1. Lecture 9 : 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)
Union of Balls
Triangulations (Delaunay) & Dual Diagrams (Voronoi)
Union of balls
Triangulation & Dual

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
<Voronoi Diagram of point set S>
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
Even when samples are
arbitrarily dense

23. Poles Farthest voronoi vertices Subset of 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