Unified heat kernel regression for diffusion, kernel smoothing and wavelets on manifolds and its application to mandible growth modeling in CT images

Abstract

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.

Characterizing mandibular growth using three-dimensional imaging techniques and anatomic landmarks

“…Such growth is in line with studies on craniofacial growth, where the 4-to 5 -year-long growth spurt during puberty tends to slow or cease around age 12–15 years for females and 15–17 years in males (Martinez-Maza et al, 2013; Bulygina et al, 2006; Nellhaus, 1968). …”

“…The application of a composite growth model, as outlined by Wang et al(2016), will likely overcome this limitation of the fourth-degree mixed-effect polynomial fit. Additionally, rendering developmental 3DCT mandible models can be used to quantify surface area growth in all planes (Chung, Qiu, Seo, & Vorperian, 2015). …”

“…Previous studies have primarily relied upon x-ray (Bulygina, Mitteroecker, &Aiello, 2006; Broadbent, Broadbent, &Golden, 1975) , lateral cephalograms (Jacob &Buschang, 2014; Broadbent et al, 1975, Walker & Kowalski, 1972)and thin plate spline analysis (Rosas &Bastir, 2002; Bulygina et al, 2006). Studies that have utilized computed tomography (CT) scans have either used geometric morphometrics to analyze sexual dimorphism of mandibular surfaces during postnatal development( Coquerelle et al, 2011), or have focused on the sexual dimorphism of the mandible up until puberty, as opposed to the growth and development over the entire adolescence period(Krarup , Darvann, Larsen, Marsh, & Kreiborg, 2005).…”

“…Studies that have utilized computed tomography (CT) scans have either used geometric morphometrics to analyze sexual dimorphism of mandibular surfaces during postnatal development( Coquerelle et al, 2011), or have focused on the sexual dimorphism of the mandible up until puberty, as opposed to the growth and development over the entire adolescence period(Krarup , Darvann, Larsen, Marsh, & Kreiborg, 2005). Although Bulyginaet al (2006) and Krarup et al (2005) included three-dimensional analyses, limited quantitative measurements were derived from their 3D data. Quantifying the typical three -dimensional growth pattern and growth rate of the mandible would help establish a normative reference and range of variability for typical growth (Björk, 1969) and would help characterize the developmental changes in sexual dimorphism.…”

Online Statistical Inference for Large-Scale Binary Images

“…Such topological defects can be easily patched by Gaussian kernel smoothing without resorting to advanced topology correction methods (Fig. 1b) [5]. …”