6,021 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
3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries
Recent advances in electron microscopy have enabled the imaging of single
cells in 3D at nanometer length scale resolutions. An uncharted frontier for in
silico biology is the ability to simulate cellular processes using these
observed geometries. Enabling such simulations requires watertight meshing of
electron micrograph images into 3D volume meshes, which can then form the basis
of computer simulations of such processes using numerical techniques such as
the Finite Element Method. In this paper, we describe the use of our recently
rewritten mesh processing software, GAMer 2, to bridge the gap between poorly
conditioned meshes generated from segmented micrographs and boundary marked
tetrahedral meshes which are compatible with simulation. We demonstrate the
application of a workflow using GAMer 2 to a series of electron micrographs of
neuronal dendrite morphology explored at three different length scales and show
that the resulting meshes are suitable for finite element simulations. This
work is an important step towards making physical simulations of biological
processes in realistic geometries routine. Innovations in algorithms to
reconstruct and simulate cellular length scale phenomena based on emerging
structural data will enable realistic physical models and advance discovery at
the interface of geometry and cellular processes. We posit that a new frontier
at the intersection of computational technologies and single cell biology is
now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies
available upon reques
A new nonlocal nonlinear diffusion equation for image denoising and data analysis
In this paper we introduce and study a new feature-preserving nonlinear
anisotropic diffusion for denoising signals. The proposed partial differential
equation is based on a novel diffusivity coefficient that uses a nonlocal
automatically detected parameter related to the local bounded variation and the
local oscillating pattern of the noisy input signal. We provide a mathematical
analysis of the existence of the solution of our nonlinear and nonlocal
diffusion equation in the two dimensional case (images processing). Finally, we
propose a numerical scheme with some numerical experiments which demonstrate
the effectiveness of the new method
Mathematical Modeling of the Human Brain
This open access book bridges common tools in medical imaging and neuroscience with the numerical solution of brain modelling PDEs. The connection between these areas is established through the use of two existing tools, FreeSurfer and FEniCS, and one novel tool, the SVM-Tk, developed for this book. The reader will learn the basics of magnetic resonance imaging and quickly proceed to generating their first FEniCS brain meshes from T1-weighted images. The book's presentation concludes with the reader solving a simplified PDE model of gadobutrol diffusion in the brain that incorporates diffusion tensor images, of various resolution, and complex, multi-domain, variable-resolution FEniCS meshes with detailed markings of anatomical brain regions. After completing this book, the reader will have a solid foundation for performing patient-specific finite element simulations of biomechanical models of the human brain
Mathematical Modeling of the Human Brain
This open access book bridges common tools in medical imaging and neuroscience with the numerical solution of brain modelling PDEs. The connection between these areas is established through the use of two existing tools, FreeSurfer and FEniCS, and one novel tool, the SVM-Tk, developed for this book. The reader will learn the basics of magnetic resonance imaging and quickly proceed to generating their first FEniCS brain meshes from T1-weighted images. The book's presentation concludes with the reader solving a simplified PDE model of gadobutrol diffusion in the brain that incorporates diffusion tensor images, of various resolution, and complex, multi-domain, variable-resolution FEniCS meshes with detailed markings of anatomical brain regions. After completing this book, the reader will have a solid foundation for performing patient-specific finite element simulations of biomechanical models of the human brain
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