Download presentation

Published bySabrina Bunton Modified over 6 years ago

1
**Proximity graphs: reconstruction of curves and surfaces**

Framework Duality between the Voronoi diagram and the Delaunay triangulation. Power diagram. Alpha shape and weighted alpha shape. The Gabriel Graph. The beta-skeleton Graph. A-shape and Crust. Local Crust and Voronoi Gabriel Graph. NN-crust. M. Melkemi

2
**Duality: Voronoi diagram and Delaunay triangulation (1)**

A Voronoi region of a point is defined by: The Voronoi diagram of the set S, DV(S), is the set of the regions A 3-cell is a Voronoi polyhedron, a 2-cell is a face,a 1-cell is an edge of DV(S).

3
**Duality: Voronoi diagram and Delaunay triangulation (2)**

is a k-simplex of the Delaunay triangulation D(S) iff there exists an open ball b such that:

4
**Duality: Voronoi diagram and Delaunay triangulation (3)**

Examples A Delaunay triangle corresponds to a Voronoi vertex. An edge of D(S) corresponds to a Voronoi edge. A Delaunay vertex corresponds to a Voronoi region.

5
**Duality: Voronoi diagram and**

Delaunay triangulation (4)

6
**Duality: Voronoi diagram and**

Delaunay triangulation (5)

7
**Power diagram and regular triangulation (1)**

A weighted point is denoted as p=(p’,p’’), with its location and its weight. For a weighted points, p=(p’,p’’), the power distance of a point x to p is defined as follows: p(p,x) x p’

8
**regular triangulation (2)**

Power diagram and regular triangulation (2) The locus of the points equidistant from two weighted points is a straight line.

9
**Power diagram and regular triangulation (3)**

1 2 1 2 R1 R2 R1 R2 1 2 1 2 R1 R2 R1 R2

10
**regular triangulation (4)**

Power diagram and regular triangulation (4) A power region of a point is defined by: The power diagram of the set S, P(S), is the set of the regions

11
**Power diagram and regular**

triangulation (5)

12
**Power diagram and regular triangulation (6)**

A power region may be empty. A power region of p may be does not contain the point p. A point on the convex hull of S has an unbounded or an empty region.

13
**Power diagram and regular triangulation (7)**

is a k- simplex of the regular triangulation of S iff

14
**Alpha-shape of a set of points (1)**

15
**Alpha-shape of a set of points: example (2)**

16
**Alpha-shape of a set of points: example(3)**

17
**Alpha-shape of a set of points: example(4)**

18
**Alpha-shape of a set of points: properties(5)**

The alpha shape is a sub-graph of the Delaunay triangulation. The convex hull is an element of the alpha shape family.

19
**Alpha-shape of a set of points (6)**

Theorem (2D case)

20
**Alpha-shape of a set of points (7)**

21
**Alpha-shape of a set of points: algorithm(8)**

Input: the point set S, output: a-shape of S Compute the Voronoi diagram of S. For each edge e compute the values amin(e) and amax(e). If (amin(e)<=a<=amax(e)) then e is in the a-shape of S.

22
**Alpha-shape of a set of points : 3D case(9)**

1-simplex 2-simplex p1 v2 v1 p3 p2

23
**Alpha-shape of a set of points (10)**

Simplicial Complex A simplicial complex K is a finite collection of simplices with the following two properties: A Delaunay triangulation is a simplicial complex.

24
**Alpha-shape of a set of points (11)**

Alpha Complex

25
**Alpha-shape of a set of points (12)**

Alpha Complex

26
**Alpha-shape of a set of points (13) Alpha Complex : example**

27
**Alpha-shape of a set of points (14)**

Curve reconstruction: definition The problem of curve reconstruction takes a set, S, of sample points on a smooth closed curve C, and requires to produce a geometric graph having exactly those edges that connect sample points adjacent in C.

28
**Alpha-shape of a set of points (15)**

Surface reconstruction A set of points S The reconstructed surface

29
**Alpha-shape of a set of points (16) Curve reconstruction : theorem**

30
**Alpha-shape of a set of points (17)**

The sampling density must be such that the center of the “disk probe” is not allowed to cross C without touching a sample point. Examples of non admissible cases of probe-manifold intersection.

31
**Weighted alpha shape (1)**

For two weighted points, (p’, p ’’) and x=(x’,x’’), we define

32
**Weighted alpha shape (2)**

x’

33
**Weighted alpha shape (3)**

34
**Weighted alpha shape (4)**

35
**Weighted alpha shape (5)**

The weighted alpha shape is a sub-graph of the regular triangulation.

