, I i = (iα, (i + 2)α), i = 0, 1, 2...;

26 α Reeb Graph We modify the equivalence relation to construct the α Reeb graph. Cover the range of d(x) with the intervals {I i } of length at most 2α, for example, I i = (iα, (i + 2)α), i = 0, 1, 2...; Define equivalence relationship x α y IFF d(x) = d(y) and x, y are in the same connected component of d 1 (I i );

27 α Reeb Graph We modify the equivalence relation to construct the α Reeb graph. Cover the range of d(x) with the intervals {I i } of length at most 2α, for example, I i = (iα, (i + 2)α), i = 0, 1, 2...; Define equivalence relationship x α y IFF d(x) = d(y) and x, y are in the same connected component of d 1 (I i ); G α = X / α.

28 Illustration I4 H 1 4 H 1 3 I3 I2 H 1 2 H 2 2 I0 I1 H 1 1 H 2 1 H d I H 1 0 V4 1 I4 V3 1 I3 V2 1 V2 2 I2 I2 V1 1 V1 2 I1 I1 V0 1 G I0 disjoint union of copies of intervals α-reeb graph

29 Geometrical Guarantee Similarly, we have: d GH (X, G α ) < (β 1 (G α ) + 1)(4(2 + N E,G (4(α + 2ɛ)))(α + 2ɛ) + ɛ)

30 Outline 1 Problem Statement Characterization of Dataset Formulation 2 Our Approaches Reeb Graph Geometrical Guarantee α Reeb Graph Topological Guarantee

31 Homotopy Equivalence Result Theorem Let r X, α > 60ɛ and I{[0, 2α), (iα, (i + 2)α) 1 i m} covers the segment [0, Diam(X )]. If no edges of G is shorter than L and no loops of G is shorter than 2L with L (17α + 9ɛ), then we have G α and G are homotopy equivalent.

32 Nerve Complex Let U = i A U i be a finite open covering of topological space T. Then the nerve complex of U, K, is a simplicial complex whose vertex set is A, and where a family {i 0, i 1,..., i k } spans a k simplex in K IFF U i0 U i1... U ik.

33 Illustration I4 H 1 4 H 1 3 I3 I2 H 1 2 H 2 2 I0 I1 H 1 1 H 2 1 H d I H 1 0 V4 1 I4 V3 1 I3 V2 1 V2 2 I2 I2 V1 1 V1 2 I1 I1 V0 1 G I0 disjoint union of copies of intervals α-reeb graph

34 Nerve Complex Let U = i A U i be a finite open covering of topological space T. Then the nerve complex of U, K, is a simplicial complex whose vertex set is A, and where a family {i 0, i 1,..., i k } spans a k simplex in K IFF U i0 U i1... U ik. Nerve Lemma Further more, if all finite intersections of the open sets in the open covering are either empty or contractible, then T is homotopy equivalent to K.

35 Illustration I4 H 1 4 H 1 3 I3 I2 H 1 2 H 2 2 I0 I1 H 1 1 H 2 1 H d I H 1 0 V4 1 I4 V3 1 I3 V2 1 V2 2 I2 I2 V1 1 V1 2 I1 I1 V0 1 G I0 disjoint union of copies of intervals α-reeb graph

36 Correspondence between X and G Let (r, g r ) C(X, G ), b(g) = d G (g r, g). Define C(U) = {g : (x, g) C, x U X }(C(V ) is defined in the same way for V G ).

37 Correspondence between X and G Let (r, g r ) C(X, G ), b(g) = d G (g r, g). Define C(U) = {g : (x, g) C, x U X }(C(V ) is defined in the same way for V G ). Path Correspondence For x 1, x 2 X, if there is a path in d 1 (l, u) connecting them, then for g i, (x i, g i ) C, we can always find a path in b 1 (l 2ɛ, u + 2ɛ) connecting g 1, g 2.

38 New Open Covering of X Ṽ 2 Ṽ2 Ṽ 1 Ṽ1 Ṽ 3 Ṽ3 N(V 0 )

39 THE END Thanks for your attention. Questions?