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Shape Dimension and Approximation from Samples

This paper explores adaptive shape reconstruction and dimension approximation from samples in smooth compact manifolds embedded in Euclidean space. Key techniques include manifold learning, medial axis analysis, and ε-sampling with bounds on local feature size. It investigates the implications of tangent and normal spaces, along with Voronoi diagrams and Delaunay triangulation for achieving accurate shape representations. Experimental results illustrate the effectiveness of the proposed algorithms on various dimensional datasets while addressing manifold extraction and potential improvements in shape reconstruction methodologies.

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Shape Dimension and Approximation from Samples

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  1. Shape Dimension and Approximation from Samples T. Dey, J. Giesen, S. Goswami, W. Zhao Dept. of CIS Ohio State University

  2. Shapes and their dimensions • Shapes for us are smooth compact manifolds embedded in an Euclidean space.

  3. Sampling

  4. Motivations • Manifold learning • Shape reconstruction

  5. Local feature size and sampling • Medial axis A • Local feature size f : Rd R, f(p) is the smallest distance to A • -sampling [ABE98] • d(p): Euclidean distance of p to nearest sample •   d(p)/f(p)

  6. Sampling and Ambiguity • (,)-sampling • -sampling and all samples are > f(p) away ([DFR01],[Erick01]) • /2 <  < 

  7. Tangent and Normal Spaces • Space spanned by tangents T(p) • Space spanned by normals N(p)

  8. Voronoi Diagram / Delaunay triangulation

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

  10. Voronoi Subpolytopes • V(p,i)  V(p), 1 i  d • pole pi+ ([AB98]) • pole vector v(p,i)

  11. Heights • pole vector v(p,i) • heights H(p,i)= ||v(p,i)||

  12. Idea • M is a k-manifold in Rd, P is an (,)-sample • heights H(p,i) are small for 1i k and big for k>i • Compare with H(p,1)

  13. Normal Lemma

  14. Height Lemma

  15. Pole-Normal Lemma

  16. Small-Height Lemma

  17. Height Theorem

  18. Ratio gap for 

  19. Algorithm Dimension

  20. Experiments • CGAL library •  =0.3

  21. Shape Reconstruction • Compute a simplicial complex K with dist(|K|,M)=O() times local feature size • In R2 and R3, |K| and M are homeomorphic

  22. Cocone C(p) • C(p): x in V(p) making angle < /8 with V(p,k) • Compute all dual simplices to (d-k)-dimensional Voronoi edges intersecting C(p)

  23. CoconeShape

  24. Shape Distance Theorem Follows from normal and small-height Lemma

  25. Manifold Extraction(?) • How to extract a k-manifold out of K? • Manifold extraction in R3 • Pruning • Walking

  26. Homeomorphism(?) ?

  27. Experimental Results

  28. Result in R4 • w=x2 + y2+ z2 • 111111 grid (1331 points) • 3d points 615, 2d points 582, 1d points 134 • Ideally 637,578,116

  29. Conclusions • We generalized the curve and surface reconstruction to shape reconstruction. • How to handle boundaries? • Manifold extraction? • Immersion instead of embedding? • Avoid Voronoi computation?

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