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

Estimating Surface Normals in Noisy Point Cloud Data

Niloy J. Mitra, An Nguyen

Stanford University

Normal Estimation for Noisy PCD Symposium on Computational Geometry

The Normal Estimation Problem Given

Noisy PCD sampled from a curve/surface


The Normal Estimation Problem Given

Noisy PCD sampled from a curve/surface

GoalCompute surface normals at each point p

Error bound the normal estimates


A Standard Solution

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


A Standard Solution

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

PROBLEM !! what neighborhood size to choose?


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.


Outline

Problem statement Related work Neighborhood Size Estimation

Analysis in 2D and 3D Applications

Future Work


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?


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

k

ii

T cpak 1

2)(1

Ti

k

ii pppp

kM

1

1

aTp=c


Deceptive Case

Collusive noise


Deceptive Cases

Collusive noise

Curvature effect


Outline

Problem statement Related work Neighborhood Size Estimation

Analysis in 2D and 3D Applications

Future Work


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


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.


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

xy

2r

O


Proof Idea

Eigen-analysis of covariance matrix

2221

1211

mm

mmM


Proof Idea

covariance matrix let,

=(|m12|+m22)/m11

2221

1211

mm

mmM


Proof Idea

covariance matrix let, =(|m12|+m22)/m11

error angle bounded by,

to bound estimation error, need to bound

2221

1211

mm

mmM

21

)1(

)1(tan


Bounding Terms of M

For evenly distributed samples it follows,

22

11 )1()(1

rxxk

m i

))(1()(1 2422

22 nryyk

m i

?|| 12 m

2221

1211

mm

mmM


Bounding m12

Evenly sampled distribution Noise and measurement are uncorrelated

E(xn)= E(x)E(n)= 0 Var(xn)= (1)r2n

2

Chebyshev Inequality bound with probability (1-)

Finally,

r

rm n)1()1(|| 312

i

ii xnEnxk

)(1

2221

1211

mm

mmM


Bounding Estimation Error

=(|m12|+m22)/m11

22

11 )1()(1

rxxk

m i

))(1()(1 2422

22 nryyk

m i

r

rm n)1()1(|| 312


Final Result in 2D

2

2

3)1()1()1(rr

r nn

• = 0,

take as large a neighborhood as possible


Final Result in 2D

2

2

3)1()1()1(rr

r nn

• = 0, take as large a neighborhood as possible

• n= 0

take as small a neighborhood as possible


Experiments in 2D

2

2

3)1()1( )1(rr

rerror nn


Result for 3D

222 /)1()/()1()1( rrrerror nn

A similar but involved analysis results in,

A good choice of r is,

3/1

221

1

n

n ccr


How can we use this result?

Need to know estimate suitable values for estimate locally

3/1

221

1

n

n ccr


Estimating c1, c2

Exact normals known at almost all points

3/1

221

1

n

n ccr

c1=1, c2=4

• same constants used for

following results


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 k saturates


Effect of Curvature on Neighborhood Size

1x noise


Effect of Noise on Neighborhood Size

2x noise1x noise


Estimation Error > 5

2x noise1x noise

o


Increasing Noise

1x noise 2x noise 4x noise

Can still get good estimates in flat areas


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


Different Noise Distribution (same variance)

uniform gaussian


Result: phone

1x noise


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

Documents

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