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