Projecting points onto a point cloud with noise

Speaker: Jun Chen

Mar 26, 2008

Data Acquisition

Point clouds

25893

Point clouds

56194

Unorganized, connectivity-free

topological

Surface Reconstruction

Noise

Definition of “onto”

Close? Which?

Applications

Rendering Parameterization Simplification Reconstruction Area computation

References

An extension on robust directed projection ofpoints onto point clouds

Ming-Cui Du, Yu-Shen Liu (CAD, In press)

Parameterization-free Projection for GeometryReconstruction Yaron Lipman, Daniel Cohen-Or, David Levin, Hillel Tal-Ezer (SIGGRAPH ’07)

An extension on robust directed projection of points onto point clouds

Ming-Cui Du, Yu-Shen Liu

CAD, In press

About the author ( 刘玉身 )

Postdoctor of Purdue University,

Ph.D. in Tsinghua University.

3 CAD, 1 The Visual Computer.

CAD, DGP .

Result

Previous work

Parameterization of clouds of unorganized points using

dynamic base surfaces (CAD, 04)

Drawing curves onto a cloud of points for point-based

Modeling (CAD, 05)

Automatic least-squares projection of points onto point

clouds with applications in reverse engineering (CAD, 06)

Weighted squared distances error

Weighted squared distances error

Proposition

Terminating criterion:

Simple, direct

Error analysis (Robustness)

True location

Independent of the cloud of points

Improved weight

distance between p

m and the axisstability

Improved weight

Reduce cloud

Setting the threshold: 1.

Reduce cloud

Setting the threshold: 1.

2. Sort the weights in a decreasing order, then choose the nth weight as threshold.

(n=N/100).

References

Robust diagnostic regression analysis. Atkinson A, Riani M. (Springer;2000)

Robust Moving Least-squares Fitting with Sharp Features Shachar Fleishman, Daniel Cohen-Or, Claudio T. Silva

(SIGGRAPH ’05)

Forward vs. backward

Backward: Start from the entire sample set, then delete bad samples.

Forward: Begins with a small outlier-free subset, then refining by adding one good sample at a time. (robust) Adding of multiple points.

Algorithm

1. Choose a small outlier-free subset Q. 2. The solution is computed to the current subset Q.

3. The point with the lowest residual in the remaining points is added into Q. (Forward)

4. Repeat steps 2 and 3 until the error is larger than a predefined threshold.

5. Compute the projection position for the final Q.

Least median of squares

( {1,7,2,5,3} 3)median

LMS algorithm

Random sampling algorithm

Robustness

P: Probability of success. g: Probability of selecting good sample. k: Number of points are selected at random.

(k = p) T: Number of iteration. (T = 1000)

Forward search

Disturbing points

Disturbing points

Limitations

Limitations

Use the first quartile (25%) instead of the median (50%)

Parameterization-free Projection for Geometry Reconstruction

Yaron Lipman, Daniel Cohen-Or, David Levin, Hillel Tal-Ezer

(SIGGRAPH ’07)

About the author (Yaron Lipman)

Ph.D. student at Tel-Aviv University. His supervisors are Prof. David Levin and Prof. Daniel Cohen-Or.

SIGGRAPH, TOG, EG, SGP

About the author (Daniel Cohen-Or)

Professor at the School of Computer Science, Tel Aviv University.

Outstanding Technical Contributions Award 2005(EG)

TOG(19), CGF,TVCG, SGP, VC

About the author (David Levin) Professor of Applied

Mathematics, Tel-Aviv University.

Major interests: Subdivision Moving Least Squares Numerical Integration CAGD Computer Graphics

About the author (David Levin) Professor of Applied

Mathematics, Tel-Aviv University.

Major interests: Subdivision Moving Least Squares Numerical Integration CAGD Computer Graphics

Results

Locally Optimal Projection (LOP)

1 2

1

2\{ }

( ) arg min { ( , , ) ( , )},

( , , ) ( ),

( , ) ( ) ( ).

X

i j i ji I j J

i i i i ii I i I i

Q G Q E X P Q E X Q

E X P Q x p q p

E X Q x q q q

θ(r), η(r) are fast decreasing functions.

Regularization

2

\{ }

\{ }

arg min ( , )

arg min ( ) ( ).

( ) ( ).

X

X i i i i ii I i I i

i i i i ii I i I i

Q E X Q

x q q q

q q q q

Multivariate L1 median

1arg min ( , , )

arg min ( ).

( )

X

X i j i ji I j J

i j i ji I j J

Q E X P Q

x p q p

q p q p

Optimization

1 2

\{ }

( , , ) ( , )

( )

( ) ( ) min.

i j i ji I j J

i i i i ii I i I i

E X P Q E X Q

q p q p

q q q q

Optimization

1 2| ( ( , , ) ( , )) 0X X Q E X P Q E X Q

The iterative LOP algorithm

Theorem

If the data set P is sampled from a C2-smooth surface S, LOP operator has an O(h2) approximation order to S , provided that Λ is carefully chosen.

Initial guess

Results

Parameters: h

Parameters: μ

Efficient simplification of point-sampled surfaces

Mark Pauly, Markus Gross, Leif P. Kobbelt

IEEE Visualization, 2002

Particle Simulation

1.Spreading Particles. 2.Repulsion.(SIG.92)

3.Projection.(MLS)

Thank you!