36
**Weighted alpha-shape (6)**

Input: the points set S, output: weighted a-shape of S. Compute the power diagram of S. For each edge e of the regular triangulation of S compute the values amin(e) and amax(e). For each edge e If (amin(e)<=a<=amax(e)) then e is in the weighted a-shape of S.

37
**Gabriel Graph: definition (1)**

38
**Gabriel Graph: example (2)**

This edge is not in the GG An edge of Gabriel

39
**Gabriel Graph: properties (3)**

1) The Gabriel graph of S is a sub graph of the Delaunay triangulation of S.

40
**Gabriel Graph: example (4)**

41
**Gabriel Graph: algorithm (5)**

Compute the Voronoi diagram of S. A Delaunay edge e belongs to the Gabriel Graph of S iff e cuts its dual Voronoi-edge.

42
**Beta skeleton (1) b-neighborhood, neighborhood,**

The Gabriel graph is an element of the b-skeleton family (b= 1). The b-skeleton is a sub-graph of the Delaunay triangulation.

43
**Examples of b-neighborhood :**

Beta skeleton (2) Examples of b-neighborhood : Forbidden regions

44
Beta skeleton (3) A beta-skeleton edge

45
Beta skeleton (4) beta = 1.1 beta = 1.4

46
**Beta skeleton : algorithm (5)**

The coordinates of these centers are:

47
Medial axis (1) The medial axis of a region, defined by a closed curves C, is the set of points p which have a same distance to at least two points of C.

48
**Medial axis and Voronoi diagram(2)**

A Delaunay disc is an approximation of a maximal ball

49
**Medial axis and Voronoi diagram (3)**

Let S be a regular sampling of C. Compute the Voronoi diagram of S. A Voronoi edge vv’ is in an approximation of the medial axis of C if it separates two non adjacent samples on C.

50
**Reconstruction : e-sampling condition(1)**

S is an e-sampling (e<1) of a curve C iff

51
**Reconstruction : e-sampling condition(2)**

52
**Reconstruction : b-skeleton (3)**

Let S e-sample a smooth curve, with e< The b-skeleton of S contains exactly the edges between adjacent vertices on the curve, for b = 1.70.

53
A-shape and Crust (1)

54
A-shape and Crust (2) An edge of A-shape

55
A-shape and Crust (3)

56
A-shape et Crust (4) Crust of S is an A-shape of S when A is the set of the vertices of the Voronoi diagram of S.

57
A-shape et Crust (5) Voronoi vertex crust Voronoi crust

58
**Crust : algorithm (6) Compute the Voronoi diagram of S, DV(S).**

Compute the Voronoi diagram of SUV, DV(SUV), V being the set of the Voronoi vertices of DV(S). A k-simplex, conv(T), of the Delaunay triangulation of SUV, belongs to the crust of S iff the points of T have a same neighbor belonging to V.

59
**Crust : reconstruction (7)**

The crust of S (S being an e-sampling of C) reconstructs the curve C if e <1/5.

60
**Local Crust : definition and properties (1)**

v v’ is the dual Voronoi edge of pp’, b(p p’ v) is the ball which circumscribes the points p, p’,v.

61
**Local Crust : definition and properties (2)**

62
**Local Crust and Gabriel Graph (3)**

Local crust of S is a sub graph of the Gabriel Graph of S.

63
**Voronoi Gabriel Graph (VGG)**

Local Crust and Gabriel Graph (4) Voronoi Gabriel Graph (VGG) [v v’] is an edge of the VGG of S iff [v v’] is the dual Voronoi edge of the Delaunay edge [pp’]. b(v v’) is the ball of diameter v v’. An edge pp’ belongs to the Local crust of S iff vv’ belongs to the VGG of S.

64
**Local Crust and Gabriel Graph (5)**

65
**Local Crust : reconstruction (6)**

The Local crust of S (S being an e-sampling of C) reconstructs the curve C, if e<0.42.

66
**Local Crust and Gabriel Graph (7)**

Voronoi Gabriel Graph

67
**NN-Crust: curve reconstruction**

Compute the Delaunay triangulation of S. E is empty. For each p in S do Compute the shortest edge pq in D(S). Compute the shortest edge ps so that the angle (pqs) more than p/2. E= E U {pq, ps}. E is the NN-crust of S.

68
**3D reconstruction: an example**

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google