Lecture 9 : Point Set Processing Acknowledgement : Prof. Amenta’s slides

Point Set (Point Cloud) Often obtained from laser snanners

Building Information Modeling (BIM)

Human Body Shapes

Triangulations (Delaunay) & Dual Diagrams (Voronoi) Union of Balls Triangulations (Delaunay) & Dual Diagrams (Voronoi) Union of balls Triangulation & Dual

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

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

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

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

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

medial axis Blum’s equivalent definitions :

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

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

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

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

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

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

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

2D Curve Reconstruction <Voronoi Diagram of point set S> Delaunay Triangulation of point set S and voronoi vertices V Black lines represent curve reconstruction

2D Medial Axis Reconstruction Pink approximate medial axis.

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

Union of Balls centered on voronoi vertices Even when samples are arbitrarily dense

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

Union of balls centered on interior poles

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

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

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

Flow Field of Shape

Shape Feature Segmentation