Optimal Bandwidth Selection for MLS Surfaces

1. Hao Wang Carlos E. Scheidegger Claudio T. Silva SCI Institute – University of Utah Optimal Bandwidth Selection for MLS Surfaces Shape Modeling International 2008 – Stony Brook University

2. Point Set Surfaces • Levin’s MLS formulation Shape Modeling International 2008 – Stony Brook University

3. Neighborhood and Bandwidth • Three parameters in both steps of Levin’s MLS: • Weight function • Neighborhood • Bandwidth Overfitting Underfitting Shape Modeling International 2008 – Stony Brook University

4. Neighborhood and Bandwidth • Common practice • Weight function: Exponential • Neighborhood: Spherical • Bandwidth: Heuristics • Problems • Optimality • Anisotropic Dataset Shape Modeling International 2008 – Stony Brook University

5. Related Work • Other MLS Formulations Alexa et al. Guennebaud et al. • Robust Feature Extraction Fleishman et al. • Bandwidth Determination Adamson et al. Lipman et al.

6. 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

7. 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

8. 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

9. 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

10. 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

11. Evaluation of Kernel Regression • MISE • Integration of MSE over the domain • Evaluate the global performance of kernel regression Shape Modeling International 2008 – Stony Brook University

12. 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

13. 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

14. 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

15. Optimal Bandwidth in 3-D • Kernel Function: with • Kernel Shape: Shape Modeling International 2008 – Stony Brook University

16. Optimal Bandwidth in 3-D • Optimal spherical bandwidth based on MSE: • Optimal spherical bandwidth based on MISE: Shape Modeling International 2008 – Stony Brook University

17. Experiments • Bandwidth selectors choose near optimal bandwidths Shape Modeling International 2008 – Stony Brook University

18. Experiments Shape Modeling International 2008 – Stony Brook University

19. Experiments Shape Modeling International 2008 – Stony Brook University

20. 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

21. Robustness Insensitivity to error in first step of Levin’s MLS Shape Modeling International 2008 – Stony Brook University

22. 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

23. Comparison • MSE/MISE-based plug-in method better than heuristic methods Shape Modeling International 2008 – Stony Brook University

24. Comparison • Heuristic methods can produce visually acceptable but not geometrically accurate reconstruction. Shape Modeling International 2008 – Stony Brook University

25. 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