1. Surface Reconstruction Using RBF Reporter : Lincong Fang 11.07.2007

2. Surface Reconstruction sample Reconstruction

3. Surface Reconstruction Delaunay/Voronoi –Alpha shape/Conformal alpha shape –Crust/Power crust –Cocone –Etc. Implicit surfaces –Signed distance function –Radial basis function(RBF) –Poisson –Fourier –MPU –Etc.

4. Implicit Surface Defined by implicit function –Such as Many topics within broad area of implicit surfaces

5. Implicit Surface Mesh independent representation - generate the desired mesh when you require it Compact representation to within any desired precision A solid model is guaranteed to produce manifold (manufacturable) surface

6. Implicit surface Tangent planes and normals can be determined analytically from the gradient of the implicit function

7. Implicit surface CSG operation

8. Implicit surface Morphing

9. Implicit surface reconstruction ReconstructionIso-surfaceReconstruction

10. Introduction to RBF Interpolation problem

11. Introduction to RBF J.Duchon. Splines minimizing rotation-invariant semi-norms in Sololev spaces. In W. Schempp and K.Zeller, editors, Constructive Theory of Functions of Several Variables, number 571 in Lecture Notes in Mathematics, pages 85-100, Berlin, 1977. Springer-Verlag.

12. Introduction to RBF An RBF is a weighted sum of translations of a radially symmetric basic function augmented by a polynomial term

13. Introduction to RBF Popular choices for include For fitting functions of three variables, good choices include

14. Introduction to RBF Matrix form

15. Introduction to RBF Matrix form

16. Reconstruction and representation of 3D objects with Radial Basis functions –J.C.Carr 1,2, R.K.Beatson 2, J.B.Cherrie 1, T.J.Mitchell 1,2 –W.R.Fright 1, B.C.McCallum 1, T.R.Evans 1 –1. Applied Research Associates NZ Ltd –2. University of Canterbury, New Zealand –Sig 2001

17. Off-surface Points

18. RBF Center Reduction

19. Greedy Algorithm Choose a subset from the interpolation nodes x i and fit an RBF only to these. Evaluate the residual, e i = f i – f(x i ), at all nodes. If max{e i } < fitting accuracy then stop. Else append new centers where e i is large. Re-fit RBF and goto 2

20. 544000 points, 80000 centers, accuracy of 5*10 -4

21. Noisy

22. Hole Filled & Non-uniformly

23. Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions –Bryan S. Morse 1, Terry S. Yoo 2, Penny Rheingans 3, –David T.Chen 2, K.R. Subramanian 4 –1. Department of CS, Brigham Young University –2. National Library of Medicine –3. Department of CS and EE, University of Maryland Baltimore County –4. Department of CS, University of North Carolina at Charlotte –Proceeding of the International Conference on Shape Modeling and Applications 2001

24. Compactly-supported RBF H. Wendland. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. AICM, 4:389-396, 1995

25. Matrix Form

26. Choice of Support Size

27. Comparison

28. Compactly supported basic functions is much more efficient. Non-compactly supported basic functions are better suited to extrapolation and interpolation of irregular, non- uniformly sampled data.

29. Modeling with implicit surfaces that interpolate –Greg Turk GVU Center, College of Computing Georgia Institute of Technology –James F.O’Brien EECS, Computer Science Division University of California, Berkeley

30. Modeling

31. Interior Constraints

32. Matrix Form

33. Exterior Constraints

34. Normal Constraints

36. Example Polygonal surface The interpolating implicit surface defined by the 800 vertices and their normals

37. A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis Functions –Yutaka Ohtake, Alexandaer Belyaev, Hans-Peter Seidel –Computer Graphics Group, Max-Planck-Institute for informatics –Germany –Proceedings of the Shape Modeling International 2003

38. Construct RBF

39. Single level Interpolation 35K points 6 seconds

40. Multi-level Interpolation

42. Coarse to Fine

43. 3D Scattered Data Approximation with Adaptive Compactly Supported Radial Basis Functions –Yutaka Ohtake, Alexandaer Belyaev, Hans-Peter Seidel –Computer Graphics Group, Max-Planck-Institute for informatics –Germany –Proceedings of the Shape Modeling International 2004

44. Construct RBF Base approximationLocal details

45. Adaptive PUNormalized RBF

47. Selection of Centers 10050010002000

48. Example

49. Compare With Multi-scale

50. Reconstructing Surfaces Using Anisotropic Basis Functions –Huong quynh Dinh, Greg Turk Georgia Institute of Technology College of Computing –Greg Slabaugh Georgia Institute of Technology Scholl of Electrical and Computer Engineering Center for Signal and Image Processing –Computer Vision, Vol 2, 2001, p606-613.

51. Basic Function

52. Direction of Anisotropy Covariance matrix –Corner point : all three eigenvalues are nearly equal –Edge point : one strong eigenvalue –Plane point : two eigenvalues are nearly equal and larger than the third

53. Noisy

56. Summary

58. Thank you !!!