Lecture 9 : Point Set Processing

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Lecture 9 Point Set Processing
Acknowledgement Prof. Amentas slides
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Point Set (Point Cloud)

• Often obtained from laser snanners

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Building Information Modeling (BIM)
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Human Body Shapes
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Triangulations (Delaunay) Dual Diagrams
(Voronoi)
Union of Balls

• Union of balls
• Triangulation Dual

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Data Structures for Point Set Data

• Voronoi Diagram
• Delaunay Triangulation
• Medial Axis

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Voronoi Diagram/Delaunay Triangulation

• Refer to Prof. Vignerons slides

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Medial Axis

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

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Applications of medial axis

• Shape matching
• Animation
• Computer Vision
• Dimension reduction (Simplification)
• Solid modeling

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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

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medial axisBlums equivalent definitions
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Medial Axis

• A set of points with more than one closest
  surface point

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3D Medial Axis

• A set of points with more than one closest
  surface point

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Medial Axis

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

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Relation to Voronoi

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

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Convergence

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

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Discrete Union of Balls

• Voronoi balls approximate the object and its
  complement.

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2D Curve Reconstruction

• Blue Delaunay edges reconstruct the curve, pink
  triangulate interior/exterior.
• Many algorithms, with proofs, for coloring edges.

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2D Curve Reconstruction
ltVoronoi Diagram of point set Sgt

• Delaunay Triangulation of point set S
• and voronoi vertices V
• Black lines represent curve reconstruction

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2D Medial Axis Reconstruction

• Pink approximate medial axis.

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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

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Union of Balls centered on voronoi vertices

• 2D
• 3D

• Even when samples are
• arbitrarily dense

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Poles

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

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Union of balls centered on interior poles
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3D surface reconstruction

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

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Results Amenta 1998,2001
Laser range data, power crust, simplified
approximate medial axis
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Shape Feature SegmentationDey 2003

• Based on Voronoi Diagram and Delaunay
  Triangulation

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Flow Field of Shape
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Shape Feature Segmentation