In this paper, the issue of averaging data on a manifold is addressed. While the Fréchet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this,$R$-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However,$R$-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call$RL$-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.

This paper considers the analysis of bivariate signals using complex Singular Spectrum Analysis (SSA). It introduces a pseudo-correlation based criterion in the grouping step of complex SSA. The advantage of using pseudo-correlation rather than correlation measures when analyzing polarized signals with complex SSA is demonstrated theoretically. This criterion is shown to be effective to extract bivariate signals modeled as particular complex Linear Recurrence Relations (LRR) of order 2. These elementary complex bricks offer a high interpretability in terms of polarization. Illustration of the proposed grouping technique is made through polarized component extraction on a real-world data example.

The article introduces an elementary cost and storage reduction method for spectral clustering and principal component analysis. The method consists in randomly “puncturing” both the data matrix $X\in\mathbb{C}^{p\times n}$ (or $\mathbb{R}^{p\times n}$) and its corresponding kernel (Gram) matrix $K$ through Bernoulli masks: $S\in\{0,1\}^{p\times n}$ for $X$ and $B\in\{0,1\}^{n\times n}$ for $K$. The resulting “two-way punctured” kernel is thus given by $K=\frac1p[(X\odot S)^\H (X\odot S)]\odot B$. We demonstrate that, for $X$ composed of independent columns drawn from a Gaussian mixture model, as $n,p\to\infty$ with $p/n\to c_0\in(0,\infty)$, the spectral behavior of $K$ – its limiting eigenvalue distribution, as well as its isolated eigenvalues and eigenvectors – is fully tractable and exhibits a series of counter-intuitive phenomena. We notably prove, and empirically confirm on various image databases, that it is possible to drastically puncture the data, thereby providing possibly huge computational and storage gains, for a virtually constant (clustering or PCA) performance. This preliminary study opens as such the path towards rethinking, from a large dimensional standpoint, computational and storage costs in elementary machine learning models.

Improperness testing for complex-valued vectors and processes has been of interest lately due to the potential applications of complex-valued time series analysis in several research areas. This paper provides exact distribution characterization of the GLRT (Generalized Likelihood Ratio Test) statistics for Gaussian complex-valued signals under the null hypothesis of properness. This distribution is a special case of the Wilks's lambda distribution, as are the distributions of the GLRT statistics in multivariate analysis of variance (MANOVA) procedures. In the high dimensional setting, i.e. when the size of the vectors grows at the same rate as the number of samples, a closed form expression is obtained for the asymptotic distribution of the GLRT statistics. This is, to our knowledge, the first exact characterization for the GLRT-based improperness testing.

The filtering estimation problem under uncertainty conditions is addressed for a class of improper quaternion signals, called widely factorizable, characterized because their augmented correlation function is a factorizable kernel. From the knowledge of the correlation functions involved, a recursive algorithm is designed for the computation of the widely linear (WL) filtering estimate and its associated mean squared error. The main advantage of the proposed solution is that it can be applied in situations where a state-space model is not readily at hand. The benefits of the proposed WL filtering algorithm is analyzed through a simulation example where WL filtering errors are compared with respect to the strictly linear (SL) counterparts, showing the superior behavior of the former over the latter.

This work concerns the definition of conditional mutual information in the framework of Algorithmic Information Theory (AIT), which is of use when no probabilistic model of the data is available, or hard to devise. We introduce a practical way to construct a conditional mutual information quantity which respects the chain rule and the data processing inequalityThe proposed implementation, named SALZA, allows to accomplish various information-theoretic tasks on sequences. The algorithmic model of the data used in this work is that of the well-known Lempel-Ziv primitive: we assume new data is to be expressed in terms of references to prior data.SALZA enables a flexible specification of prior data and extracts information quantities based on the significance of the references to these prior data. The tool readily implements the computation of an information measure based on LZ77 and a universal classifier based on the Ziv-Merhav relative coder for the universal clustering of sequences.Illustration of the proposed implementation is provided on clustering and causality inference examples.

