Sparse principal component analysis (PCA) aims at mapping large dimensional data to a linear subspace of lower dimension. By imposing loading vectors to be sparse, it performs the double duty of dimension reduction and variable selection. Sparse PCA algorithms are usually expressed as a trade-off between explained variance and sparsity of the loading vectors (i.e., number of selected variables). As a high explained variance is not necessarily synonymous with relevant information, these methods are prone to select irrelevant variables. To overcome this issue, we propose an alternative formulation of sparse PCA driven by the false discovery rate (FDR). We then leverage the Terminating-Random Experiments (T-Rex) selector to automatically determine an FDR-controlled support of the loading vectors. A major advantage of the resulting T-Rex PCA is that no sparsity parameter tuning is required. Numerical experiments and a stock market data example demonstrate a significant performance improvement.

In this study, we consider the realm of covariance matrices in machine learning, particularly focusing on computing Fréchet means on the manifold of symmetric positive definite matrices, commonly referred to as Karcher or geometric means. Such means are leveraged in numerous machine learning tasks. Relying on advanced statistical tools, we introduce a random matrix theory based method that estimates Fréchet means, which is particularly beneficial when dealing with low sample support and a high number of matrices to average. Our experimental evaluation, involving both synthetic and real-world EEG and hyperspectral datasets, shows that we largely outperform state-of-the-art methods.

Though very popular, it is well known that the Expectation-Maximisation (EM) algorithm for the Gaussian mixture model performs poorly for non-Gaussian distributions or in the presence of outliers or noise. In this paper, we propose a Flexible EM-like Clustering Algorithm (FEMCA): a new clustering algorithm following an EM procedure is designed. It is based on both estimations of cluster centers and covariances. In addition, using a semi-parametric paradigm, the method estimates an unknown scale parameter per data point. This allows the algorithm to accommodate heavier tail distributions, noise, and outliers without significantly losing efficiency in various classical scenarios. We first present the general underlying model for independent, but not necessarily identically distributed, samples of elliptical distributions. We then derive and analyze the proposed algorithm in this context, showing in particular important distribution-free properties of the underlying data distributions. The algorithm convergence and accuracy properties are analyzed by considering the first synthetic data. Finally, we show that FEMCA outperforms other classical unsupervised methods of the literature, such as k-means, EM for Gaussian mixture models, and its recent modifications or spectral clustering when applied to real data sets as MNIST, NORB, and 20newsgroups.

This work deals with elliptical Wishart distributions on the set of symmetric positive definite matrices. It contains two major contributions. First, the information geometry associated with elliptical Wishart distributions is derived. Second, this geometry is leveraged to propose Riemannian-optimization-based maximum likelihood estimators of any elliptical Wishart distribution. Particular attention is given to two specific distributions: the t- and Kotz Wishart ones. The performance of the proposed methods is assessed through numerical experiments on simulated data.

Expectation-Maximization (EM) algorithm is a widely used iterative algorithm for computing (local) maximum likelihood estimate (MLE). It can be used in an extensive range of problems, including the clustering of data based on the Gaussian mixture model (GMM). Numerical instability and convergence problems may arise in situations where the sample size is not much larger than the data dimensionality. In such low sample support (LSS) settings, the covariance matrix update in the EM-GMM algorithm may become singular or poorly conditioned, causing the algorithm to crash. On the other hand, in many signal processing problems, a priori information can be available indicating certain structures for different cluster covariance matrices. In this paper, we present a regularized EM algorithm for GMM-s that can make efficient use of such prior knowledge as well as cope with LSS situations. The method aims to maximize a penalized GMM likelihood where regularized estimation may be used to ensure positive definiteness of covariance matrix updates and shrink the estimators towards some structured target covariance matrices. We show that the theoretical guarantees of convergence hold, leading to better performing EM algorithm for structured covariance matrix models or with low sample settings.

The article proposes and theoretically analyses a computationally efficient multi-task learning (MTL) extension of popular principal component analysis (PCA)-based supervised learning schemes. The analysis reveals that (i) by default, learning may dramatically fail by suffering from negative transfer, but that (ii) simple counter-measures on data labels avert negative transfer and necessarily result in improved performances. Supporting experiments on synthetic and real data benchmarks show that the proposed method achieves comparable performance with state-of-the-art MTL methods but at a significantly reduced computational cost.

