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Department of Computer Science and Engineering Normal Estimation for Point Clouds: A Comparison Study for a Voronoi Based Method Tamal K. DeyGang LiJian Sun (presenting)

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2/10 Department of Computer Science and Engineering 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]

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3/10 Department of Computer Science and Engineering 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:

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4/10 Department of Computer Science and Engineering 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:

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5/10 Department of Computer Science and Engineering 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

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6/10 Department of Computer Science and Engineering 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

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7/10 Department of Computer Science and Engineering Normal lemmas

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8/10 Department of Computer Science and Engineering 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

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9/10 Department of Computer Science and Engineering Mean error plot

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10/10 Department of Computer Science and Engineering Standard deviation and timing for global scale noise

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11/10 Department of Computer Science and Engineering Standard deviation and timing for local scale noise

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12/10 Department of Computer Science and Engineering Special Case I: uneven sampling Sample the surface densely along some curves

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13/10 Department of Computer Science and Engineering Special Case II: the surface with high curvature A very thin ellipsoid

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14/10 Department of Computer Science and Engineering 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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Chapter 3 (continued) Nutan S. Mishra. Exercises 3.11-3.15 Size of the data set = 12 for all the five problems In 3.11 variable x 1 = monthly rent of.

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