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Advanced Point Set Processing: Medial Axis Reconstruction and Shape Segmentation

This lecture covers point set processing techniques including Delaunay triangulations, Voronoi diagrams, medial axis reconstruction, and shape segmentation. Learn about extracting new shape descriptors, skeleton applications, solid modeling, and more. Dive into topics like convergence, discrete union of balls, 2D/3D curve reconstruction, and results of surface reconstruction. Explore the relation between Voronoi balls and medial balls, poles, and their impact on shape segmentation. Discover how shape feature segmentation is influenced by flow fields and advanced geometric algorithms.

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Advanced Point Set Processing: Medial Axis Reconstruction and Shape Segmentation

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  1. Lecture 9 : 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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