We propose estimating the scale parameter (mean of the eigenvalues) of the scatter matrix of an unspecified elliptically symmetric distribution using weights obtained by solving Tyler's M-estimator of the scatter matrix. The proposed Tyler's weights-based estimate (TWE) of scale is then used to construct an affine equivariant Tyler's M-estimator as a weighted sample covariance matrix using normalized Tyler's weights. We then develop a unified framework for estimating the unknown tail parameter of the elliptical distribution (such as the degrees of freedom (d.o.f.)$\nu$of the multivariate$t$(MVT) distribution). Using the proposed TWE of scale, a new robust estimate of the d.o.f. parameter of MVT distribution is proposed with excellent performance in heavy-tailed scenarios, outperforming other competing methods. R-package is available that implements the proposed method.

Linear and Quadratic Discriminant Analysis are well-known classical methods but can heavily suffer from non-Gaussian distributions and/or contaminated datasets, mainly because of the underlying Gaussian assumption that is not robust. To fill this gap, this paper presents a new robust discriminant analysis where each data point is drawn by its own arbitrary Elliptically Symmetrical (ES) distribution and its own arbitrary scale parameter. Such a model allows for possibly very heterogeneous, independent but non-identically distributed samples. After deriving a new decision rule, it is shown that maximum-likelihood parameter estimation and classification are very simple, fast and robust compared to state-of-the-art methods.

The multivariate power exponential (MEP) distribution can model a broad range of signals. In noisy scenarios, the robust estimation of the MEP parameters has been traditionally addressed by a fixed-point approach associated with a nonconvex optimization problem. Establishing convergence properties for this approach when the distribution mean is unknown is still an open problem. As an alternative, this paper presents a novel convex formulation for robustly estimating MEP parameters in the presence of multiplicative perturbations. The proposed approach is grounded on a re-parametrization of the original likelihood function in a way that ensures convexity. We also show that this property is preserved for several typical regularization functions. Compared with the robust Tyler’s estimator, the proposed method shows a more accurate precision matrix estimation, with similar mean and covariance estimation performance.

This paper aims at providing an original Riemannian geometry to derive robust covariance matrix estimators in spiked models (i.e. when the covariance matrix has a low-rank plus identity structure). The considered geometry is the one induced by the product of the Stiefel manifold and the manifold of Hermitian positive definite matrices, quotiented by the unitary group. One of the main contributions is to consider a Riemannian metric related to the Fisher information metric of elliptical distributions, leading to new representations for the tangent spaces and a new retraction. A new robust covariance matrix estimator is then obtained as the minimizer of Tyler's cost function, redefined directly on the set of low-rank plus identity matrices, and computed with the aforementioned tools. The main interest of this approach is that it appears well suited to the cases where the sample size is lower than the dimension, as illustrated by numerical experiments.

A popular regularized (shrinkage) covariance estimator is the shrinkage sample covariance matrix (SCM) which shares the same set of eigenvectors as the SCM but shrinks its eigenvalues toward its grand mean. In this paper, a more general approach is considered in which the SCM is replaced by an M-estimator of scatter matrix and a fully automatic data adaptive method to compute the optimal shrinkage parameter with minimum mean squared error is proposed. Our approach permits the use of any weight function such as Gaussian, Huber's, or t weight functions, all of which are commonly used in M-estimation framework. Our simulation examples illustrate that shrinkage M-estimators based on the proposed optimal tuning combined with robust weight function do not loose in performance to shrinkage SCM estimator when the data is Gaussian, but provide significantly improved performance when the data is sampled from a heavy-tailed distribution.

In this paper, we propose a new regularized (penalized) covariance matrix estimator which encourages grouping of the eigenvalues by penalizing large differences (gaps) between successive eigenvalues. This is referred to as fusing eigenvalues (eFusion). The proposed penalty function utilizes Tukey's biweight function that is widely used in robust statistics. The main advantage of the proposed method is that it has very small bias for sufficiently large values of penalty parameter. Hence, the method provides accurate grouping of eigenvalues. Such benefits of the proposed method are illustrated with a numerical example, where the method is shown to perform favorably compared to a state-of-art method.

