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Projecting points onto a point cloud with noise Speaker: Jun Chen Mar 26, 2008.

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Presentation on theme: "Projecting points onto a point cloud with noise Speaker: Jun Chen Mar 26, 2008."— Presentation transcript:

1 Projecting points onto a point cloud with noise Speaker: Jun Chen Mar 26, 2008

2 Data Acquisition

3 Point clouds 25893

4 Point clouds 56194

5 Unorganized, connectivity-free topological

6 Surface Reconstruction

7 Noise

8 Definition of “onto” Close? Which?

9 Applications Rendering Parameterization Simplification Reconstruction Area computation

10 References An extension on robust directed projection of points onto point clouds Ming-Cui Du, Yu-Shen Liu (CAD, In press) Parameterization-free Projection for Geometry Reconstruction Yaron Lipman, Daniel Cohen-Or, David Levin, Hillel Tal-Ezer (SIGGRAPH ’07)

11 An extension on robust directed projection of points onto point clouds Ming-Cui Du, Yu-Shen Liu CAD, In press

12 About the author ( 刘玉身 ) Postdoctor of Purdue University, Ph.D. in Tsinghua University. 3 CAD, 1 The Visual Computer. CAD, DGP.

13

14 Result

15 Previous work Parameterization of clouds of unorganized points using dynamic base surfaces (CAD, 04) Drawing curves onto a cloud of points for point-based Modeling (CAD, 05) Automatic least-squares projection of points onto point clouds with applications in reverse engineering (CAD, 06)

16 Weighted squared distances error

17

18 Proposition Terminating criterion: Simple, direct

19 Error analysis (Robustness) True location Independent of the cloud of points

20 Improved weight distance between p m and the axis stability

21 Improved weight

22 Reduce cloud Setting the threshold: 1.

23

24 Reduce cloud Setting the threshold: 1. 2. Sort the weights in a decreasing order, then choose the nth weight as threshold. (n=N/100).

25 References Robust diagnostic regression analysis. Atkinson A, Riani M. (Springer;2000) Robust Moving Least-squares Fitting with Sharp Features Shachar Fleishman, Daniel Cohen-Or, Claudio T. Silva (SIGGRAPH ’05)

26 Forward vs. backward Backward: Start from the entire sample set, then delete bad samples. Forward: Begins with a small outlier-free subset, then refining by adding one good sample at a time. (robust) Adding of multiple points.

27 Algorithm 1. Choose a small outlier-free subset Q. 2. The solution is computed to the current subset Q. 3. The point with the lowest residual in the remaining points is added into Q. (Forward) 4. Repeat steps 2 and 3 until the error is larger than a predefined threshold. 5. Compute the projection position for the final Q.

28 Least median of squares

29 LMS algorithm

30 Random sampling algorithm

31 Robustness P: Probability of success. g: Probability of selecting good sample. k: Number of points are selected at random. (k = p) T: Number of iteration. (T = 1000)

32 Forward search

33

34

35

36 Disturbing points

37

38 Limitations

39 Use the first quartile (25%) instead of the median (50%)

40 Parameterization-free Projection for Geometry Reconstruction Yaron Lipman, Daniel Cohen-Or, David Levin, Hillel Tal-Ezer (SIGGRAPH ’07)

41 About the author ( Yaron Lipman) Ph.D. student at Tel-Aviv University. His supervisors are Prof. David Levin and Prof. Daniel Cohen-Or. SIGGRAPH, TOG, EG, SGP

42 About the author (Daniel Cohen-Or) Professor at the School of Computer Science, Tel Aviv University. Outstanding Technical Contributions Award 2005(EG) TOG(19), CGF,TVCG, SGP, VC

43 About the author (David Levin) Professor of Applied Mathematics, Tel-Aviv University. Major interests: Subdivision Moving Least Squares Numerical Integration CAGD Computer Graphics

44 About the author (David Levin) Professor of Applied Mathematics, Tel-Aviv University. Major interests: Subdivision Moving Least Squares Numerical Integration CAGD Computer Graphics

45 Results

46 Locally Optimal Projection (LOP) θ(r), η(r) are fast decreasing functions.

47 Regularization

48 Multivariate L 1 median

49 Optimization

50

51

52 The iterative LOP algorithm

53 Theorem If the data set P is sampled from a C 2 -smooth surface S, LOP operator has an O(h 2 ) approximation order to S, provided that Λ is carefully chosen.

54 Initial guess

55

56

57 Results

58 Parameters: h

59 Parameters: μ

60 Efficient simplification of point- sampled surfaces Mark Pauly, Markus Gross, Leif P. Kobbelt IEEE Visualization, 2002

61 Particle Simulation 1.Spreading Particles. 2.Repulsion.(SIG.92) 3.Projection.(MLS)

62 Thank you!

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