流形上扩散、核平滑和小波的统一热核回归及其在CT图像下颌骨生长建模中的应用
Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images
摘要：
我们提出了一种新的核回归框架平滑标量表面数据使用Laplace-Beltrami特征函数。从特征函数构造的热核开始，我们将一个新的二元核回归框架表述为以热核为权重的加权特征函数展开。新的核回归在数学上等价于各向同性热扩散，核平滑和最近流行的扩散小波。与许多以前的偏微分方程为基础的方法，涉及扩散，我们的方法表示扩散解析的解决方案，减少数值不准确性和收敛速度慢。数值实现验证了一个单位球使用球谐函数。作为一个例子，我们已经应用该方法在表征的局部生长模式的下颌骨表面从0岁至20岁之间的受试者的CT图像中获得的位移向量的长度相对于模板表面回归。
英文摘要：
We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of displacement vectors with respect to the template surface.
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