The geometry of proper quaternion random variables

Bihan, Nicolas Le

doi:10.1016/j.sigpro.2017.03.017

Cited by 15 publications

(12 citation statements)

References 29 publications

(65 reference statements)

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“…It is usual to describe the full secondorder statistical structure of a quaternion RV q by the four covariances E {qq µ }, µ = i, j, k. These covariances often obey some symmetries characterized by the notion of properness. Properness levels of quaternion RV have been investigated by several authors [32]- [34] and reviewed recently in [35]. The spectral increments of a C i -valued process x[t] satisfy the symmetry (16) and thus property 3) of theorem 1 with…”

Section: A Main Resultsmentioning

confidence: 99%

“…Eq. (18) shows that the spectral increments dX(ν) are (1, j)proper in the classification of [35], also denoted as C jproperness in [33].…”

Section: A Main Resultsmentioning

confidence: 99%

“…C. Estimation of the degree of polarization 1) Theoretical properties: The estimation of the degree of polarization (35) has attracted interest in the signal processing community [45], [46] in relation to many fields [39], [40]. A naive estimator (at frequencies where the polarization periodogram is nonzero) based on the polarization periodogram would be trivial since:…”

Section: B Multitaperingmentioning

confidence: 99%

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Spectral Analysis of Stationary Random Bivariate Signals

Flamant

Bihan

Chainais

2017

IEEE Trans. Signal Process.

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Abstract-A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Using the Quaternion Fourier Transform, we introduce a quaternionvalued 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.

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Section: A Main Resultsmentioning

confidence: 99%

“…Eq. (18) shows that the spectral increments dX(ν) are (1, j)proper in the classification of [35], also denoted as C jproperness in [33].…”

Section: A Main Resultsmentioning

confidence: 99%

Section: B Multitaperingmentioning

confidence: 99%

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Spectral Analysis of Stationary Random Bivariate Signals

Flamant

Bihan

Chainais

2017

IEEE Trans. Signal Process.

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“…The initial source factor W 0 has i.i.d. entries drawn from the quaternion circular unit Gaussian distribution [64] projected onto the constraint (S). We then alternate Eqs.…”

Section: B Spectro-polarimetric Blind Unmixing Using Qalsmentioning

confidence: 99%

Quaternion Non-Negative Matrix Factorization: Definition, Uniqueness, and Algorithm

Flamant

Miron

Brie

2020

IEEE Trans. Signal Process.

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This article introduces quaternion non-negative matrix factorization (QNMF), which generalizes the usual nonnegative matrix factorization (NMF) to the case of polarized signals. Polarization information is represented by Stokes parameters, a set of 4 energetic parameters widely used in polarimetric imaging. QNMF relies on two key ingredients: (i) the algebraic representation of Stokes parameters thanks to quaternions and (ii) the exploitation of physical constraints on Stokes parameters. These constraints generalize non-negativity to the case of polarized signals, encoding positive semi-definiteness of the covariance matrix associated which each source. Uniqueness conditions for QNMF are presented. Remarkably, they encompass known sufficient uniqueness conditions from NMF. Meanwhile, QNMF further relaxes NMF uniqueness conditions requiring sources to exhibit a certain zero-pattern, by leveraging the complete polarization information. We introduce a simple yet efficient algorithm called quaternion alternating least squares (QALS) to solve the QNMF problem in practice. Closed-form quaternion updates are obtained using the recently introduced generalized HR calculus. Numerical experiments on synthetic data demonstrate the relevance of the approach. QNMF defines a promising generic low-rank approximation tool to handle polarization, notably for blind source separation problems arising in imaging applications.

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“…In this decomposition, K must be sampled from a uniform distribution on Sp(n), and R with diagonal elements r i from the multivariate density (22). Sampling from a uniform distribution on Sp(n) can be achieved as follows : let Z be an n × n quaternion matrix whose elements are independent normal proper quaternion random variables [11], and write Z = KP for the polar decomposition of Z [9]. Then, K has a uniform distribution on Sp(n).…”

Section: Sampling and Inferencementioning

confidence: 99%

Riemannian Gaussian Distributions on the Space of Positive-Definite Quaternion Matrices

Said¹,

Bihan

Manton

2017

Lecture Notes in Computer Science

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Abstract. 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.

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The geometry of proper quaternion random variables

Cited by 15 publications

References 29 publications

Spectral Analysis of Stationary Random Bivariate Signals

Spectral Analysis of Stationary Random Bivariate Signals

Quaternion Non-Negative Matrix Factorization: Definition, Uniqueness, and Algorithm

Riemannian Gaussian Distributions on the Space of Positive-Definite Quaternion Matrices

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