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Estimating Surface Normals in Noisy Point Cloud Data

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Estimating Surface Normals in Noisy Point Cloud Data. Niloy J. Mitra, An Nguyen. Stanford University. The Normal Estimation Problem. Given Noisy PCD sampled from a curve/surface. The Normal Estimation Problem. Given Noisy PCD sampled from a curve/surface

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Estimating Surface Normals in Noisy Point Cloud Data

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

  2. The Normal Estimation Problem • GivenNoisy PCD sampled from a curve/surface

  3. The Normal Estimation Problem • GivenNoisy PCD sampled from a curve/surface • GoalCompute surface normals at each point p Error bound the normal estimates

  4. A Standard Solution Use least square fit to a neighborhood of radius r around point p

  5. A Standard Solution Use least square fit to a neighborhood of radius r around point p PROBLEM !!what neighborhood size to choose?

  6. 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. Outline • Problem statement • Related work • Neighborhood Size Estimation • Analysis in 2D and 3D • Applications • Future Work

  8. 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. aTp=c Least Square Fit • Assume best fit hyperplane: aTp=c • Minimize • Reduces to the eigen-analysis ofthe covariance matrix • Smallest eigenvector of M is the estimate of the normal

  10. Collusive noise Deceptive Case

  11. Collusive noise Curvature effect Deceptive Cases

  12. Outline • Problem statement • Related work • Neighborhood Size Estimation • Analysis in 2D and 3D • Applications • Future Work

  13. 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. 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. O y x 2r 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

  16. Proof Idea • Eigen-analysis of covariance matrix

  17. Proof Idea • covariance matrix • let, =(|m12|+m22)/m11

  18. Proof Idea • covariance matrix • let, =(|m12|+m22)/m11 • error angle bounded by, • to bound estimation error, need to bound 

  19. Bounding Terms of M • For evenly distributed samples it follows,

  20. Bounding m12 • Evenly sampled distribution • Noise and measurement are uncorrelated • E(xn)= E(x)E(n)= 0 • Var(xn)= (1)r2n2 • Chebyshev Inequality • bound with probability (1-) • Finally,

  21. Bounding Estimation Error =(|m12|+m22)/m11

  22. Final Result in 2D •  = 0, • take as large a neighborhood as possible

  23. Final Result in 2D •  = 0, take as large a neighborhood as possible • n= 0 • take as small a neighborhood as possible

  24. Experiments in 2D

  25. Result for 3D A similar but involved analysis results in, A good choice of r is,

  26. How can we use this result? • Need to • know • estimate suitable values for • estimate locally

  27. Estimating c1, c2 Exact normals known at almost all points • c1=1, c2=4 • same constants used for • following results

  28. 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 rnew • knew = rnew2 old • Stop if • k>threshold • ksaturates

  29. Effect of Curvature on Neighborhood Size 1x noise

  30. Effect of Noise on Neighborhood Size 2x noise 1x noise

  31. Estimation Error > 5 o 2x noise 1x noise

  32. Increasing Noise Can still get good estimates in flat areas 1x noise 2x noise 4x noise

  33. Future Work • How to find a suitable neighborhood size for good curvature estimation • Find a better way for estimating c1, c2 • Design of a sparse query data structure for quick extraction of normal, curvature, etc from PCDs

  34. Different Noise Distribution (same variance) gaussian uniform

  35. Result: phone 1x noise

  36. 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

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