Lecture 7 : 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 ballsTriangulation & Dual

Union of Balls

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 axisBlum’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

• 2D

• 3D

• Even when samples are arbitrarily dense

Poles

• Farthest 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