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Shape Reconstruction from Samples with Cocone

Shape Reconstruction from Samples with Cocone. Tamal K. Dey Dept. of CIS Ohio State University. A point cloud and reconstruction. Surface meshing from sample. A point set from satelite imaging. A reconstruction with and without noise. Why Sample Based Modeling?.

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Shape Reconstruction from Samples with Cocone

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  1. Shape Reconstruction from Samples with Cocone Tamal K. Dey Dept. of CIS Ohio State University

  2. A point cloud and reconstruction

  3. Surface meshing from sample

  4. A point set from satelite imaging

  5. A reconstruction with and without noise

  6. Why Sample Based Modeling? • Sampling is easy and convenient with advanced technology • Automatization (no manual intervention for meshing) • Uniform approach for variety of inputs (laser scanner, probe digitizer, MRI,scientific simulations) • Robust algorithms are available

  7. Challenges • Nonuniform data • Boundaries • Undersampling • Large data • Noise

  8. Nonuniform data

  9. Boundaries

  10. Undersampling

  11. Large data 3.4 million points

  12. Cocone • Cocone meets the challenges • It guarantees geometrically close surface with same topological type • Detects boundaries • Detects undersampling • Handles large data (Supercocone) • Watertight surface (Tight Cocone)

  13. e-Sampling (ABE98) f(x) is the distance to medial axis Each x has a sample within ef(x)

  14. Voronoi/Delaunay

  15. Surface and Voronoi Diagram • Restricted Voronoi • Restricted Delaunay • skinny Voronoi cell • poles

  16. Cocone algorithm • Cocone Space spanned by vectors making angle q /8 with horizontal

  17. Radius, height and neighbors • p is the farthest point from p in the cocone. • radius r(p): p radius of cocone • height h(p): min distance to the poles • cocone neighbors Np

  18. Flatness condition • Vertex p is flat if 1. Ratio condition: r(p)   h(p) 2. Normal condition: v(p),v(q)   q with pNq

  19. Boundary detection Boundary(P,,) Compute the set R of flat vertices; while pR and pNq with qR and r(p)h(p) and v(p),v(q)  R:=Rp; endwhile return P\R end

  20. Detected Boundary Samples

  21. Detected Boundary Samples

  22. Undersampling repaired

  23. Holes are created

  24. Tight Cocone Guarantee: A water tight surface no matter how the input is.

  25. Tight Cocone output

  26. Holes are created

  27. Hole filling

  28. Time

  29. Time

  30. Large Data • Delaunay takes space and time • Exact computation is necessary. Doubles the time. Floating point Exact arithmetic

  31. Large Data (Supercocone) • Octree subdivision

  32. Cracks • Cracks appear in surface computed from octree boxes

  33. Surface matching

  34. David’s Head 2 mil points, 93 minutes

  35. Lucy25 3.5 million points, 198 mints

  36. Shape of arbitrary dimension

  37. Tangent and Normal Polytopes • T(p) = V(p)T(p) • N(p) = V(p)N(p)

  38. Experiments

  39. Sample Decimation Original 40K points • = 0.33 12K points • = 0.4 8K points

  40. Rocker •  0.33 11K points Original 35K points

  41. Bunny •  0.4 7K points •  0.33 11K points Original 35K points

  42. Bunny •  0.4 7K points •  0.33 11K points Original 35K points

  43. Triangle Aspect Ratio

  44. Medial axis

  45. Medial axis

  46. Noise Cleaned Outliers

  47. Noise (Local) This is a challenge unsolved. Perturbation by very tiny amount is tolerated by Cocone.

  48. Boundaries Engineering Medical

  49. Geometric Models Sports Drug design

  50. Undersampling for Nonsmoothness

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