Projecting points onto a point cloud with noise Speaker: Jun Chen Mar 26, 2008.
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Transcript of Projecting points onto a point cloud with noise Speaker: Jun Chen Mar 26, 2008.
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Projecting points onto a point cloud with noise
Speaker: Jun Chen
Mar 26, 2008
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Data Acquisition
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Point clouds
25893
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Point clouds
56194
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Unorganized, connectivity-free
topological
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Surface Reconstruction
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Noise
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Definition of “onto”
Close? Which?
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Applications
Rendering Parameterization Simplification Reconstruction Area computation
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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)
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An extension on robust directed projection of points onto point clouds
Ming-Cui Du, Yu-Shen Liu
CAD, In press
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About the author ( 刘玉身 )
Postdoctor of Purdue University,
Ph.D. in Tsinghua University.
3 CAD, 1 The Visual Computer.
CAD, DGP .
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Result
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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)
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Weighted squared distances error
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Weighted squared distances error
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Proposition
Terminating criterion:
Simple, direct
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Error analysis (Robustness)
True location
Independent of the cloud of points
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Improved weight
distance between p
m and the axisstability
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Improved weight
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Reduce cloud
Setting the threshold: 1.
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Reduce cloud
Setting the threshold: 1.
2. Sort the weights in a decreasing order, then choose the nth weight as threshold.
(n=N/100).
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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)
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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.
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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.
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Least median of squares
( {1,7,2,5,3} 3)median
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LMS algorithm
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Random sampling algorithm
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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)
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Forward search
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Disturbing points
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Disturbing points
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Limitations
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Limitations
Use the first quartile (25%) instead of the median (50%)
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Parameterization-free Projection for Geometry Reconstruction
Yaron Lipman, Daniel Cohen-Or, David Levin, Hillel Tal-Ezer
(SIGGRAPH ’07)
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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
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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
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About the author (David Levin) Professor of Applied
Mathematics, Tel-Aviv University.
Major interests: Subdivision Moving Least Squares Numerical Integration CAGD Computer Graphics
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About the author (David Levin) Professor of Applied
Mathematics, Tel-Aviv University.
Major interests: Subdivision Moving Least Squares Numerical Integration CAGD Computer Graphics
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Results
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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.
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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
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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
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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
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Optimization
1 2| ( ( , , ) ( , )) 0X X Q E X P Q E X Q
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The iterative LOP algorithm
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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.
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Initial guess
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Results
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Parameters: h
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Parameters: μ
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Efficient simplification of point-sampled surfaces
Mark Pauly, Markus Gross, Leif P. Kobbelt
IEEE Visualization, 2002
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Particle Simulation
1.Spreading Particles. 2.Repulsion.(SIG.92)
3.Projection.(MLS)
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Thank you!