1. CS175 2003 1 CS 175 – Week 3 Triangulating Point Clouds VD, DT, MA, MAT, Crust

2. CS175 2003 2 Overview Voronoi Diagrams Delaunay triangulations medial axis medial axis transform crust

3. CS175 2003 3 Voronoi-Diagram points P = {p 1, …, p n } ½ R d Voronoi cell V i = all points x 2 R d closest to p i partition of space intersections of V i form Voronoi vertices, edges, etc.

4. CS175 2003 4 Delaunay-Triangulation D(S) Delaunay cell for S ½ {1,…,n} D(S) = convex hull of {p i }, i 2 S, if all V i, i 2 S intersect usually, D(S) =  for #S ¸ d+2 D(S) are points, edges, triangles, … partition of P’s convex hull

5. CS175 2003 5 Delaunay-Triangulation Properties dual to the Voronoi diagram global circumcircle criterion local circumcircle criterion maximum minimum angle

6. CS175 2003 6 Delaunay-Triangulation Algorithms edge-flipping incremental divide and conquer plane sweep triangle program http://www-2.cs.cmu.edu/~quake/triangle.html

7. CS175 2003 7 MA and MAT for a smooth, closed curve F medial axis MA(F) = all points that have more than one closest point on F medial axis transform MAT(F) = all pairs (x,r) where x 2 MA(F) and r the radius of the maximal disc at x

8. CS175 2003 8 LFS and r-sample local feature size for x 2 F : LFS (x) = distance from x closest point on MA(F) r-sample set of points P ½ F r-sample, if distance from x 2 F to P is smaller then r ¢ LFS(x)

9. CS175 2003 9 Crust P = set of 2D points V = Voronoi vertices T = DT of P [ V crust (P) = edges in T with endpoints in P

10. CS175 2003 10 Crust Properties P is r-sample of F L = polygon with points from P r · 0.39 ) crust(P) ¾ L r · 0.25 ) crust(P) = L x 2 F : dist(x, crust(P)) · r 2 LFS(x)/2

11. CS175 2003 11 3D-Crust Voronoi vertices NOT necessarily close to MA(F) take maximal vertices from V(p i ) requires r · 0.06

12. CS175 2003 12 3D-Crust Problems not necessarily manifold normal filtering boundaries sharp features sampling criterion usually not met holes