Bivariate signals are commonly processed with the usual Fourier transform, using methods such as the rotary spectrum analysis. We show that bivariate signals can be efficiently processed using the Quaternion Fourier transform. A bivariate counterpart of the analytic signal is introduced, the quaternion embedding of a complex signal. It leads to identify natural parameters describing polarization properties, amplitude and phase of the signal. The properties of the quaternion short-term Fourier transform are studied and the polarization spectrogram is introduced. A synthetic example illustrates the relevance of the proposed approach.

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Single-particle imaging experiments using X-ray Free-Electron Lasers (XFEL) belong to a new generation of X-ray imaging techniques potentially allowing high resolution images of non-crystallizable molecules to be obtained. One of the challenges of single-particle imaging is the reconstruction of the 3D intensity function from only a few samples collected on a planar detector after the interaction of a free falling molecule and the X-ray beam. In this paper, we take advantage of the symmetries of the intensity function to propose an original low-resolution reconstruction algorithm based on an Expansion Maximization Compression (EMC) approach. We study the problem of adequate sampling of the rotation group via simulation to illustrate the potential of the approach.

This paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres Sp-1. It allows one to derive the Fourier expansions on Sp-1of these processes. A result of unimodality for the multiconvolution of symmetrical probability density functions on Sp-1is also introduced. Such processes are then studied in the case where the scattering distribution is von Mises-Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs on Sp-1are obtained. Both Fourier expansion and asymptotic approximation allow us to compute estimation bounds for the parameters of compound cox processes on Sp-1.

This paper considers the problem of parameter estimation for multiple scattering process on the sphere. Using harmonic analysis, a Fourier expansion of the pdf of the process is obtained. Based on the Fourier coefficient statistics, we consider the problem of estimating the parameter of the process using an Approximate Bayesian Computation (ABC) approach. Simulations show the ability of the proposed approach for the density estimation of intensity and concentration parameters for the von Mises Fisher multiple scattering process.

Partial observation of stochastic processes can occur for various reasons, ranging from faulty sensors to occultation issues. In this paper, we consider the problem of estimating the angular velocity of a rotating system from partial observation corrupted by noise. The system is assumed to evolve on the rotation group SO(n), and only k noisy measurements with k <; n are available. We propose an optimal filter to track the angular velocity. We show that, under some conditions, it is possible to recover the angular velocity of the rotating system and we propose a solution based on a Monte-Carlo method (particle filter). In particular, we show that if the angular velocity is stepwise constant, our algorithm succeed in estimating it. Simulations illustrate the proposed approach.

This paper presents a “method of moments” estimation technique for the study of multiple scattering on the hypersphere. The proposed model is similar to a compound Poisson process evolving on a special manifold: the unit hypersphere. The presented work makes use of an approximation result for multiply convolved von Mises-Fisher distributions on hyperspheres. Comparison with other approximations show the accuracy of the proposed model to provide estimators for the mean free path and concentration parameters when studying a multiple scattering process. Such a process is classically used to model the propagation of waves or particles in random media.

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. © 2011 IEEE.

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.

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.

For polarized signals, which arise in many application fields, a statistical framework in terms of quaternionic random processes is proposed. Based on it, the ability of real-, complex- and quaternionic-valued multi-layer perceptrons (MLPs) of performing classification tasks for such signals is evaluated. For the multi-dimensional neural networks the relevance of class label representations is discussed. For signal to noise separation it is shown that the quaternionic MLP yields an optimal solution. Results on the classification of two different polarized signals are also reported.

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

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.

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.

The problem of separation of instantaneous mixtures of narrowband signals impinging on sensors often arises in signal processing. In geophysics, the aims of signal processing are the separation and the identification of waves or sources to get a better understanding of the onshore. For a couple of years, multicomponent sensors are used in acquisition and allow one access to a physical property: the polarization. This paper shows that, taking the advantages from the multicomponent sensors, it is possible to have complete wavefield separation without a priori information. We propose to perform the estimation of wave's polarization using an algorithm derived from the Jointly Approximated Diagonalization of Eigen-elements (JADE) technique, which is based on the properties of the 4th order cumulants. From a real seismic acquisition, we present results where Rayleigh wave separation is performed. We also present the analysis of physical properties of the surface waves.