[show abstract][hide abstract] ABSTRACT: Noncommutative harmonic analysis is used to solve a nonparametric estimation problem stated in terms of compound Poisson processes on compact Lie groups. This problem of decompounding is a generalization of a similar classical problem. The proposed solution is based on a characteristic function method. The treated problem is important to recent models of the physical inverse problem of multiple scattering.
IEEE Transactions on Information Theory 07/2010; · 2.62 Impact Factor
[show abstract][hide abstract] ABSTRACT: Motivated by applications in multiple scattering, we study the problem of decompounding on compact Lie groups. Employing tools from harmonic analysis, we give a nonparametric approach to this problem. The case of the special orthogonal group SO(3) is discussed in detail.
Acoustics, Speech and Signal Processing, 2009. ICASSP 2009. IEEE International Conference on; 05/2009 · 4.63 Impact Factor
[show abstract][hide abstract] ABSTRACT: 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 reduced Stokes vector throughout the propagation. It also allows an exact description of the evolution of averaged quantities, such as the degree of polarization. We 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.
Waves in Random and Complex Media 05/2008; 18(2):275-292. · 0.94 Impact Factor
[show abstract][hide abstract] ABSTRACT: In this paper, we consider the extension of the Fourier transform to biquaternion-valued signals. We introduce a transform that we call the biquaternion Fourier transform (BiQFT). After giving some general properties of this transform, we show how it can be used to generalize the notion of analytic signal to complex-valued signals. We introduce the notion of hyperanalytic signal. We also study the Hermitian symmetries of the BiQFT and their relation to the geometric nature of a biquaternion-valued signal. Finally, we present a fast algorithm for the computation of the BiQFT. This algorithm is based on a (complex) change of basis and four standard complex FFTs.
IEEE Transactions on Signal Processing 05/2008; · 2.81 Impact Factor
[show abstract][hide abstract] ABSTRACT: 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.
IEEE Transactions on Signal Processing 04/2006; · 2.81 Impact Factor
[show abstract][hide abstract] ABSTRACT: PGA, or Principal Geodesic Analysis, is an extension of the classical PCA (Principal Component Analysis) to the case of data taking values on a Riemannian manifold. In this paper a new and original algorithm, for the exact computation of the PGA of data on the rotation group SO(3), is presented. Some properties of this algorithm are illustrated, with tests on simulated and real data, and its possible applications are finally discussed.
Proceedings of the 15th European Signal Processing Conference, EUSIPCO-2007.
[show abstract][hide abstract] ABSTRACT: Nous proposons un modele de type Processus de Poisson sur un groupe de Lie compact pour la description et l'etude de la diffusion multiple dans les milieuax aleatoires. Cette approche permet d'inferer sur le milieu et en particulier d'estimer la fonction de phase du milieu qui donne acces au spectre d'heterogeneite, grendeur caracteristique du milieu. La technique d'inference proposee consiste en une decomposition du processus de Poisson modelisant le phenomene de diffusion multiple. Nous validons la technique proposee en testant les estimateurs sur des donnees de simulation Monte Carlo.
[show abstract][hide abstract] ABSTRACT: This is a note containing the most recent results we have arrived at in our work on polarized light in random media. After an introduction fixing the problem we are dealing with, we start by pointing out the connection between the Poincare sphere formalism and noncommutative harmonic analysis. This leads us to the main equations of our formalism. We use these equation to generalize the notions of Stokes vector, degree of polarization (DOP) and Mueller matrix to higher order statistics of the electric field. We also use them to give a formalism for the propagation of a state of polarisation (SOP) in a random medium. We show how a more detailed description of this propagation is then made possible.