Presentation is loading. Please wait.

Presentation is loading. Please wait.

Proximity graphs: reconstruction of curves and surfaces  Duality between the Voronoi diagram and the Delaunay triangulation.  Power diagram.  Alpha.

Similar presentations


Presentation on theme: "Proximity graphs: reconstruction of curves and surfaces  Duality between the Voronoi diagram and the Delaunay triangulation.  Power diagram.  Alpha."— Presentation transcript:

1 Proximity graphs: reconstruction of curves and surfaces  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. Framework M. Melkemi

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

3 is a k-simplex of the Delaunay triangulation D(S) iff there exists an open ball b such that: Duality: Voronoi diagram and Delaunay triangulation (2)

4 Duality: Voronoi diagram and Delaunay triangulation (3) 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. Examples

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 andits weight.For a weighted points, p=(p’,p’’), the power distance of a point x to p is defined as follows:  (p,x) x p’

8 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) R1R2 R1 R2 R1R2 R1R2

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

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) alpha = 10alpha = 20 alpha = 40alpha = 60

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

18 The alpha shape is a sub-graph of the Delaunay triangulation. The convex hull is an element of the alpha shape family. Alpha-shape of a set of points: properties(5)

19 Theorem (2D case) Alpha-shape of a set of points (6)

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

21 Input: the point set S, output:  -shape of S Compute the Voronoi diagram of S. For each edge e compute the values  min (e) and  max (e). For each edge e If (  min (e)<=  <=  max (e)) then e is in the  - shape of S. Alpha-shape of a set of points: algorithm(8)

22 Alpha-shape of a set of points : 3D case(9) p1 p2 p3 v1 v2 2-simplex 1-simplex

23 Simplicial Complex Alpha-shape of a set of points (10) A simplicial complex K is a finite collection of simplices with the following two properties: A Delaunay triangulation is a simplicial complex.

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

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

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 A set of points S The reconstructed surface Alpha-shape of a set of points (15) Surface reconstruction

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

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 For two weighted points, (p’, p ’’) and x=(x’,x’’), we define Weighted alpha shape (1)

32 p’ x’ Weighted alpha shape (2)

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 Input: the points set S, output: weighted  -shape of S. Compute the power diagram of S. For each edge e of the regular triangulation of S compute the values  min (e) and  max (e). For each edge e If (  min (e)<=  <=  max (e)) then e is in the weighted  - shape of S. Weighted alpha-shape (6)

37 Gabriel Graph: definition (1)

38 Gabriel Graph: example (2) An edge of Gabriel This edge is not in the GG

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 Compute the Voronoi diagram of S. A Delaunay edge e belongs to the Gabriel Graph of S iff e cuts its dual Voronoi-edge. Gabriel Graph: algorithm (5)

42 Beta skeleton (1)  -neighborhood, neighborhood, The Gabriel graph is an element of the  -skeleton family (  = 1). The  -skeleton is a sub-graph of the Delaunay triangulation.

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

44 A beta-skeleton edge (3)Beta skeleton

45 Beta skeleton (4) beta = 1.1beta = 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 S is an  -sampling (  <1) of a curve C iff Reconstruction :  -sampling condition(1)

51 Reconstruction :  -sampling condition(2)

52 Reconstruction :  -skeleton (3) Let S  -sample a smooth curve, with  < The  -skeleton of S contains exactly the edges between adjacent vertices on the curve, for  = 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 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. Crust : algorithm (6)

59 The crust of S (S being an  -sampling of C) reconstructs the curve C if  <1/5. Crust : reconstruction (7)

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) [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. [v v’] is an edge of the VGG of S iff

64 Local Crust and Gabriel Graph (5)

65 The Local crust of S (S being an  -sampling of C) reconstructs the curve C, if  <0.42. Local Crust : reconstruction (6)

66 Local Crust and Gabriel Graph (7) Local crust Voronoi Gabriel Graph

67 NN-Crust: curve reconstruction 1.Compute the Delaunay triangulation of S. E is empty. 2.For each p in S do 1.Compute the shortest edge pq in D(S). 2.Compute the shortest edge ps so that the angle (pqs) more than . E= E U {pq, ps}. 3.E is the NN-crust of S.

68 3D reconstruction: an example


Download ppt "Proximity graphs: reconstruction of curves and surfaces  Duality between the Voronoi diagram and the Delaunay triangulation.  Power diagram.  Alpha."

Similar presentations


Ads by Google