This paper introduces an improved Low Rank Adaptive Normalized Matched Filter (LR-ANMF) detector in a high dimensional (HD) context where the observation dimension is large and of the same order of magnitude than the sample size. To that end, the statistical analysis of the LR-ANMF, in a context where the target signal is disturbed by a spatially correlated Gaussian clutter and a spatially white Gaussian noise, is addressed. More specifically, the asymptotic distribution under the null hypothesis is derived, in the regime where both the dimension M of the observations and the number N of samples converge to infinity at the same rate and when the clutter covariance matrix has fixed rank K. In particular, it is shown that the LR-ANMF test statistic does not exhibit the CFAR property in the previous asymptotic regime. A correction to the LR-ANMF test is then proposed to ensure the asymptotic CFAR property, providing the improved LR-ANMF, termed as HD-LR-ANMF. Its asymptotic distribution is derived under both the null and the alternative hypotheses. Numerical simulations illustrate the fact that, despite the asymptotic nature of the analysis, the results obtained are accurate for reasonable values of M, N.

International audience

This paper presents a new algorithm to estimate the number of sources embedded in a correlated Complex Elliptically Distributed (CES) noise in the context of large dimensional regime. The proposed method is a two-steps ones: first the data covariance matrix is estimated with a robust and consistent estimator exploiting the Toeplitz structure assumption of the true scatter matrix. Then, after whitening the signal thanks to the first estimator, the distribution of its Tyler estimator eigenvalues is studied, as in [1]. This allows to derive a threshold, estimated thanks to the Marchenko-Pastur law, to separate the eigenvalues corresponding to the noise and those corresponding to the sources. The number of sources can therefore be deduced. The proposed method is compared to classical ones as the Akaike Information Criterion (AIC) or other algorithms recently developed.

International audience

One of the most general and acknowledged models for background statistics characterization is the family of elliptically symmetric distributions. They account for heterogeneity and non-Gaussianity of real data. Today, although nonGaussian models are assumed for background modeling and design of detectors, the parameters estimation is usually performed using classical Gaussian-based estimators. This letter analyzes robust estimation techniques in a nonGaussian environment and highlights their interest as an alternative to classical procedures for target detection purposes. The goal of this letter is to extend well-known detection methodologies to nonGaussian framework, when the statistical mean is nonnull and unknown. Furthermore, a theoretical closed-form expression for false-alarm regulation is derived and the Constant False Alarm Rate property is pursued to allow the detector to be independent of nuisance parameters. The experimental validation is conducted on simulations.

Classical target detection schemes are usually obtained by deriving the likelihood ratio under Gaussian hypothesis and replacing the unknown background parameters by their estimates. In most applications, interference signals are assumed to be Gaussian with zero mean [or with a known mean vector (MV)] and with an unknown covariance matrix (CM). When the MV is unknown, it has to be jointly estimated with the CM. In this paper, adaptive versions of the classical matched filter (MF) and the normalized MF, as well as two versions of the Kelly detector are first derived and then analyzed for the case where the MV of the background is unknown. More precisely, theoretical closed-form expressions for false alarm (FA) regulation are derived and the constant FA rate property is pursued to allow the detector to be independent of nuisance parameters. Finally, the theoretical contributions are validated through simulations.

In recent years, remote sensing of the Earth surface using images acquired from aircraft or satellites has gained a lot of attention. The acquisition technology has been evolving fast and, as a consequence, many different kinds of sensors (e.g., optical, radar, multispectral, and hyperspectral) are now available to capture different features of the observed scene. One of the main objectives of remote sensing is to monitor changes on the Earth surface. Change detection has been thoroughly studied in the case of images acquired by the same sensors (mainly optical or radar sensors). However, due to the diversity and complementarity of the images, change detection between images acquired with different kinds of sensors (sometimes referred to as heterogeneous sensors) is clearly an interesting problem. A statistical model and a change detection strategy were recently introduced in [J. Prendes, M. Chabert, F. Pascal, A. Giros, and J.-Y. Tourneret, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, Florence, Italy, 2014; IEEE Trans. Image Process., 24 (2015), pp. 799--812] to deal with images captured by heterogeneous sensors. The main idea of the suggested strategy was to model the objects contained in an analysis window by mixtures of distributions. The manifold defined by these mixtures was then learned using training data belonging to unchanged areas. The changes were finally detected by thresholding an appropriate distance to the estimated manifold. This paper goes a step further by introducing a Bayesian nonparametric framework allowing us to deal with an unknown number of objects in analysis windows without specifying an upper bound for this number. A Markov random field is also introduced to account for the spatial correlation between neighboring pixels. The proposed change detector is validated using different sets of synthetic and real images (including pairs of optical images and pairs of optical and radar images) showing a significant improvement when compared to existing algorithms.

