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Computing Shapes and Their Features from Point Samples

This paper discusses various algorithms for surface reconstruction, medial axis computation, and shape segmentation and matching from point cloud data. It explores Voronoi-based algorithms such as Alpha-shapes, Crust, Natural Neighbors, Cocone, and Power Crust. The paper also introduces the concept of the medial axis and its computation using the Cocone algorithm. It concludes with an analysis of undersampling and the Tight Cocone algorithm for generating watertight surfaces.

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Computing Shapes and Their Features from Point Samples

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  1. Computing Shapes and Their Features from Point Samples Tamal K. Dey The Ohio State University

  2. Problems • Surface reconstruction (Cocone) • Medial axis (Medial) • Shape segmentation and matching (SegMatch)

  3. ` Surface Reconstruction Point Cloud Surface Reconstruction

  4. Voronoi based algorithms • Alpha-shapes (Edelsbrunner, Mucke 94) • Crust (Amenta, Bern 98) • Natural Neighbors (Boissonnat, Cazals 00) • Cocone (Amenta, Choi, Dey, Leekha, 00) • Tight Cocone (Dey, Goswami, 02) • Power Crust (Amenta, Choi, Kolluri 01)

  5. Medial Axis

  6. f(x) • f(x) is the distance to medial axis Local Feature Size[Amenta-Bern-Eppstein 98]

  7. x -Sampling[ABE98] • Each x has a sample within f(x) distance

  8. Voronoi/Delaunay

  9. Normal and Voronoi Cells(3D) [Amenta-Bern SoCG98]

  10. P+ P- Poles

  11. P+ P- Normal Lemma • The angle between the pole vector vp and the normal np is O(). vp np

  12. Cocone Algorithm[Amenta-Choi-Dey-Leekha SoCG00] • Simplified/improved the Crust • Only single Voronoi computation • Analysis is simpler • No normal filtering step • Proof of homeomorphism

  13. Cocone • vp= p+ - pis the pole vector • Space spanned by vectors within the Voronoi cell making angle > 3/8 with vp or -vp

  14. Cocone Algorithm

  15. Cocone Guarantees Theorem: Any point x S is within O(e)f(x) distance from a point in the output. Conversely, any point of output surface has a point x S within O(e)f(x) distance. Theorem: The output surface computed by Cocone from an e -sample is homeomorphic to the sampled surface for sufficiently small e.

  16. Undersampling [Dey-Giesen SoCG01] • Boundaries • Small features • Non-smoothness

  17. Boundaries

  18. Small Features • High curvature regions are often undersampled

  19. Data Set Engine

  20. Nonsmoothness

  21. Watertight Surfaces

  22. Tight Cocone [Dey-Goswami SM03]

  23. Tight COCONE Principle • Compute the Delaunay triangulation of the input point set. • Use COCONE along with detection of undersampling to get an initial surface with undersampled regions identified. • Stitch the holes from the existing Delaunay triangles without inserting any new point. • Effectively, the output surface bounds one or more solids.

  24. Result • Sharp corners and edges of AutoPart can be reconstructed.

  25. Timings PIII, 933Mhz, 512MB

  26. Rear view Front view Noisy Data – Ram Head

  27. Example movie file Mannequin

  28. Point data Tight Cocone Robust Cocone Bunny data • Bunny

  29. Medial axis from point sampleDey-Zhao SM02 • [Hoffman-Dutta 90],[Culver-Keyser-Manocha 99],[Giblin-Kimia 00], [Foskey-Lin-Manocha 03] • Voronoi based [Attali-Montanvert-Lachaud 01] • Power shape : guarantees topology, uses power diagram [Amenta-Choi-Kolluri 01] • Medial : Approximates the medial axis as a Voronoi subcomplex and has converegence guarantee. [Dey-Zhao 02]

  30. Medial Axis • Medial Ball • Medial Axis •  -Sampling

  31. Geometric Definitions • Delaunay Triangulation • Voronoi Diagram • Pole and Pole Vector • Tangent Polygon • Umbrella Up

  32. Filtering conditions Our goal: approximate the medial axis as a subset of Voronoi facets. • Medial axis point m • Medial angle θ • Angle and Ratio Conditions

  33. Angle Condition • Angle Condition [θ ]: 

  34. Ratio Condition • Ratio Condition []:

  35. Algorithm

  36. Theorem • Let F be the subcomplex computed by MEDIAL. As  approaches zero: • Each point in F converges to a medial axis point. • Each point in the medial axis is converged upon by a point in F.

  37. Experimental Results

  38. Experimental Results

  39. Experimental Results

  40. Computation Time • Pentium PC • 933 MHz CPU • 512 MB memory • CGAL 2.3 • C++ • O1 optimization

  41. CAD model Point Sampling Medial Axis from a CAD model Medial Axis

  42. Medial Axis from a CAD model CAD model Medial Axis Point Sampling

  43. Example movie file Anchor Medial

  44. Segmentation and matching • Siddiqui-Shokoufandeh-Dickinson-Zucker 99 (Shock graphs) • Hilaga-Shinagawa-Kohmura-Kunni 01 (Reeb graph) • Osada-Funkhouser-Chazelle-Dobkin 01 (Shape distribution) • Bespalov-Shokoufandeh-Regli-Sun 03(spectral decomposition) • Dey-Giesen-Goswami 03 (Morse theory)

  45. Segmentation and matchingDey-Giesen-Goswami 03 • Segment a shape into `features’ • Match two shapes based on the segmentation

  46. Continuous Flow Discretization Discrete flow Feature definition

  47. Anchor set • Height fuinction: • Anchor set:

  48. Driver and critical points • Anchor Hull : H(x) is convex hull of A(x) • Driver : d(x) is the closest point on the anchor hull • Critical points

  49. Flow • Vector field v : if x is regular and 0 otherwise • Flow  induced by v Fix points of  are the critical points of h

  50. Features • F(x)= closure(S(x)) for a maximum x

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