Spectral Analysis of Stationary Random Bivariate Signals

Flamant, Julien; Bihan, Nicolas Le; Chainais, Pierre

doi:10.1109/tsp.2017.2736494

Cited by 15 publications

(38 citation statements)

References 42 publications

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“…More recently [18], [21], it has been shown that (2) exhibits a general relevance for arbitrary bivariate signals. Our choice for the ordering of Stokes parameters in (2) adopts that of [21].…”

Section: Quaternion Representation Of Stokes Parametersmentioning

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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Section: Quaternion Representation Of Stokes Parametersmentioning

confidence: 99%

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

Flamant

Miron

Brie

2020

IEEE Trans. Signal Process.

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“…We briefly survey the Quaternion Fourier Transform (QFT) first introduced in [26] and further studied in [17]. Recent works [17], [18] have demonstrated the relevance of this QFT to process bivariate signals. In particular the QFT decomposes directly bivariate signals into a sum of polarized monochromatic signals.…”

Section: B Quaternion Fourier Transformmentioning

confidence: 99%

“…Many signals however are random and only of finite power, which makes the spectral density definition (8) no longer applicable. Fortunately thanks to a spectral representation theorem based on the QFT [18] one can extend the definition (8) to define a quaternion power spectral density for stationary random bivariate signals. In short the standard QFT X(ν) is replaced by the spectral increment dX(ν): see Appendix D for details.…”

Section: Quaternion Spectral Density Of Bivariate Signalsmentioning

confidence: 99%

“…In contrast we provide frequency-dependent expressions for unitary and Hermitian filters to deal with generic wideband bivariate signals. It must be pointed out 2 The term "total" refers to the fact that S 0,x (ν) contains power contributions from the unpolarized and polarized part, see [18] i, S3 S0 j, S1 gives the orientation θ and ellipticity χ. The radius Φ gives the degree of polarization.…”

Section: Lti Filtering For Bivariate Signalsmentioning

confidence: 99%

“…We have recently introduced a powerful alternative approach to bivariate signal processing [17], [18] using a tailored quaternion Fourier transform (QFT). The proposed framework exhibits an unifying structure by directly connecting usual physical quantities from polarization to well-defined mathematical (quaternion-valued) quantities such as spectral densities, covariances, time-frequency representations, etc.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Complete Framework for Linear Filtering of Bivariate Signals

Flamant

Chainais

Bihan

2018

IEEE Trans. Signal Process.

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A complete framework for the linear time-invariant (LTI) filtering theory of bivariate signals is proposed based on a tailored quaternion Fourier transform. This framework features a direct description of LTI filters in terms of their eigenproperties enabling compact calculus and physically interpretable filtering relations in the frequency domain. The design of filters exhibiting fondamental properties of polarization optics (birefringence, diattenuation) is straightforward. It yields an efficient spectral synthesis method and new insights on Wiener filtering for bivariate signals with prescribed frequency-dependent polarization properties. This generic framework facilitates original descriptions of bivariate signals in two components with specific geometric or statistical properties. Numerical experiments support our theoretical analysis and illustrate the relevance of the approach on synthetic data.

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Eigenstructure and fractionalization of the quaternion discrete Fourier transform

Ribeiro

Lima

2020

Optik

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Spectral Analysis of Stationary Random Bivariate Signals

Cited by 15 publications

References 42 publications

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

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

A Complete Framework for Linear Filtering of Bivariate Signals

Eigenstructure and fractionalization of the quaternion discrete Fourier transform

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