Department of Computer Science and Engineering

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

Tamal K. Dey Gang Li Jian Sun (presenting)

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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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Plane fitting method [HDD*92]

cxnT

k

ii

T cpncne1

)(),(

1nnT

• Assume the best fitting plane at point p:

• Minimize the error term

under the constraint• Reduce to an eigenvalue problem:

k

i

T

ii pppp1

))((

4/10Department of Computer Science and Engineering

Weighted plane fitting method (WPF)[PKKG03]

)()(),(1

i

k

ii

T ppcpncne

• Observation: the best fitting plane should respect the nearby points than the distant points

• Define the error term:

• Weighting function: 2

2

)( h

pp

i

i

epp

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Adaptive plane fitting method (APF)[MNG04]

n

3/1221 ))(

1( n

n ccr

r

• Consider the points within a ball of radius

• Noise assumption mean: , standard deviation:

• An optimal radius

• Compute in an iterative manner

r

0

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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

pcr

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Normal lemmas

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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

er nn ,

rn

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Mean error plot

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Special Case I: uneven sampling

• Sample the surface densely along some curves

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Special Case II: the surface with high curvature

• A very thin ellipsoid

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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.