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CS 175 – Week 3 Triangulating Point Clouds VD, DT, MA, MAT, Crust

CS 175 – Week 3 Triangulating Point Clouds VD, DT, MA, MAT, Crust. Overview. Voronoi Diagrams Delaunay triangulations medial axis medial axis transform crust. 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

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CS 175 – Week 3 Triangulating Point Clouds VD, DT, MA, MAT, Crust

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  1. CS 175 – Week 3TriangulatingPoint CloudsVD, DT, MA, MAT, Crust

  2. Overview • Voronoi Diagrams • Delaunay triangulations • medial axis • medial axis transform • crust

  3. Voronoi-Diagram • points P = {p1, … , pn} ½ Rd • Voronoi cell Vi =all points x 2 Rd closest to pi • partition of space • intersections of Vi form Voronoi vertices, edges, etc.

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

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

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

  7. MA and MAT • for a smooth, closed curve F • medial axis MA(F) =all points that have more than oneclosest 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. 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. Crust • P = set of 2D points • V = Voronoi vertices • T = DT of P [ V • crust (P) =edges in T with endpoints in P

  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)) · r2 LFS(x)/2

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

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

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