Approximating Riemannian Voronoi Diagrams and Delaunay triangulations.

Transcription

1 Approximating Riemannian Voronoi Diagrams and Delaunay triangulations. Jean-Daniel Boissonnat, Mael Rouxel-Labbé and Mathijs Wintraecken Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

2 Outline Introduction and previous work Power Protection and stability Discrete approximations of the Voronoi diagram Combinatorial correctness of the discrete Voronoi Diagram: Protection and Sperner s lemma Distortion and straight simplices; Delaunay triangulations Experimental results Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

3 Introduction and previous work Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

4 How to prescribe anisotropy Metric field g Continuous map g : p Ω g p, positive symmetric definite matrix Normally: length(γ) = γ, γ 1/2 g dt = γ t (t)g γ(t) γ(t)dt d M (p, q) = inf γ length(γ) g p also defines an anisotropic distance if domain R d : d gp (a, b) = d p (a, b) = (a b) t g p (a b) Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

5 Local approximations Locally d gp (a, b) approximates d M (p, q). Moreover g p defines and inner product, which after a linear transformation is the Euclidean. Metric distortion ψ: x, y in Ω, we have 1 ψ d G (x, y) d G(x, y) ψd G (x, y). For G = g p and G = g the Metric distortion decreases if Ω smaller, g p g x small, for x, p Ω. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

6 Blanket assumption: point sets (ɛ, µ)-nets! Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

7 Riemannian Voronoi diagrams and approximations Riemannian Voronoi diagrams The Riemannian Voronoi cell of the site p is given by V M (p i ) = {x Ω d M (p i, x) d M (p j, x), p j P \ p i }. Anisotropic Voronoi diagrams The anisotropic Voronoi cell of the site p is given by: Labelle and Shewchuk Du and Wong V LS (p) = {x R d : d p (p, x) d q (q, x), q P, q p} V DW (p) = {x R d : d x (p, x) d x (q, x), q P, q p} Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

8 Anisotropic Voronoi diagrams: Labelle and Shewchuk and Du and Wong Each site is within its cell Possibility of orphans (non-connected cells) The dual may not be a triangulation Termination of up sampling and quality bounds can be proven in 2D [Labelle, Shewchuk 03] Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

9 Riemannian Voronoi Diagrams Each site is within its cell No orphans (non-connected cells) Theoretical guarantees for dual to be a triangulation (Boissonnat, Dyer, Gosh) Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

10 Protection [Boissonnat et al. 13] If B(c, ρ) circum ball, then no alien vertices in B(c, ρ + δ). Gives: lower bounds on height B p c τ ρ B c B+δ r stability of the circumcentre q s Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

11 Power Protection and stability Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

12 Power protection and height H B q q θ θ c c B B +δ H B and B circumspheres of simplices, that share a face. Common face lies in H. Distance H, H lower bounds height. Lower bound height, gives bounds on angles, because (ɛ, µ)-net. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

13 H E H E H E aff(τ) q 1 ψ r r ψr c q q c 1 ψ r r ψr 1 ψ r 2 + δ 2 r 2 + δ 2 ψ r 2 + δ 2 Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

14 Stability circumcentre The circumcentre is stable with respect to perturbations and even with metric distortion Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

15 Stability in higher dimensions With metric distortion intersection of the bisectors of 3 vertices of a face is constraint to a cylinder (in fact a parallel piped) orthogonal to the face, the circumcentre lies in the intersection of such cylinders. Induction on dimension is now possible. Boissonnat, Rouxel-Labbe, Wintraecken Approximating Voronoi Diagrams October / 1

16 Discrete approximations of the Voronoi diagram Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

17 Discrete Riemannian Voronoi diagrams Geodesic distances cannot be computed exactly: must discretize Canvas The underlying structure used to compute geodesic distances (typically, an isotropic triangulation) Many methods exist to compute geodesic distances and paths: Fast marching methods [Konukoglu et al. 07] Heat-kernel based methods [Crane et al. 13] Short-term vector Dijkstra [Campen et al. 13] Computing geodesics from multiple sites is a natural extension from the propagation of geodesic distances from one site.

18 Construction of the discrete diagram Canvas generated fine enough so that the discrete diagram has the correct dual Color each vertex of the canvas with the closest site New sites of the discrete diagram are inserted through a farthest point refinement algorithm driven by a sizing field Connectivity is extracted Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

19 Construction of the discrete diagram Canvas generated fine enough so that the discrete diagram has the correct dual Color each vertex of the canvas with the closest site New sites of the discrete diagram are inserted through a farthest point refinement algorithm driven by a sizing field Connectivity is extracted Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

20 Construction of the discrete diagram Canvas generated fine enough so that the discrete diagram has the correct dual Color each vertex of the canvas with the closest site New sites of the discrete diagram are inserted through a farthest point refinement algorithm driven by a sizing field Connectivity is extracted Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

21 Construction of the discrete diagram Canvas generated fine enough so that the discrete diagram has the correct dual Color each vertex of the canvas with the closest site New sites of the discrete diagram are inserted through a farthest point refinement algorithm driven by a sizing field Connectivity is extracted Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

