Projecting points onto a point cloud

1. Projecting points onto a point cloud Speaker: Jun Chen Mar 22, 2007

2. Data Acquisition

3. Point clouds 25893

4. Point clouds 56194

5. topological Unorganized, connectivity-free

6. Surface Reconstruction

7. Applications • Reverse Engineering • Virtual Engineering • Rapid Prototyping • Simulation • Particle systems

8. Definition of “onto”

9. References Parameterization of clouds of unorganized points using dynamic base surfaces Phillip N. Azariadis(CAD,2004) Drawing curves onto a cloud of points for point-based modeling Phillip N. Azariadis, Nickolas S. Sapidis (CAD,2005)

10. References Automatic least-squares projection of points onto point clouds with applications in reverse engineering Yu-Shen Liu, Jean-Claude Paul et al. (CAD,2006)

11. Parameterization of clouds of unorganized points using dynamicbase surfaces Phillip N. Azariadis CAD, 2004, 36(7): p607-623

12. About the author Instructor of the University of the Aegean, director of the Greek research institute “ELKEDE Technology & Design Centre SA”. CAD , Design for Manufacture, Reverse Engineering, CG and Robotics.

13. Parameterization each point well parameterized cloud adequate parameter accurate surface fitting

14. 2 D

15. Previous work • Mesh－－Starting from an underlying 3D triangulation of the cloud of points. Ref.[17] • Unorganized • Projecting data points onto the base surface • Hoppe’s method, ‘simplicial’ surfaces approximating an unorganized set of points • Piegl and Tiller’s method, base surfaceis fitted to the given boundary curves and to some of the data points. no safe, universal

16. (0.3,1) (0,1)

17. Work of this paper

18. Algorithm Step 1 • Initial base surface---- a Coons bilinear blended patch: To get the n×m grid points, define: Ri(v)=S(ui,v), Rj(u)=S(u,vj), pi,j= Ri(v)∩ Rj(u)=S(ui,vj), so ni,j, Su(ui,vj, ), Sv(ui,vj, ) can be computed.

19. Step 2: Squared distances error Error function: it is suitable for the point set with noise and irregular samples.

20. Step 2: Squared distances error

21. Step 2: Squared distances error Letpi,j* be the result of the projection of the pointpi,jonto the cloud of points following an associated directionni,j.

22. Proposition 1

23. Step 3: Minimizing the length of the projected grid sections • No crossovers or self-loops. • Define: pi0,j(1<j<m-2) is a row. identity closeness tridiagonal and symmetric length

24. Step 3: Minimizing the length of the projected grid sections • Combined projection : Bigger－>smoother O(m)

25. Step 4:Fitting the DBS to the grid • Given the set of n×m grid points, a (p,q)th-degree tensor product B-spline interpolating surface is Ref.[26,9.2.5]:

26. Step 5: Crossovers checking • If it fails • 1. Terminate the algorithm. • 2. Compute geodesic grid sections.The DBS is re-fitted to the new grid. • 3. Increase smoothing term. • 4. Remove the grid sections which are stamped as invalid.

27. Step 5:Terminating criterion • 1. The DBS approximates the cloud of points with an accepted accuracy.

28. Step 5:Terminating criterion • 1. The DBS approximates the cloud of points with an accepted accuracy. • 2. The dimension of the grid has reached a predefined threshold. • 3. The maximum number of iterations is surpassed.

29. A final refinement

30. Advantage Contrarily to existing methods, there is no restriction regarding the density thin dense Only assumption: 4 boundary curves

31. Conclusions • Error functions • Smoothing • Crossovers checking

32. Drawing curves onto a cloud of points for point-based modelling Phillip N. Azariadis, Nickolas S. Sapidis CAD, 2005, 37(1): p109-122

33. About the authors • Instructor of the University of the Aegean, the Advisory Editorial Board of CAD. • curve and surface modeling/fairing/visualization, discrete solid models, finite-element meshing, reverse engineering, solid modeling

34. Work of this paper

35. Projection vectors pf pn

36. Previous work • Dealing with 2D point set. Ref.[7,19,21,26] • Appeared in Ref.[21,37] • (a) selection of the starting point is accomplished by trial and error, • (b) it involves four parameters that the user must specify, • (c) no proof of converge is presented, neither any measure for the required execution time.

37. Note！ • Reconstructing an interpolating or fitting surface is meaningless. • Surface reconstruction is not make sense. • They are not always work well. (smooth, closed,density, complexity) • Require the expenditure of large amounts of time and space. • Approximation causes some loss of information.

38. Error function

39. Error analysis True location Independent of the cloud of points

40. Weight function distance between pm and the axis stability

41. Weight function distance between pm and the axis stability

42. Weight function

43. Projection vectors pf pn

44. Algorithm

45. increase

46. Conclusions • Accuracy and robustness, directly without any reconstruction. • Method improved: • Error analysis • Weight function • Iterative algorithm

47. Projection of polylines onto a cloud of points

48. Smoothing

49. Automatic least-squares projection of points onto point clouds with applications in reverse engineering Yu-Shen Liua, Jean-Claude Paul, Jun-Hai Yong, Pi-Qiang Yu, Hui Zhang, Jia-Guang Sun, Karthik Ramanib CAD, 2006, 37(12): p1251-1263

50. About the authors • Postdoctor of Purdue University • CAD • Senior researcher at CNRS • CAD, numerical analysis • Associate professor of Tsinghua University, • CAD, CG