robust moving least-squares fitting with sharp features
DESCRIPTION
Robust Moving Least-squares Fitting with Sharp Features. Shachar Fleishman* Daniel Cohen-Or § Claudio T. Silva*. * University of Utah § Tel-Aviv university. Surface reconstruction. Noise Smooth surface Smooth sharp features - PowerPoint PPT PresentationTRANSCRIPT
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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
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Surface reconstructionSurface reconstruction
• Noise
• Smooth surface
• Smooth sharp features
• Method for identifying and reconstructing sharp features
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Point set surfaces (Levin ’98)Point set surfaces (Levin ’98)
• Defines a smooth surface using a projection operator
)(' xPx
x
'x
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Point set surfacesPoint set surfaces
• Defines a smooth surface using a projection operator
• Noisy point set
• The surface S is defined:
)(| xPxx
)(' xPx
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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
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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
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Sharp featuresSharp features
• Smoothed out
• Ambiguous
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Sharp featuresSharp features
• Smoothed out
• Ambiguous
– Classify
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Projection near sharp featureProjection near sharp feature
)(' xPx
'x
x
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Projection near sharp featureProjection near sharp feature
)(' xPx 'x
x
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Projection near sharp featureProjection near sharp feature
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ClassificationClassification
Using outlier identification algorithm
That fits a polynomial patch to a neighborhood
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ClassificationClassification
Using outlier identification algorithm
That fits a polynomial patch to a neighborhood
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Statistics 101Statistics 101
• Find the center of a set of points
xmean
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Statistics 101Statistics 101
• Find the center of a set of points
• Robustly using median
xmeanmedian
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Regression with backward searchRegression with backward search
• Loop
– Fit a model
– Remove point withmaximal residual
• Until no more outliers x
y
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Regression with backward searchRegression with backward search
• Outliers can have a significant influence of the fitted model
x
y
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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
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Monitoring the forward searchMonitoring the forward search
x
y
samples#
residualsResidual plot
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Monitoring the forward searchMonitoring the forward search
samples#
residualsResidual plot
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ResultsResults
Polynomial fit allows reconstruction of curved edges
Input with missing data
Reconstructed
and corners
Smooth MLS
MLS w. edges
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ResultsResults
Noisy input Reconstructed
input smooth sharp
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ResultsResults
Outliers are ignored Misaligned regions are determined to be two regions
Local decision may cause inconsistencies
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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
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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