1. Estimating Surface Normals in Noisy Point Cloud Data Niloy J. Mitra, An Nguyen Stanford University

2. Symposium on Computational Geometry Normal Estimation for Noisy PCD The Normal Estimation Problem Given Noisy PCD sampled from a curve/surface

3. Symposium on Computational Geometry Normal Estimation for Noisy PCD The Normal Estimation Problem Given Noisy PCD sampled from a curve/surface Goal Compute surface normals at each point p Error bound the normal estimates

4. Symposium on Computational Geometry Normal Estimation for Noisy PCD A Standard Solution Use least square fit to a neighborhood of radius r around point p

5. Symposium on Computational Geometry Normal Estimation for Noisy PCD A Standard Solution Use least square fit to a neighborhood of radius r around point p PROBLEM !! what neighborhood size to choose?

6. Symposium on Computational Geometry Normal Estimation for Noisy PCD Contributions of this paper Study the effects of curvature, noise, sampling density on the choice of neighborhood size. Use this insight to choose an optimal neighborhood size. Compute bound on the estimation error.

7. Symposium on Computational Geometry Normal Estimation for Noisy PCD Outline Problem statement Related work Neighborhood Size Estimation  Analysis in 2D and 3D  Applications Future Work

8. Symposium on Computational Geometry Normal Estimation for Noisy PCD Related Work Surface reconstruction  crust, cocone, etc  Guarantees about the surface normals  Mostly works in absence of noise Curve/Surface fitting  pointShop3D, point-set  Works in presence of noise  Performance guarantees?

9. Symposium on Computational Geometry Normal Estimation for Noisy PCD Least Square Fit Assume best fit hyperplane: a T p=c Minimize Reduces to the eigen-analysis of the covariance matrix Smallest eigenvector of M is the estimate of the normal a T p=c

10. Symposium on Computational Geometry Normal Estimation for Noisy PCD Deceptive Case Collusive noise

11. Symposium on Computational Geometry Normal Estimation for Noisy PCD Deceptive Cases Collusive noise Curvature effect

12. Symposium on Computational Geometry Normal Estimation for Noisy PCD Outline Problem statement Related work Neighborhood Size Estimation  Analysis in 2D and 3D  Applications Future Work

13. Symposium on Computational Geometry Normal Estimation for Noisy PCD Assumptions Noise  Independent of measurement  Zero mean  Variance is known (noise need not be bounded) Data  Sampling criterion satisfied  Evenly distributed data To prevent biased estimates  Curvature is bounded

14. Symposium on Computational Geometry Normal Estimation for Noisy PCD Sampling Criteria (2D) Sampling density lower bound (like Nyquist rate) upper bound (to prevent biased fits) Evenly distributed Number of points in a disc of radius r bounded above and below by  (1)r  ( ,  ) sampling condition [Dey et. al.] implies evenly distributed.

15. Symposium on Computational Geometry Normal Estimation for Noisy PCD Modeling (2D) At a point O  Points of PCD inside a disc of radius r comes from a segment of the curve  y = g(x) define the curve for all x  [-r,r]  Bounded curvature: |g’’(x)|<  for all x  Additive Noise(n) in y-direction (x,g(x)+n)   r,  n /r assumed to be small x y 2r O

16. Symposium on Computational Geometry Normal Estimation for Noisy PCD Proof Idea Eigen-analysis of covariance matrix

17. Symposium on Computational Geometry Normal Estimation for Noisy PCD Proof Idea covariance matrix let,  =(|m 12 |+m 22 )/m 11

18. Symposium on Computational Geometry Normal Estimation for Noisy PCD Proof Idea covariance matrix let,  =(|m 12 |+m 22 )/m 11 error angle bounded by, to bound estimation error, need to bound 

19. Symposium on Computational Geometry Normal Estimation for Noisy PCD Bounding Terms of M For evenly distributed samples it follows,

20. Symposium on Computational Geometry Normal Estimation for Noisy PCD Bounding m 12 Evenly sampled distribution Noise and measurement are uncorrelated  E(xn)= E(x)E(n)= 0  Var(xn)=  (1)r 2  n 2 Chebyshev Inequality  bound with probability (1-  ) Finally,

21. Symposium on Computational Geometry Normal Estimation for Noisy PCD Bounding Estimation Error  =(|m 12 |+m 22 )/m 11

22. Symposium on Computational Geometry Normal Estimation for Noisy PCD Final Result in 2D  = 0, take as large a neighborhood as possible

23. Symposium on Computational Geometry Normal Estimation for Noisy PCD Final Result in 2D  = 0, take as large a neighborhood as possible  n = 0 take as small a neighborhood as possible

24. Symposium on Computational Geometry Normal Estimation for Noisy PCD Experiments in 2D

25. Symposium on Computational Geometry Normal Estimation for Noisy PCD Result for 3D A similar but involved analysis results in, A good choice of r is,

26. Symposium on Computational Geometry Normal Estimation for Noisy PCD How can we use this result? Need to  know  estimate suitable values for  estimate locally

27. Symposium on Computational Geometry Normal Estimation for Noisy PCD Estimating c 1, c 2 Exact normals known at almost all points c 1 =1, c 2 =4 same constants used for following results

28. Symposium on Computational Geometry Normal Estimation for Noisy PCD Algorithm For each point, start with k =15 Iterate and refine (maximum of 10 steps)  Compute r, ,  [Gumhold et al.] locally  Use them to compute r new  k new =  r new 2  old  Stop if k>threshold k saturates

29. Symposium on Computational Geometry Normal Estimation for Noisy PCD Effect of Curvature on Neighborhood Size 1x noise

30. Symposium on Computational Geometry Normal Estimation for Noisy PCD Effect of Noise on Neighborhood Size 2x noise 1x noise

31. Symposium on Computational Geometry Normal Estimation for Noisy PCD Estimation Error > 5 2x noise 1x noise o

32. Symposium on Computational Geometry Normal Estimation for Noisy PCD Increasing Noise 1x noise2x noise4x noise Can still get good estimates in flat areas

33. Symposium on Computational Geometry Normal Estimation for Noisy PCD Future Work How to find a suitable neighborhood size for good curvature estimation Find a better way for estimating c 1, c 2 Design of a sparse query data structure for quick extraction of normal, curvature, etc from PCDs

34. Symposium on Computational Geometry Normal Estimation for Noisy PCD Different Noise Distribution (same variance) uniform gaussian

35. Symposium on Computational Geometry Normal Estimation for Noisy PCD Result: phone 1x noise

36. Symposium on Computational Geometry Normal Estimation for Noisy PCD Varying neighborhood size Neighborhood size at all points being shown using color-coding. Purple denotes the smallest neighborhood and turns blue as the neighborhood size increases