Robust Moving Least-squares Fitting with Sharp Features
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Robust Moving Least-squares Fitting with Sharp FeaturesRobust Moving Least-squares Fitting with Sharp Features
Shachar Fleishman*
Daniel Cohen-Or§
Claudio T. Silva*
* University of Utah § Tel-Aviv university
Surface reconstructionSurface reconstruction
• Noise
• Smooth surface
• Smooth sharp features
• Method for identifying and reconstructing sharp features
Point set surfaces (Levin ’98)Point set surfaces (Levin ’98)
• Defines a smooth surface using a projection operator
)(' xPx
x
'x
Point set surfacesPoint set surfaces
• Defines a smooth surface using a projection operator
• Noisy point set
• The surface S is defined:
)(| xPxx
)(' xPx
The MLS projection: overviewThe MLS projection: overview
• Find a point q on the surfaces whose normal goes through the projected point x
• q is the projection of x
The MLS projection: overviewThe MLS projection: overview
• Find a point q on the surfaces whose normal goes through the projected point x
• q is the projection of x
• Improve approximation order using polynomial fit
'x
Sharp featuresSharp features
• Smoothed out
• Ambiguous
Sharp featuresSharp features
• Smoothed out
• Ambiguous
– Classify
Projection near sharp featureProjection near sharp feature
)(' xPx
'x
x
Projection near sharp featureProjection near sharp feature
)(' xPx 'x
x
Projection near sharp featureProjection near sharp feature
ClassificationClassification
Using outlier identification algorithm
That fits a polynomial patch to a neighborhood
ClassificationClassification
Using outlier identification algorithm
That fits a polynomial patch to a neighborhood
Statistics 101Statistics 101
• Find the center of a set of points
xmean
Statistics 101Statistics 101
• Find the center of a set of points
• Robustly using median
xmeanmedian
Regression with backward searchRegression with backward search
• Loop
– Fit a model
– Remove point withmaximal residual
• Until no more outliers x
y
Regression with backward searchRegression with backward search
• Outliers can have a significant influence of the fitted model
x
y
Regression with forward search (Atkinson and Riani)Regression with forward search (Atkinson and Riani)
• Start with an initial good but crude surface
– LMS (least median of squares)
• Incrementally improve the fit
• Monitor the search x
y
Monitoring the forward searchMonitoring the forward search
x
y
samples#
residualsResidual plot
Monitoring the forward searchMonitoring the forward search
samples#
residualsResidual plot
ResultsResults
Polynomial fit allows reconstruction of curved edges
Input with missing data
Reconstructed
and corners
Smooth MLS
MLS w. edges
ResultsResults
Noisy input Reconstructed
input smooth sharp
ResultsResults
Outliers are ignored Misaligned regions are determined to be two regions
Local decision may cause inconsistencies
SummarySummary
• Classification of noisy point sets to smooth regions
• Application to PSS
– Reconstruct surfaces with sharp features from noisy data
– Improve the stability of the projection
• Local decisions may result different neighborhoods for adjacent points
• Can be applied to other surface reconstruction methods such as the MPU
AcknowledgementsAcknowledgements
• Department of Energy under the VIEWS program and the MICS office
• The National Science Foundation under grants CCF-0401498, EIA-0323604, and OISE-0405402
• A University of Utah Seed Grant
• The Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities), and the Israeli Ministry of Science