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Nicolas Le Bihan
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Nicolas Le Bihan

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Mathematics - Numerical Analysis (4)
Mathematics - Statistics (4)
Physics - Data Analysis; Statistics and Probability (4)
Statistics - Theory (4)
Mathematics - Rings and Algebras (3)
Computer Science - Information Theory (3)
Mathematics - Information Theory (3)
Physics - Optics (3)
Statistics - Methodology (2)
Physics - Classical Physics (1)
Mathematics - Functional Analysis (1)
Computer Science - Other (1)
Mathematics - General Mathematics (1)
Mathematical Physics (1)
Mathematics - Mathematical Physics (1)

Publications Authored By Nicolas Le Bihan

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. Read More

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. Read More

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. The time-frequency analysis of bivariate signals is usually carried out by analyzing two separate quantities, e.g. Read More

A complete set of practical estimators for the conditional, simple and joint algorihmic complexities is presented, from which a semi-metric is derived. Also, new directed information estimators are proposed that are applied to causality inference on Directed Acyclic Graphs. The performances of these estimators are investigated and shown to compare well with respect to the state-of-the-art Normalized Compression Distance (NCD). Read More

In 3D single particle imaging with X-ray free-electron lasers, particle orientation is not recorded during measurement but is instead recovered as a necessary step in the reconstruction of a 3D image from the diffraction data. Here we use harmonic analysis on the sphere to cleanly separate the angu- lar and radial degrees of freedom of this problem, providing new opportunities to efficiently use data and computational resources. We develop the Expansion-Maximization-Compression algorithm into a shell-by-shell approach and implement an angular bandwidth limit that can be gradually raised during the reconstruction. Read More

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. Read More

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. Read More

This paper considers the problem of optimal filtering for partially observed signals taking values on the rotation group. More precisely, one or more components are considered not to be available in the measurement of the attitude of a 3D rigid body. In such cases, the observed signal takes its values on a Stiefel manifold. Read More

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. Read More

The ideas of instantaneous amplitude and phase are well understood for signals with real-valued samples, based on the analytic signal which is a complex signal with one-sided Fourier transform. We extend these ideas to signals with complex-valued samples, using a quaternion-valued equivalent of the analytic signal obtained from a one-sided quaternion Fourier transform which we refer to as the hypercomplex representation of the complex signal. We present the necessary properties of the quaternion Fourier transform, particularly its symmetries in the frequency domain and formulae for convolution and the quaternion Fourier transform of the Hilbert transform. Read More

We present a stochastic description of multiple scattering of polarized waves in the regime of forward scattering. In this regime, if the source is polarized, polarization survives along a few transport mean free paths, making it possible to measure an outgoing polarization distribution. We solve the direct problem using compound Poisson processes on the rotation group SO(3) and non-commutative harmonic analysis. Read More

We report the experimental observation of a geometric phase for elastic waves in a waveguide with helical shape. The setup reproduces the experiment by Tomita and Chiao [A. Tomita, R. Read More

This paper is concerned with the study of a circular random distribution called geodesic Normal distribution recently proposed for general manifolds. This distribution, parameterized by two real numbers associated to some specific location and dispersion concepts, looks like a standard Gaussian on the real line except that the support of this variable is $[0,2\pi)$ and that the Euclidean distance is replaced by the geodesic distance on the circle. Some properties are studied and comparisons with the von Mises distribution in terms of intrinsic and extrinsic means and variances are provided. Read More

The concept of the analytic signal is extended from the case of a real signal with a complex analytic signal to a complex signal with a hypercomplex analytic signal (which we call a hyperanalytic signal) The hyperanalytic signal may be interpreted as an ordered pair of complex signals or as a quaternion signal. The hyperanalytic signal contains a complex orthogonal signal and we show how to obtain this by three methods: a pair of classical Hilbert transforms; a complex Fourier transform; and a quaternion Fourier transform. It is shown how to derive from the hyperanalytic signal a complex envelope and phase using a polar quaternion representation previously introduced by the authors. Read More

The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. Read More

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 char- acteristic function method. Read More

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. Read More

We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex 'modulus' and a complex 'argument'. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of -1), but the complex phase is multiplied by a different complex root of -1 in the exponential function. Read More

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. Read More

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. Read More

We present a practical and efficient means to compute the singular value decomposition (svd) of a quaternion matrix A based on bidiagonalization of A to a real bidiagonal matrix B using quaternionic Householder transformations. Computation of the svd of B using an existing subroutine library such as lapack provides the singular values of A. The singular vectors of A are obtained trivially from the product of the Householder transformations and the real singular vectors of B. Read More