1. Lecture 7 : Point Set Processing Acknowledgement : Prof. Amenta’s slides

2. Point Set (Point Cloud) Often obtained from laser snanners

3. Building Information Modeling (BIM)

4. Human Body Shapes

5. Triangulations (Delaunay) & Dual Diagrams (Voronoi) äUnion of balls äTriangulation & Dual Union of Balls

6. Data Structures for Point Set Data Voronoi Diagram Delaunay Triangulation Medial Axis

7. Voronoi Diagram/Delaunay Triangulation Refer to Prof. Vigneron’s slides

8. Medial Axis A Transformation For Extracting New Descriptors of Shape Locus of points equidistant from contour Skeleton

9. Applications of medial axis Shape matching Animation Computer Vision Dimension reduction (Simplification) Solid modeling

10. Definitions of medial axis Locus of points equidistant from contour Grass-fire, prairie-fire, wave-front collision Locus of centers of maximal circles Local maxima in distance transform Result of topology preserving thinning

11. medial axis Blum’s equivalent definitions :

12. Medial Axis A set of points with more than one closest surface point

13. 3D Medial Axis A set of points with more than one closest surface point

14. Medial Axis Maximal ball avoiding surface is a medial ball Every solid is a union of balls

15. Relation to Voronoi Voronoi balls approximate medial balls For dense surface samples in 2D, all voronoi vertices lie near axis

16. Convergence In 2D, set of Voronoi vertices converges to the medial axis as sampling density increases.

17. Discrete Union of Balls Voronoi balls approximate the object and its complement.

18. 2D Curve Reconstruction Blue Delaunay edges reconstruct the curve, pink triangulate interior/exterior. Many algorithms, with proofs, for coloring edges.

19. 2D Curve Reconstruction Delaunay Triangulation of point set S and voronoi vertices V Black lines represent curve reconstruction

20. 2D Medial Axis Reconstruction Pink approximate medial axis.

21. 3D Case Different from 2D In 3D, some Voronoi vertices are not near medial axis Red : voronoi cell of the blue point Notice the red voronoi vertex that is far from the medial axis

22. Union of Balls centered on voronoi vertices 2D 3D Even when samples are arbitrarily dense

23. Poles Farthest voronoi vertices –p+, p- (opposite sides) Subset of voronoi vertices, the poles, approximate near medial axis

24. Union of balls centered on interior poles

25. 3D surface reconstruction Similar to 2D surface reconstruction, but use poles instead of entire voronoi vertices

26. Results [Amenta 1998,2001] Laser range data, power crust, simplified approximate medial axis

27. Shape Feature Segmentation [Dey 2003] Based on Voronoi Diagram and Delaunay Triangulation

28. Flow Field of Shape

29. Shape Feature Segmentation