1. Shape Reconstruction From Planar Cross Sections CAI Hongjie | May 28, 2008

2. Motivation S. Hahmann, et al. CAGD 2000, GMP 2008

3. Reconstruction From Cross Sections Triangulated tiling

4. Reconstruction From Cross Sections Smooth skinning

5. Papers List T.W. Sederberg, K.S. Klimaszewski, et al Triangulation of Branching Contours using Area Minimization,1997 Jean-Daniel Boissonnat Shape Reconstruction From Planar Cross-Sections, 1988 L.G. Nonato, A.J. Cuadros-Vargas, et al Beta-Connection: Generating a Family of Models from Planar Cross Sections, 2005 N.C. Gabrielides, A.I. Ginnis, et al G 1 -Smooth Branching Surface Construction From Cross Sections, 2007

6. Triangulation of Branching Contours using Area Minimization T.W. Sederberg, K.S. Klimaszewski Mu Hong, Kazufumi Kaneda International Journal of Computational Geometry & Applications, 1997

7. Thomas W. Sederberg Main Post Professor of Computer Science, Brigham Young University Research Interests Computer aided geometric design Computer graphics Image morphing Publications SIGGRAPH 11; CAGD 26; CAD 6; …

8. Ambiguous Triangulation

9. Previous Works Graph representation Some heuristics Volume maximized (E. Keppel, 1975) Faces area minimized (H. Fuchs et al, 1977)

10. Branching and Link-Edge

11. Bad Case and Remedy

12. Polygonal Bridge & Area Minimization

13. More Results

14. Shape Reconstruction From Planar Cross-Sections Jean-Daneil Boissonat INRIA Computer Vision, Graphics, and Image Processing, 1988

15. Jean-Daneil Boissonat Main Posts Research Director at INRIA Sophia-Antipolis Head of the Geometrica projectGeometrica Chair of the Evaluation Board of INRIAEvaluation Board Research Interests Discrete and Computational Geometry Voronoi diagrams & Delaunay triangulation Surface reconstruction

16. 3D Delaunay Triangulation A B C D F E a b c

17. Computing 3D Delaunay Triangulation

18. Subdivision of Contours

19. Result

20. Beta-Connection: Generating a Family of Models from Planar Cross Sections L.G. Nonato, A.J. Cuadros-Vargas R. Minghim, M.C. F.DE Oliveira ACM Transactions on Graphics, 2005

21. Different Correspondence

22. β -Connection

23. Tetrahedron Types Internal tetrahedron External tetrahedron Redundant tetrahedron Reverse tetrahedron

24. Graph Representation Graph G from Delaunay triangulation Sphere node: region Cylinder node: redundant tetrahedron Cone node: external tetrahedron

25. β -Components Definition 1: a, b region nodes in G, d G ( a,b ) is the length of the shortest path connecting a and b. Definition 2:,region nodes a and b are said to be β -connected, denoted, if G, where,such that d G ( σ i, σ i+1 ) ≤ β, i=1,…,k-1. Lemma 1: is an equivalence relation. Definition 3: β -components are defined by the equivalence class generated by.

26. Algorithm 3D Delaunay triangulation Create graph representation and determine β -components Remove cylinder and cone nodes connecting different β -components For each β -component, tackle with the singularity

27. Examples

28. More Example

29. G 1 -Smooth Branching Surface Construction From Cross Sections N.C Gabrielides, A.I. Ginnis, P.D. Kaklis, M.I. Karavelas Computer-Aided Design, 2007

30. Menelaos Karavelas Curriculum Vitae Ph.D. at Stanford UniversityStanford University Post-doc at INRIA Sophia-AntipolisINRIA Sophia-Antipolis Assistant professor at the Department of Applied Mathematics of the University of CreteDepartment of Applied MathematicsUniversity of Crete Research Interests Voronoi diagrams and Delaunay triangulations Shape-preserving interpolation Shape reconstruction

31. Transfinite Interpolation Surface Given parametric surface f ( u, v ), 0 ≤ u, v ≤ 1.Let u- and v- isoparametric boundaries be 0v,1v ; u0,u1. Then are transfinite interpolation surfaces. Coons, MIT Project MAC-TR-41,1967

32. Coons Bi-cubic Blending Surface Let boundaries hodograph of f ( u, v ) be 0v u,1v u ; u0 v,u1 v (0v u =∂ f ( u, v )/ ∂u| u=0 ), then Coons surface is where

33. Sketch of the Algorithm Step 1: tangent vector estimation Step 2: planar contours and tangent ribbons Step 3: surrounding surface Step 4: trimming Step 5: hole filling

34. Step 1: tangent vector estimation

35. Step 2: Planar Contours and Tangent Ribbons

36. Step 3: Surrounding Surface

37. Step 4: Trimming Trimming curve Y(t)=S X(t) Cross-tangent of trimming curve

38. Step 5: Hole Filling Selecting center point K xy coordinate centroid of M k z coordinate K z K K MkMk M k-1 M k+1

39. Step 5: Hole Filling Guide curves Gordon-Coons surface K MkMk M k-1 M k+1 GkGk G k+1 G k-1

40. The “many-to-many” Problem

41. The Separating Strip

42. Symmetric Data Set

43. Symmetric Construction

44. Non-Symmetric Case

45. Integral Construction

46. Open Set Case

47. The Bulbous Hull Example

48. Failure of the Algorithm

49. Improvement by Voronoi Diagram

50. Conclusions Ensuring G 1 continuity Preserving data symmetry Automatic solution is available Topological instability Failure for multiple-connected residuals

51. Thanks! Q&A