This paper presents two approaches to derive an asymptotic distribution of the robust Adaptive Normalized Matched Filter (ANMF). More precisely, the ANMF has originally been derived under the assumption of Gaussian distributed noise where the variance is different between the observation under test and the set of secondary data. We propose in this work to relax the Gaussian hypothesis: we analyze the ANMF built with robust estimators, namely the M-estimators and the Tyler's estimator, under the Complex Elliptically Symmetric (CES) distributions framework. In this context, we derive two asymptotic distributions for this robust ANMF. Firstly, we combine the asymptotic properties of the robust estimators and the Gaussian-based distribution of the ANMF at finite distance. Secondly, we directly derive the asymptotic distribution of the robust ANMF. Then, Monte-Carlo simulations show the good approximation provided by the proposed methods. Moreover, for a non-asymptotic regime, the simulations provide very promising results.

The paper addresses the problem of approximating the detector distribution used in target detection embedded in a disturbance composed of a low rank Gaussian noise and a white Gaussian noise. In this context, it is interesting to use an adaptive version of the Low Rank Normalized Matched Filter (LR-ANMF) detector, which is a function of the estimated projector onto the low rank noise subspace. We will show that the traditional approximation of the LR-ANMF detector distribution is not always the better one. In this paper, we propose to perform its limits when the number of secondary data K and the data dimension m both tend to infinity at the same rate m/K → c∈2 (0;∞). Then, we give the theoretical distributions of these limits in the large dimensional regime and approximate the LR-ANMF detector distribution by them. The comparison of empirical and theoretical distributions on a jamming application shows the interest of our approach.

This paper lies in the lineage of recent works studying the asymptotic behaviour of robust-scatter estimators in the case where the number of observations and the dimension of the population covariance matrix grow at infinity with the same pace. In particular, we analyze the fluctuations of bilinear forms of the robust shrinkage estimator of covariance matrix. We show that this result can be leveraged in order to improve the design of robust detection methods. As an example, we provide an improved generalized likelihood ratio based detector which combines robustness to impulsive observations and optimality across the shrinkage parameter, the optimality being considered for the false alarm regulation.

This paper introduces a Bayesian non parametric (BNP) model associated with a Markov random field (MRF) for detecting changes between remote sensing images acquired by homogeneous or heterogeneous sensors. The proposed model is built for an analysis window which takes advantage of the spatial information via an MRF. The model does not require any a priori knowledge about the number of objects contained in the window thanks to the BNP framework. The change detection strategy can be divided into two steps. First, the segmentation of the two images is performed using a region based approach. Second, the joint statistical properties of the objects in the two images allows an appropriate manifold to be defined. This manifold describes the relationships between the different sensor responses to the observed scene and can be learnt from a training unchanged area. It allows us to build a similarity measure between the images that can be used in many applications such as change detection or image registration. Simulation results conducted on synthetic and real optical and synthetic aperture radar (SAR) images show the efficiency of the proposed method for change detection.

In this letter, we propose an alternative proof for the uniqueness of Maronna's M-estimator of scatter for N vector observations y1, ..., yN∈ Rmunder a mild constraint of linear independence of any subset of m of these vectors. This entails in particular almost sure uniqueness for random vectors yi with a density as long as N > m. This approach allows to establish further relations that demonstrate that a properly normalized Tyler's M-estimator of scatter can be considered as a limit of Maronna's M-estimator. More precisely, the contribution is to show that each M-estimator, verifying some mild conditions, converges towards a particular Tyler's M-estimator. These results find important implications in recent works on the large dimensional (random matrix) regime of robust M-estimation.

Remote sensing images are commonly used to monitor the earth surface evolution. This surveillance can be conducted by detecting changes between images acquired at different times and possibly by different kinds of sensors. A representative case is when an optical image of a given area is available and a new image is acquired in an emergency situation (resulting from a natural disaster for instance) by a radar satellite. In such a case, images with heterogeneous properties have to be compared for change detection. This paper proposes a new approach for similarity measurement between images acquired by heterogeneous sensors. The approach exploits the considered sensor physical properties and specially the associated measurement noise models and local joint distributions. These properties are inferred through manifold learning. The resulting similarity measure has been successfully applied to detect changes between many kinds of images, including pairs of optical images and pairs of optical-radar images.

