Optimal Bandwidth Selection for MLS Surfaces

Hao Wang

Carlos E. Scheidegger

Claudio T. Silva

SCI Institute – University of Utah

Shape Modeling International 2008 – Stony Brook University

Point Set Surfaces

• Levin’s MLS formulation

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Neighborhood and Bandwidth

• Three parameters in both steps of Levin’s MLS:– Weight function– Neighborhood– Bandwidth

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Overfitting Underfitting

Neighborhood and Bandwidth

• Common practice

– Weight function: Exponential

– Neighborhood: Spherical

– Bandwidth: Heuristics

• Problems

– Optimality

– Anisotropic Dataset

Shape Modeling International 2008 – Stony Brook University

Related Work

• Other MLS Formulations

Alexa et al.

Guennebaud et al.

• Robust Feature ExtractionFleishman et al.

• Bandwidth Determination

Adamson et al.

Lipman et al.

Locally Weighted Kernel Regression

• Problem– Points sampled from functional with white noise added– White noise are i.i.d. random variables– Reconstruct the functional with least squares criterion

• Approach– Consider each point p individually– p is reconstructed by utilizing information of its neighborhood– Influence of each neighboring point is related to its distance

from p

Shape Modeling International 2008 – Stony Brook University

Kernel Regression v.s MLS Surfaces

• Kernel Regression is mostly the same as the second step in Levin’s MLS.

• The only difference is between kernel weighting and MLS weighting.

Shape Modeling International 2008 – Stony Brook University

Kernel Regression v.s MLS Surfaces

• Difference– Kernel weighting for functional data

– MLS weighting for manifold data

• Advantages of Kernel Regression– More mature technique for processing noisy sample points

– Behavior of the neighborhood and kernel better studied

• Goal– Adapt techniques in kernel regression to MLS surfaces – Extend theoretical results of kernel regression to MLS surfaces

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Weight Function

• Common choices of weight functions in kernel regression:– Epanechnikov– Normal– Biweight

• Optimal weight function: Epanechnikov

• Choice of weight function not important

• Implication:– Optimality

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Evaluation of Kernel Regression

• MSE– MSE = Mean Squared Error

– Evaluate result of the functional fitting at each point

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Evaluation of Kernel Regression

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•MISE

–Integration of MSE over the domain

–Evaluate the global performance of kernel regression

Optimal Bandwidth

• Optimality– Leading to minimum MSE / MISE

– Each point with a different optimal bandwidth

• Computation– MSE / MISE approximated by Taylor Polynomial

– Solve for the minimizing bandwidth

Shape Modeling International 2008 – Stony Brook University

Optimal Bandwidth

• Unknown quantities in computation– Derivatives of underlying functional

– Variance of random noise variables

– Density of point set

• Approach– Derivatives: Ordinary Least Squares Fitting

– Variance: Statistical Inference

– Density: Kernel Density Estimation

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Optimal Bandwidth in 2-D

• Optimal bandwidth based on MSE:

• Interpretation– Higher noise level : larger bandwidth– Higher curvature : smaller bandwidth– Higher density : smaller bandwidth– More point samples : smaller bandwidth

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Optimal Bandwidth in 3-D• Kernel Function:

with

• Kernel Shape:

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Optimal Bandwidth in 3-D

• Optimal spherical bandwidth based on MSE:

• Optimal spherical bandwidth based on MISE:

Shape Modeling International 2008 – Stony Brook University

Experiments

• Bandwidth selectors choose near optimal bandwidths

Shape Modeling International 2008 – Stony Brook University

Experiments

Shape Modeling International 2008 – Stony Brook University

Experiments

Shape Modeling International 2008 – Stony Brook University

Optimal Bandwidth in MLS

• From functional domain to manifold domain

– Choose a functional domain

– Use kernel regression with modification

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Robustness

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Insensitivity to error in first step of Levin’s MLS

Comparison

• Constant h: uniform v.s non-uniform sampling

• k-NN: sampling v.s feature

• MSE/MISE based plug-in method: most robust and flexible

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Comparison

• MSE/MISE-based plug-in method better than heuristic methods

Shape Modeling International 2008 – Stony Brook University

Comparison

• Heuristic methods can produce visually acceptable but not geometrically accurate reconstruction.

Shape Modeling International 2008 – Stony Brook University

Future Work

• Nonlinear kernel regression bandwidth selector in 3-D

• Compute optimal bandwidth implicitly

• Extend the method to other MLS formulations

Shape Modeling International 2008 – Stony Brook University