247 research outputs found
Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images
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.Comment: Accepted in Medical Image Analysi
Embedding of Functional Human Brain Networks on a Sphere
Human brain activity is often measured using the blood-oxygen-level dependent
(BOLD) signals obtained through functional magnetic resonance imaging (fMRI).
The strength of connectivity between brain regions is then measured and
represented as Pearson correlation matrices. As the number of brain regions
increases, the dimension of matrix increases. It becomes extremely cumbersome
to even visualize and quantify such weighted complete networks. To remedy the
problem, we propose to embedded brain networks onto a hypersphere, which is a
Riemannian manifold with constant positive curvature
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