22 Combinatorial correctness of the discrete Voronoi diagram: Protection and Sperner s lemma Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

23 Combinatorial correctness: 2D v 5 v 4 s p v 3 θ l v 2 v 0 v 1 Thanks to protected (ɛ, µ)-net well shaped Voronoi cells; distance between foreign objects is large. It is impossible for triangle to be coloured in a way that does not correspond to a Voronoi vertex. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

24 Combinatorial correctness: 2D with metric distortion η η p 0 DV +η0 E (p0) V E(p 0) EV η0 E (p0) With metric distortion one needs a margin. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

25 Combinatorial correctness: 2D with metric distortion DV +η0 g0 (p 0 ) η 0 V g0 (p 0 ) EV η0 g0 (p 0 ) DV +η0 g0 (p 0 ) V g0 (p 0 ) η 0 EV η0 g0 (p 0 ) c c θ χ η 0 χ θ c c η 0 The Riemannian Voronoi vertices are close to the Euclidean vertices if the distortion is small. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

26 Combinatorial correctness: 2D with metric distortion The canvas has to be dense Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

27 Combinatorial correctness: 2D q 4 q 1 q 2 q 6 v 0 q 0 q 3 q 5 Thanks to induction one finds a three coloured simplex for every Voronoi vertex. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

28 Higher dimensions Combinatorial correctness: You have everything: Sperner s lemma You don t get more than this (power) protection (same as in 2D) constant metric approximation Riemannian p l s l v Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

29 Sperner s Lemma Theorem (Sperner s lemma) Let σ = (e 0,..., e n ) be an n-simplex and T σ a triangulation of the simplex. Let e T σ be colored such that: The vertices e i of σ all have different colors. If e lies on a k-face (e i0,... e ik ) of σ, then e has the same color as one of the vertices of the face, that is e ij. Then, there exists at least one simplex in T σ whose vertices are colored with all n + 1 colors. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

30 Construction of Sperner simplex Find safe points (thanks to protection) on Voronoi objects. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

31 Construction of Sperner simplex v The simplex with the canvas. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

32 Construction of Sperner simplex p µ 2 v r Π v v q δ2 8ɛ Technical details: shift away from the voronoi vertices. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

33 Construction of Sperner simplex p µ 2 v r v q δ2 8ɛ Π v Technical details: shift away from the voronoi vertices. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

34 Distortion and straight simplices; Delaunay triangulations Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

35 Embeddability of the straight dual: 2D Low distortion: geodesics are close to straight edges Straight Riemannian R l P a x r x s Q Embeddability (ɛ, µ)-net, geodesics can be straightened without creating inversions. Thus: embeddability of the Riemannian dual = embeddability of the straight dual Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

36 Embeddability of the straight dual: 2D Low distortion: geodesics are close to straight edges Straight Riemannian R l P a x r x s Q Embeddability (ɛ, µ)-net, geodesics can be straightened without creating inversions. Thus: embeddability of the Riemannian dual = embeddability of the straight dual Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

37 Centres of mass in Euclidean space Weighted average of points µi p i, with µ i = 1. Generalizes to p dµ(p), which is where the minimum of P R n(x) = 1 2 x p 2 dµ(p) is attained. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

38 Theorem (Karcher) Let M be a manifold whose sectional curvature K is bounded, that is Λ l K Λ u. Let P M the function on B ρ defined by P M (x) = 1 d M (x, p) 2 dµ(p), 2 where dµ is a positive measure and the support of dµ is contained in B ρ. We now give two conditions on ρ: ρ is less than half the injectivity radius, if Λ u > 0 then ρ < π 2 Λ u. If these conditions are met then P M has a unique critical point in B ρ, which is a minimum. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

39 Riemannian simplices using Riemannian centres of mass Definition (Riemannian simplex) E λ (x) = 1 λ i d M (x, p i ) 2 2 barycentric coordinates: λ i 0, λ i = 1 B σ j : j M i λ argmin E λ (x) x B ρ j the standard Euclidean j-simplex, σ M image Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

40 Smooth map Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

41 Distortion We assume that d g (x, y) 2 = x y 2 + δd 2 g(x, y), λ i x p i 2 = x i i argmin i λ id g (x, p i ) 2 close to argmin i λ i x p i 2 Works also for continuous distributions and negative weights! λ i p i 2+const. Riemannian simplex close to Euclidean simplex: straight simplex triangulation. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

42 Experimental results Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

43 Discrete Riemannian Voronoi diagram: Result Voronoi diagram of 750 sites

44 Straight Delaunay triangulation (Straight) dual of the previous diagram

45 Riemannian Delaunay triangulation Anisotropic and curved elements

46 Approximating Riemannian simplices Piecewise flat approximation of Riemannian simplices

47 Acknowledgments Thanks to Ramsay Dyer for suggesting Sperner s lemma, instead of dimension theory. Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1

48 Questions? Boissonnat, Rouxel-Labbé, Wintraecken Approximating Voronoi Diagrams October / 1