3,249 research outputs found
Gaussian process regression can turn non-uniform and undersampled diffusion MRI data into diffusion spectrum imaging
We propose to use Gaussian process regression to accurately estimate the
diffusion MRI signal at arbitrary locations in q-space. By estimating the
signal on a grid, we can do synthetic diffusion spectrum imaging:
reconstructing the ensemble averaged propagator (EAP) by an inverse Fourier
transform. We also propose an alternative reconstruction method guaranteeing a
nonnegative EAP that integrates to unity. The reconstruction is validated on
data simulated from two Gaussians at various crossing angles. Moreover, we
demonstrate on non-uniformly sampled in vivo data that the method is far
superior to linear interpolation, and allows a drastic undersampling of the
data with only a minor loss of accuracy. We envision the method as a potential
replacement for standard diffusion spectrum imaging, in particular when
acquistion time is limited.Comment: 5 page
Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition
Hankel tensors arise from applications such as signal processing. In this
paper, we make an initial study on Hankel tensors. For each Hankel tensor, we
associate it with a Hankel matrix and a higher order two-dimensional symmetric
tensor, which we call the associated plane tensor. If the associated Hankel
matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel
tensor. We show that an order -dimensional tensor is a Hankel tensor if
and only if it has a Vandermonde decomposition. We call a Hankel tensor a
complete Hankel tensor if it has a Vandermonde decomposition with positive
coefficients. We prove that if a Hankel tensor is copositive or an even order
Hankel tensor is positive semi-definite, then the associated plane tensor is
copositive or positive semi-definite, respectively. We show that even order
strong and complete Hankel tensors are positive semi-definite, the Hadamard
product of two strong Hankel tensors is a strong Hankel tensor, and the
Hadamard product of two complete Hankel tensors is a complete Hankel tensor. We
show that all the H-eigenvalue of a complete Hankel tensors (maybe of odd
order) are nonnegative. We give some upper bounds and lower bounds for the
smallest and the largest Z-eigenvalues of a Hankel tensor, respectively.
Further questions on Hankel tensors are raised
Nonnegative Definite EAP and ODF Estimation via a Unified Multi-Shell HARDI Reconstruction
International audienceIn High Angular Resolution Diffusion Imaging (HARDI), Orientation Distribution Function (ODF) and Ensemble Average Propagator (EAP) are two important Probability Density Functions (PDFs) which reflect the water diffusion and fiber orientations. Spherical Polar Fourier Imaging (SPFI) is a recent model-free multi-shell HARDI method which estimates both EAP and ODF from the diffusion signals with multiple b values. As physical PDFs, ODFs and EAPs are nonnegative definite respectively in their domains S^2 and R^3 . However, existing ODF / EAP estimation methods like SPFI seldom consider this natural constraint. Although some works considered the nonnegative constraint on the given discrete samples of ODF / EAP, the estimated ODF/EAP is not guaranteed to be nonnegative definite in the whole continuous domain. The Riemannian framework for ODFs and EAPs has been proposed via the square root parameterization based on pre-estimated ODFs and EAPs by other methods like SPFI. However, there is no work on how to estimate the square root of ODF / EAP called as the wavefuntion directly from diffusion signals. In this paper, based on the Riemannian framework for ODFs / EAPs and Spherical Polar Fourier (SPF) basis representation, we propose a unified model-free multi-shell HARDI method, named as Square Root Parameterized Estimation (SRPE), to simultaneously estimate both the wavefunction of EAPs and the nonnegative definite ODFs and EAPs from diffusion signals. The experiments on synthetic data and real data showed SRPE is more robust to noise and has better EAP reconstruction than SPFI, especially for EAP profiles at large radius
Stochastic models associated to a Nonlocal Porous Medium Equation
The nonlocal porous medium equation considered in this paper is a degenerate
nonlinear evolution equation involving a space pseudo-differential operator of
fractional order. This space-fractional equation admits an explicit,
nonnegative, compactly supported weak solution representing a probability
density function. In this paper we analyze the link between isotropic transport
processes, or random flights, and the nonlocal porous medium equation. In
particular, we focus our attention on the interpretation of the weak solution
of the nonlinear diffusion equation by means of random flights.Comment: Published at https://doi.org/10.15559/18-VMSTA112 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
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