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 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?