In the context of an heterogeneous disturbance with a Low Rank (LR) structure (called clutter), one may use the LR approximation for filtering and detection process. These methods are based on the projector onto the clutter subspace instead of the noise covariance matrix. In such context, adaptive LR schemes have been shown to require less secondary data to reach equivalent performances as classical ones. The main problem is then the estimation of the clutter subspace instead of the noise covariance matrix itself. Maximum Likelihood estimator (MLE) of the clutter subspace has been recently studied for a noise composed of a LR Spherically Invariant Random Vector (SIRV) plus a white Gaussian Noise (WGN). This paper focuses on environments with a high Clutter to Noise Ratio (CNR). An original MLE of the clutter subspace is proposed in this context. A cross-interpretation of this new result and previous ones is provided. Validity and interest - in terms of performance and robustness - of the different approaches are illustrated through simulation results.

This paper introduces a new statistical model for homogeneous images acquired by the same kind of sensor (e.g., two optical images) and heterogeneous images acquired by different sensors (e.g., optical and synthetic aperture radar (SAR) images). The proposed model assumes that each image pixel is distributed according to a mixture of multi-dimensional distributions depending on the noise properties and on the transformation between the actual scene and the image intensities. The parameters of this new model can be estimated by the classical expectation-maximization algorithm. The estimated parameters are finally used to learn the relationships between the different images. This information can be used in many image processing applications, particularly those requiring a similarity measure (e.g., change detection or registration). Simulation results on synthetic and real images show the potential of the proposed model. A brief application to change detection between optical and SAR images is finally investigated.

The Reed-Xiaoli (RX) is considered as the benchmark algorithm in multidimensional anomaly detection (AD). However, the RX detector performance decreases when the statistical parameters estimation is poor. This could happen when the background is non-homogeneous or the noise independence assumption is not fulfilled. For a better performance, the statistical parameters are estimated locally using a sliding window approach. In this approach, called adaptive RX, a window is centered over the pixel under the test (PUT), so the background mean and covariance statistics are estimated using the data samples lying inside the window's spatial support, named the secondary data. Sometimes, a smaller guard window prevents those pixels close to the PUT to be used, in order to avoid the presence of outliers in the statistical estimation. The size of the window is chosen large enough to ensure the invertibility of the covariance matrix and small enough to justify both spatial and spectral homogeneity. We present here an alternative methodology to select the secondary data for a PUT by means of a binary partition tree (BPT) representation of the image. We test the proposed BPT-based adaptive hyperspectral RX AD algorithm using a real dataset provided by the Target Detection Blind Test project.

Anomaly Detection methods are used when there is not enough information about the target to detect. These methods search for pixels in the image with spectral characteristics that differ from the background. The most widespread detection test, the RX-detector, is based on the Mahalanobis distance and on the background statistical characterization through the mean vector and the covariance matrix. Although non-Gaussian distributions have already been introduced for background modeling in Hyperspectral Imaging, the parameters estimation is still performed using the Maximum Likelihood Estimates for Gaussian distribution. This paper describes robust estimation procedures more suitable for non-Gaussian environment. Therefore, they can be used as plug-in estimators for the RX-detector leading to some great improvement in the detection process. This theoretical improvement has been evidenced over two real hyperspectral images.

This paper studies the limiting behavior of a class of robust population covariance matrix estimators, originally due to Maronna in 1976, in the regime where both the number of available samples and the population size grow large. Using tools from random matrix theory, we prove that, for sample vectors made of independent entries having some moment conditions, the difference between the sample covariance matrix and (a scaled version of) such robust estimator tends to zero in spectral norm, almost surely. This result can be applied to various statistical methods arising from random matrix theory that can be made robust without altering their first order behavior.

An original interface between robust estimation theory and random matrix theory for the estimation of population covariance matrices is proposed. Consider a random vector x = ANy ∈ CNwith y ∈ CMmade of M ≥ N independent entries, E[y] = 0, and E[yy*] = IN. It is shown that a class of robust estimators ĈNof CN= ANA*N, obtained from n independent copies of x, is (N, n)-consistent with the traditional sample covariance matrix r̂Nin the sense that ∥ĈN- αr̂N∥ → 0 in spectral norm for some α > 0, almost surely, as N, n → ∞ with N/n and M/N bounded. This result, in general not valid in the fixed N regime, is used to propose improved subspace estimation techniques, among which an enhanced direction-of-arrival estimator called robust G-MUSIC.

