optimal bandwidth selection for mls surfaces hao wang carlos e. scheidegger claudio t. silva sci...

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

Shape Modeling International 2008 – Stony Brook University

Neighborhood and Bandwidth

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

Shape Modeling International 2008 – Stony Brook University

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

Shape Modeling International 2008 – Stony Brook University

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

Shape Modeling International 2008 – Stony Brook University

Evaluation of Kernel Regression

• MSE– MSE = Mean Squared Error

– Evaluate result of the functional fitting at each point

Shape Modeling International 2008 – Stony Brook University

Evaluation of Kernel Regression

Shape Modeling International 2008 – Stony Brook University

•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

Shape Modeling International 2008 – Stony Brook University

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

Shape Modeling International 2008 – Stony Brook University

Optimal Bandwidth in 3-D• Kernel Function:

with

• Kernel Shape:

Shape Modeling International 2008 – Stony Brook University

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

Shape Modeling International 2008 – Stony Brook University

Robustness

Shape Modeling International 2008 – Stony Brook University

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

Shape Modeling International 2008 – Stony Brook University

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