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Normal Estimation for Point Clouds: A Comparison Study for a Voronoi Based Method

Normal Estimation for Point Clouds: A Comparison Study for a Voronoi Based Method. Tamal K. Dey Gang Li Jian Sun (presenting). The normal estimation problem and some existing methods. Problem:

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Normal Estimation for Point Clouds: A Comparison Study for a Voronoi Based Method

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  1. Normal Estimation for Point Clouds: A Comparison Study for a Voronoi Based Method Tamal K. Dey Gang Li Jian Sun (presenting)

  2. The normal estimation problem and some existing methods • Problem: given a possibly noisy point cloud sampled from a surface, estimate the surface normals from input points • Methods: • Numerical methods: plane fitting [HDD*92] and its variations [PKKG03][MNG04] • Combinatorial methods: Voronoi based [AB99] [DG04, DS05]

  3. Plane fitting method [HDD*92] • Assume the best fitting plane at point p: • Minimize the error term under the constraint • Reduce to an eigenvalue problem:

  4. Weighted plane fitting method (WPF)[PKKG03] • Observation: the best fitting plane should respect the nearby points than the distant points • Define the error term: • Weighting function:

  5. Adaptive plane fitting method (APF)[MNG04] • Consider the points within a ball of radius • Noise assumption mean: , standard deviation: • An optimal radius • Compute in an iterative manner

  6. Voronoi based method • Noise-free Point Cloud [AB99] • The line through p and its pole, the furthest Voronoi vertex of Voronoi cell of p, approximates the normal line at p • Noisy Point Cloud — Big Delaunay ball method (BDB) [DG04, DS05] • The line through p and its pole, the furthest Voronoi vertex of Voronoi cell of p whose dual Delaunay ball is “big”, approximates the normal at p • A Delaunay ball is big if

  7. Normal lemmas

  8. Experimental setup • Add noise to the original noise-free point cloud • The x, y and z components of the noise are independent and uniformly distributed • Noise level • Global scale: the amplitude is a factor (0, 0.005, 0.01, 0.02) of the largest side of the axis parallel bounding box • Local scale: the amplitude is a factor (0, 0.5, 1, 2) of the average distance to the five nearest neighbors • Compute a referential normal from the original noise-free point cloud • Estimation error = • Specially sampled point clouds

  9. Mean error plot

  10. Special Case I: uneven sampling • Sample the surface densely along some curves

  11. Special Case II: the surface with high curvature • A very thin ellipsoid

  12. Summary • In case where the noise level is low, all three methods works almost equally well though WPF gives the best result. • In case where the noise level is high or the sample is skewed along some curves, BDB method gives the best result. • Timing • When #pts ~ million, BDB is safer to use. Otherwise WPF or APF is preferred.

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