Proximity graphs: reconstruction of curves and surfaces

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) • 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 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. 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. 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 point is 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 = 10 alpha = 20 alpha = 40 alpha = 60

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). • For each edge 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) p’ 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, • The Gabriel graph is an element of the b-skeleton family (b= 1). • The b-skeleton is a sub-graph of the Delaunay triangulation. neighborhood,

43. 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