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