Bessel Functions of Matrix Argument

Herz, Carl

doi:10.2307/1969810

Cited by 417 publications

(274 citation statements)

References 14 publications

Supporting

Mentioning

266

Contrasting

Unclassified

Order By: Relevance

“…The Laplace transform of f : n → , denoted by f, at the symmetric matrix Z∈ n×n is defined by Herz (1955): (17) For a sufficiently nice function f, the integral above converges in the right half plane, Re(Z) >X 0 (Re(Z) denotes the real part of Z), meaning that Re(Z) − X 0 ∈ n and the inversion formula for this Laplace transform is: (18) Here dZ = Πdz ij and the integral is over symmetric matrices Z with fixed real part.…”

Section: Definitionmentioning

confidence: 99%

A novel tensor distribution model for the diffusion-weighted MR signal

et al. 2007

View full text Add to dashboard Cite

Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

show abstract

Section: Definitionmentioning

confidence: 99%

A novel tensor distribution model for the diffusion-weighted MR signal

et al. 2007

View full text Add to dashboard Cite

show abstract

“…with 1 F 1 a confluent hypergeometric function of matrix argument as defined by [12]. Thus, the probability concentration is controlled by the eigenvalues of Z, which might be interpreted as concentration parameters.…”

Section: The Bingham Distributionmentioning

confidence: 99%

On the use of the Bingham statistical distribution in microsphere-based constitutive models for arterial tissue

Alastrué

Saéz

Martı́nez

et al. 2010

Mechanics Research Communications

View full text Add to dashboard Cite

Constitutive models for arterial tissue have been an active research field during the last years. The main micro-constituents of blood vessels are different types of cells and the extra-cellular matrix formed by an isotropic high water content ground substance and a network composed of elastin and collagen fibres. Usually the arterial tissue has been modelled as a hyperelastic material within the framework of continuum mechanics, whereas inclusion of structural tensors into constitutive laws is the most widely used technique to introduce the anisotropy induced by the fibres. Though the different existing fibre bundles present a clear preferential direction, the dispersion inherent to biological tissue advices using of constitutive models including representative structural information associated to the spatial probabilistic distribution of the fibres. Lately, microsphere-based models have demonstrated to be a powerful tool to incorporate this information. The fibre dispersion is incorporated by means of an Orientation Density Function (ODF) that weights the contribution of each fibre in each direction of the micro-sphere. In previous works the rotationally symmetric von Mises ODF was successfully applied to the modelling of blood vessels. In this study the inclusion of the Bingham ODF into microsphere-based model is analysed. This ODF exhibits some advantages with respect to the von Mises one, like a greater versatility and a comparable response to uniaxial and equibiaxial tension tests.

show abstract

“…for n → ∞ with 8) where P X denotes the law of a random variable X. Moreover, as Q is a isomorphism from…”

Section: Lemmamentioning

confidence: 99%

“…[14]). Moreover, in case (2), Bessel functions of matrix argument and the associated Bessel convolutions on matrix cones appear (see [6], [8], [16]), and in case (3), Jacobi functions and Jacobi convolutions on [0, ∞[ appear (see the survey [15]). Finally, in the compact case, spheres and projective spaces lead to Jacobi polynomials and Jacobi convolutions on [−1, 1], and the finite examples associated with Hamming schemes or Johnson schemes lead to Krawtchouk and Meixner polynomials respectively.…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for radial random walks on homogeneous spaces with growing dimensions

Voit

2008

Infinite Dimensional Harmonic Analysis IV

View full text Add to dashboard Cite

Bessel Functions of Matrix Argument

Cited by 417 publications

References 14 publications

A novel tensor distribution model for the diffusion-weighted MR signal

A novel tensor distribution model for the diffusion-weighted MR signal

On the use of the Bingham statistical distribution in microsphere-based constitutive models for arterial tissue

Limit theorems for radial random walks on homogeneous spaces with growing dimensions

Contact Info

Product

Resources

About