N. Le Bihan

Quaternion-MUSIC for vector-sensor array processing

This paper considers the problem of direction of arrival (DOA) and polarization parameters estimation in the case of multiple polarized sources impinging on a vector-sensor array. The quaternion model is used, and a data covariance model is proposed using quaternion formalism. A comparison between long vector orthogonality and quaternion vector orthogonality is also performed, and its implications for signal subspace estimation are discussed. Consequently, a MUSIC-like algorithm is presented, allowing estimation of waves DOAs and polarization parameters. The algorithm is tested in numerical simulations, and performance analysis is conducted. When compared with other MUSIC-like algorithms for vector-sensor array, the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort, for equivalent performance. This paper also illustrates a compact and elegant way of dealing with multicomponent complex-valued data.

Quaternion principal component analysis of color images

In this paper, we present quaternion matrix algebra techniques that can be used to process the eigen analysis of a color image. Applications of principal component analysis (PCA) in image processing are numerous, and the proposed tools aim to give material for color image processing, that take into account their particular nature. For this purpose, we use the quaternion model for color images and introduce the extension of two classical techniques to their quaternionic case: singular value decomposition (SVD) and Karhunen-Loeve transform (KLT). For the quaternionic version of the KLT, we also introduce the problem of eigenvalue decomposition (EVD) of a quaternion matrix. We give the properties of these quaternion tools for color images and present their behavior on natural images. We also present a method to compute the decompositions using complex matrix algebra. Finally, we start a discussion on possible applications of the proposed techniques in color images processing.

MUSIC Algorithm for Vector-Sensors Array Using Biquaternions

In this paper, we use a biquaternion formalism to model vector-sensor signals carrying polarization information. This allows a concise and elegant way of handling signals with eight-dimensional (8-D) vector-valued samples. Using this model, we derive a biquaternionic version of the well-known array processing MUSIC algorithm, and we show its superiority to classically used long-vector approach. New results on biquaternion valued matrix spectral analysis are presented. Of particular interest for the biquaternion MUSIC (BQ-MUSIC) algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces. We propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition (EVD) of a Hermitian biquaternionic matrix by means of a newly defined quantity, the quaternion adjoint matrix. The BQ-MUSIC estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing (LV-MUSIC and PSA-MUSIC). The proposed algorithm is shown to be superior in several aspects to the existing approaches. Compared with LV-MUSIC, the BQ-MUSIC algorithm is more robust to modelization errors and coherent noise while it can detect less sources. In comparaison with PSA-MUSIC, our approach exhibits more accurate estimation of direction of arrival (DOA) for a small number of sources, while keeping the polarization information accessible.

Color image watermarking using quaternion Fourier transform

The paper presents a digital color image watermarking scheme using a hypercomplex numbers representation and the quaternion Fourier transform (QFT). Previous color image watermarking methods are first presented and the quaternion representation is then described. In this framework, RGB pixel values are associated with a unique quaternion number having three imaginary parts. The QFT is presented; this transform depends on an arbitrary unit pure quaternion, /spl mu/. The value of /spl mu/ is selected to provide embedding spaces having robustness and/or perceptual properties. In our approach, /spl mu/ is a function of the mean color value of a block and a perceptual component. A watermarking scheme based on the QFT and the quantization index modulation scheme is then presented. This scheme is evaluated for different color image filtering processes (JPEG, blur). The fact that perceptive QFT embedding can offer robustness to luminance filtering techniques is outlined.

Fast Complexified Quaternion Fourier Transform

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.

Rotor fault detection of electrical machines by low frequency magnetic stray field analysis

This paper shows the reliability of fault detection on electrical machines by analysis of the low frequency magnetic stray field. It is based on our own experience about magnetic discretion of naval electrical propulsion machine. We try to apply the techniques developed in previous works on the subject to faults detection. In this paper we focus on rotor defaults in a synchronous generator (eccentricity and short-circuit in rotor). Two kinds of study are performed. The first one is numerical. Firstly, an adapted finite elements method is used to compute the stray field around the device. However, this approach is difficult to apply to fault detection and not well-adapted. A new model, simpler and faster, is developed. Results are compared for both modelling. The second one is experimental and is driven thanks to a laboratory machine representative of a real high power generator and to fluxgate magnetometers located around the device. Both studies show good agreement and demonstrate the reliability of the approach.

High resolution vector-sensor array processing using quaternions

The aim of this paper is to introduce a novel MUSIC-like algorithm for polarized sources characterization based on a quaternion model for two-component sensor-array signal. The associated data covariance matrix is described and a comparison with the classical long-vector approach is made. We show that the use of quaternions improves the signal subspace estimation accuracy and reduces the computational burden. Additionally, the proposed algorithm presents a better resolution power for direction of arrival (DOA) estimation than the long-vector approach, for equivalent statistical performances

Blind extraction of improper quaternion sources

Blind extraction of quaternion-valued latent sources is addressed based on their local temporal properties. The extraction criterion is based on the minimum mean square widely linear prediction error, thus allowing for the extraction of both proper and improper quaternion sources. The use of the widely linear adaptive predictor is justified by the relationship between the mean square prediction error and the crosscorrelation and cross-pseudocorrelations of the source signals. Simulations on benchmark improper quaternion sources together with a real-world example of EEG artifact removal illustrate the usefulness of the proposed methodology.

High Resolution Vector-Sensor Array Processing Based on Biquaternions

This paper presents a version of MUSIC algorithm for linear vector-sensor arrays based on a complexified quaternionic (biquaternionic) modelization of the output three-components vector-signals. A way of computing the eigenvalue decomposition of a biquaternion valued matrix is introduced and the subspace decomposition of the biquaternionic spectral matrix of the observations is used to define the biquaternionic MUSIC estimator (BQ-MUSIC). Performances of the BQ-MUSIC are compared with classical long-vector technique

MUSIQUE: A quaternion-based array processing technique for surface wave polarization analysis

In this paper, we present MUSIQUE, an algorithm designed to analyze seismic signals recorded by arrays of three-component seismic sensors. Using the original MUSIC algorithm [1], azimuth and slowness (or velocity) of incident waves are estimated. The quaternion-MUSIC algorithm [2, 3] is then used to characterize the polarization properties of the waves. In this way, Love and Rayleigh waves are distinguished and their respective properties retrieved. This allows the characterization of the seismic wave field and the soil structure underneath the seismic array, which are important parameters for the estimation of seismic hazards.