3,159 research outputs found

### The geometry of proper quaternion random variables

Second order circularity, also called properness, for complex random
variables is a well known and studied concept. In the case of quaternion random
variables, some extensions have been proposed, leading to applications in
quaternion signal processing (detection, filtering, estimation). Just like in
the complex case, circularity for a quaternion-valued random variable is
related to the symmetries of its probability density function. As a
consequence, properness of quaternion random variables should be defined with
respect to the most general isometries in $4D$, i.e. rotations from $SO(4)$.
Based on this idea, we propose a new definition of properness, namely the
$(\mu_1,\mu_2)$-properness, for quaternion random variables using invariance
property under the action of the rotation group $SO(4)$. This new definition
generalizes previously introduced properness concepts for quaternion random
variables. A second order study is conducted and symmetry properties of the
covariance matrix of $(\mu_1,\mu_2)$-proper quaternion random variables are
presented. Comparisons with previous definitions are given and simulations
illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure

### Nonparametric estimation of the heterogeneity of a random medium using Compound Poisson Process modeling of wave multiple scattering

In this paper, we present a nonparametric method to estimate the
heterogeneity of a random medium from the angular distribution of intensity
transmitted through a slab of random material. Our approach is based on the
modeling of forward multiple scattering using Compound Poisson Processes on
compact Lie groups. The estimation technique is validated through numerical
simulations based on radiative transfer theory.Comment: 23 pages, 8 figures, 21 reference

### Higher Order Statistsics of Stokes Parameters in a Random Birefringent Medium

We present a new model for the propagation of polarized light in a random
birefringent medium. This model is based on a decomposition of the higher order
statistics of the reduced Stokes parameters along the irreducible
representations of the rotation group. We show how this model allows a detailed
description of the propagation, giving analytical expressions for the
probability densities of the Mueller matrix and the Stokes vector throughout
the propagation. It also allows an exact description of the evolution of
averaged quantities, such as the degree of polarization. We will also discuss
how this model allows a generalization of the concepts of reduced Stokes
parameters and degree of polarization to higher order statistics. We give some
notes on how it can be extended to more general random media

### Spectral analysis of stationary random bivariate signals

A novel approach towards the spectral analysis of stationary random bivariate
signals is proposed. Using the Quaternion Fourier Transform, we introduce a
quaternion-valued spectral representation of random bivariate signals seen as
complex-valued sequences. This makes possible the definition of a scalar
quaternion-valued spectral density for bivariate signals. This spectral density
can be meaningfully interpreted in terms of frequency-dependent polarization
attributes. A natural decomposition of any random bivariate signal in terms of
unpolarized and polarized components is introduced. Nonparametric spectral
density estimation is investigated, and we introduce the polarization
periodogram of a random bivariate signal. Numerical experiments support our
theoretical analysis, illustrating the relevance of the approach on synthetic
data.Comment: 11 pages, 3 figure

### Riemannian Gaussian distributions on the space of positive-definite quaternion matrices

Recently, Riemannian Gaussian distributions were defined on spaces of
positive-definite real and complex matrices. The present paper extends this
definition to the space of positive-definite quaternion matrices. In order to
do so, it develops the Riemannian geometry of the space of positive-definite
quaternion matrices, which is shown to be a Riemannian symmetric space of
non-positive curvature. The paper gives original formulae for the Riemannian
metric of this space, its geodesics, and distance function. Then, it develops
the theory of Riemannian Gaussian distributions, including the exact expression
of their probability density, their sampling algorithm and statistical
inference.Comment: 8 pages, submitted to GSI 201

### Isotropic Multiple Scattering Processes on Hyperspheres

This paper presents several results about isotropic random walks and multiple
scattering processes on hyperspheres ${\mathbb S}^{p-1}$. It allows one to
derive the Fourier expansions on ${\mathbb S}^{p-1}$ of these processes. A
result of unimodality for the multiconvolution of symmetrical probability
density functions (pdf) on ${\mathbb S}^{p-1}$ is also introduced. Such
processes are then studied in the case where the scattering distribution is von
Mises Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs
on ${\mathbb S}^{p-1}$ are obtained. Both Fourier expansion and asymptotic
approximation allows us to compute estimation bounds for the parameters of
Compound Cox Processes (CCP) on ${\mathbb S}^{p-1}$.Comment: 16 pages, 4 figure

### Density estimation on the rotation group using diffusive wavelets

This paper considers the problem of estimating probability density functions
on the rotation group $SO(3)$. Two distinct approaches are proposed, one based
on characteristic functions and the other on wavelets using the heat kernel.
Expressions are derived for their Mean Integrated Squared Errors. The
performance of the estimators is studied numerically and compared with the
performance of an existing technique using the De La Vall\'ee Poussin kernel
estimator. The heat-kernel wavelet approach appears to offer the best
convergence, with faster convergence to the optimal bound and guaranteed
positivity of the estimated probability density function

### Fast complexified quaternion Fourier transform

A discrete complexified quaternion Fourier transform is introduced. This is a
generalization of the discrete quaternion Fourier transform to the case where
either or both of the signal/image and the transform kernel are complex
quaternion-valued. It is shown how to compute the transform using four standard
complex Fourier transforms and the properties of the transform are briefly
discussed

### Asymptotic regime for impropriety tests of complex random vectors

Impropriety testing for complex-valued vector has been considered lately due
to potential applications ranging from digital communications to complex media
imaging. This paper provides new results for such tests in the asymptotic
regime, i.e. when the vector dimension and sample size grow commensurately to
infinity. The studied tests are based on invariant statistics named impropriety
coefficients. Limiting distributions for these statistics are derived, together
with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in
the Gaussian case. This characterization in the asymptotic regime allows also
to identify a phase transition in Roy's test with potential application in
detection of complex-valued low-rank subspace corrupted by proper noise in
large datasets. Simulations illustrate the accuracy of the proposed asymptotic
approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS

- â€¦