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Lecture 7 : Point Set Processing

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 Balls. Union of balls

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Lecture 7 : Point Set Processing

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

  2. Point Set (Point Cloud) • Often obtained from laser snanners

  3. Building Information Modeling (BIM)

  4. Human Body Shapes

  5. Triangulations (Delaunay) & Dual Diagrams (Voronoi) Union of Balls • Union of balls • Triangulation & Dual

  6. Data Structures for Point Set Data • Voronoi Diagram • Delaunay Triangulation • Medial Axis

  7. Voronoi Diagram/Delaunay Triangulation • Refer to Prof. Vigneron’s slides

  8. Medial Axis • A Transformation For Extracting New Descriptors of Shape • Locus of points equidistant from contour • Skeleton

  9. Applications of medial axis • Shape matching • Animation • Computer Vision • Dimension reduction (Simplification) • Solid modeling

  10. 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

  11. medial axisBlum’s equivalent definitions :

  12. Medial Axis • A set of points with more than one closest surface point

  13. 3D Medial Axis • A set of points with more than one closest surface point

  14. Medial Axis • Maximal ball avoiding surface is a medial ball • Every solid is a union of balls

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

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

  17. Discrete Union of Balls • Voronoi balls approximate the object and its complement.

  18. 2D Curve Reconstruction • Blue Delaunay edges reconstruct the curve, pink triangulate interior/exterior. • Many algorithms, with proofs, for coloring edges.

  19. 2D Curve Reconstruction <Voronoi Diagram of point set S> • Delaunay Triangulation of point set S • and voronoi vertices V • Black lines represent curve reconstruction

  20. 2D Medial Axis Reconstruction • Pink approximate medial axis.

  21. 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

  22. Union of Balls centered on voronoi vertices • 2D • 3D • Even when samples are • arbitrarily dense

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

  24. Union of balls centered on interior poles

  25. 3D surface reconstruction • Similar to 2D surface reconstruction, but use poles instead of entire voronoi vertices

  26. Results [Amenta 1998,2001] Laser range data, power crust, simplified approximate medial axis

  27. Shape Feature Segmentation[Dey 2003] • Based on Voronoi Diagram and Delaunay Triangulation

  28. Flow Field of Shape

  29. Shape Feature Segmentation

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