Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

摘要

We present a novel kernel regression framework for smoothing scalar surfacedata using the Laplace-Beltrami eigenfunctions. Starting with the heat kernelconstructed from the eigenfunctions, we formulate a new bivariate kernelregression framework as a weighted eigenfunction expansion with the heat kernelas the weights. The new kernel regression is mathematically equivalent toisotropic heat diffusion, kernel smoothing and recently popular diffusionwavelets. Unlike many previous partial differential equation based approachesinvolving diffusion, our approach represents the solution of diffusionanalytically, reducing numerical inaccuracy and slow convergence. The numericalimplementation is validated on a unit sphere using spherical harmonics. As anillustration, we have applied the method in characterizing the localized growthpattern of mandible surfaces obtained in CT images from subjects between ages 0and 20 years by regressing the length of displacement vectors with respect tothe template surface.