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Lecture 7 : Point Set Processing Acknowledgement : Prof. Amenta’s 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) äUnion of balls äTriangulation & Dual Union of Balls
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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. Vigneron’s 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 axis Blum’s 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 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 Segmentation [Dey 2003] Based on Voronoi Diagram and Delaunay Triangulation
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Flow Field of Shape
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Shape Feature Segmentation
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