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
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The Normal Estimation Problem

• GivenNoisy PCD sampled from a curve/surface

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The Normal Estimation Problem

• GivenNoisy PCD sampled from a curve/surface
• GoalCompute surface normals at each point p
• Error bound the normal estimates

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A Standard Solution

• Use least square fit to a neighborhood of
• radius r around point p

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A Standard Solution

• Use least square fit to a neighborhood of
• radius r around point p
• PROBLEM !! what neighborhood size to choose?

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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.

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Outline

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

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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?

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Least Square Fit

• Assume
• best fit hyperplane aTpc
• Minimize
• Reduces to the eigen-analysis ofthe covariance
  matrix
• Smallest eigenvector of M is the estimate of the
  normal

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Deceptive Case
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Deceptive Cases
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Outline

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

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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

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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.
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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)lt? for all x
• Additive Noise(n) in y-direction (x,g(x)n)
• ?r, ?n/r assumed to be small

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Proof Idea

• Eigen-analysis of covariance matrix

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Proof Idea

• covariance matrix
• let, ?(m12m22)/m11

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Proof Idea

• covariance matrix
• let, ?(m12m22)/m11
• error angle bounded by,
• to bound estimation error, need to bound ?

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Bounding Terms of M

• For evenly distributed samples it follows,

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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,

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Bounding Estimation Error

• ?(m12m22)/m11

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Final Result in 2D

• ? 0,
• take as large a neighborhood as possible

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Final Result in 2D

• ? 0, take as large a neighborhood as possible
• ?n 0
• take as small a neighborhood as possible

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Experiments in 2D
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Result for 3D

• A similar but involved analysis results in,
• A good choice of r is,

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How can we use this result?

• Need to
• know
• estimate suitable values for
• estimate locally

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Estimating c1, c2
Exact normals known at almost all points

• c11, c24
• same constants used for
• following results

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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
• kgtthreshold
• k saturates

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Effect of Curvature on Neighborhood Size
1x noise
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Effect of Noise on Neighborhood Size
2x noise
1x noise
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Estimation Error gt 5
o
2x noise
1x noise
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Increasing Noise
Can still get good estimates in flat areas
1x noise
2x noise
4x noise
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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

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Different Noise Distribution (same variance)
gaussian
uniform
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Result phone
1x noise
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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