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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
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Point Set Surfaces
• Levin’s MLS formulation
Shape Modeling International 2008 – Stony Brook University
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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
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Neighborhood and Bandwidth
• Common practice
– Weight function: Exponential
– Neighborhood: Spherical
– Bandwidth: Heuristics
• Problems
– Optimality
– Anisotropic Dataset
Shape Modeling International 2008 – Stony Brook University
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Related Work
• Other MLS Formulations
Alexa et al.
Guennebaud et al.
• Robust Feature ExtractionFleishman et al.
• Bandwidth Determination
Adamson et al.
Lipman et al.
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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
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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
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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
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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
Shape Modeling International 2008 – Stony Brook University
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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
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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
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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
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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
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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
Shape Modeling International 2008 – Stony Brook University
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Optimal Bandwidth in 3-D• Kernel Function:
with
• Kernel Shape:
Shape Modeling International 2008 – Stony Brook University
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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
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Experiments
• Bandwidth selectors choose near optimal bandwidths
Shape Modeling International 2008 – Stony Brook University
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Experiments
Shape Modeling International 2008 – Stony Brook University
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Experiments
Shape Modeling International 2008 – Stony Brook University
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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
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Robustness
Shape Modeling International 2008 – Stony Brook University
Insensitivity to error in first step of Levin’s MLS
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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
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Comparison
• MSE/MISE-based plug-in method better than heuristic methods
Shape Modeling International 2008 – Stony Brook University
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Comparison
• Heuristic methods can produce visually acceptable but not geometrically accurate reconstruction.
Shape Modeling International 2008 – Stony Brook University
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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