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

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

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

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