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