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
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
• 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