Recently, a general approach for high-resolution polarimetric SAR (POLSAR) data classification in heterogeneous clutter was presented, based on a statistical test of equality of covariance matrices. Here, we extend that approach by taking advantage of the Constant False Alarm Ratio (CFAR) property of the statistical test in order to improve the clustering process. We show that the CFAR property can be used in the hierarchical segmentation of the POLSAR data images to automatically detect the number of clusters. The proposed method will be applied on a high-resolution polarimetric data set acquired by the ONERA RAMSES system.

When accounting for heterogeneity and non-Gaussianity of real hyperspectral data, elliptical distributions provide reliable models for background characterization. Through these assumptions, this paper highlights the fact that robust estimation procedures are an interesting alternative to classical methods and can bring some great improvement to the detection process. The goal of this paper is then not only to recall well-known methodologies of target detection but also to propose ways to extend them for taking into account the heterogeneity and non-Gaussianity of the hyperspectral images.

This paper studies the performance of the maximum likelihood estimators (MLE) for the parameters of multivariate generalized Gaussian distributions. When the shape parameter belongs to ]0, 1[, we have proved that the scatter matrix MLE exists and is unique up to a scalar factor. After providing some elements about this proof, an estimation algorithm based on a Newton-Raphson recursion is investigated. Some experiments illustrate the convergence speed of this algorithm. The bias and consistency of the scatter matrix estimator are then studied for different values of the shape parameter. The performance of the shape parameter estimator is finally addressed by comparing its variance to the Cramér-Rao bound.

When dealing with impulsive background echoes, Gaussian model is no longer pertinent. We study in this paper the class of elliptically contoured (EC) distributions. They provide a multivariate location-scatter family of distributions that primarily serve as long tailed alternatives to the multivariate normal model. They are proven to represent a more accurate characterization of HSI data than models based on the multivariate Gaussian assumption. For data in ℝk, robust proposals for the sample covariance estimate are the M-estimators. We have also analyzed the performance of an adaptive non- Gaussian detector built with these improved estimators. Constant False Alarm Rate (CFAR) is pursued to allow the detector independence of nuisance parameters and false alarm regulation.

We discuss in the paper the use of the Riemannian mean given by the differential geometric tools. This geometric mean is used in this paper for computing the centers of class in the polarimetric H/α unsupervised classification process. We can show that the centers of class will remain more stable during the iteration process, leading to a different interpretation of the H/α/A classification. This technique can be applied both on classical SCM and on Fixed Point covariance matrices. Used jointly with the Fixed Point CM estimate, this technique can give nice results when dealing with high resolution and highly textured polarimetric SAR images classification.

This paper deals with hyperspectral detection in impulsive and/or non homogeneous background contexts. In hyperspectral imaging applications, the detection performance of the detectors (target detection or anomaly detection like Mahalanobis distance) is typically evaluated on Gaussian assumption. However, it is well known that hyperspectral imaging data exhibit spatial heterogeneity and non-Gaussian behavior leading to a poor performance for all the conventional Gaussian detectors. Many works have been already derived in the context of radar detection in non-homogeneous and non-Gaussian clutter. These works can be easily extended in the context of hyperspectral detection. The aim of this pa per is twofold. In the context of Spherically Invariant Random Vectors (SIRV) modeling for the background, we re call some properties of different non-Gaussian detectors built with a nice and robust estimate of the background Covariance Matrix. Secondly, we present some results on regulation of false alarm obtained on experimental background hyper spectral data. These results demonstrate the interest of the proposed detection scheme, and show an excellent correspondence between experimental and theoretical results.

This paper presents a new estimation scheme for optimally deriving clutter parameters with high resolution POLSAR data. The heterogeneous clutter in POLSAR data is described by the Spherically Invariant Random Vectors model. Three parameters are introduced for the high resolution POLSAR data clutter: the span, the normalized texture and the speckle normalized covariance matrix. The asymptotic distribution of the novel span estimator is investigated. A novel heterogeneity test for the POLSAR clutter is also discussed. The proposed method is tested with airborne POLSAR images provided by the ONERA RAMSES system.

This letter presents a new estimation scheme for optimally deriving clutter parameters with high-resolution polarimetric synthetic aperture radar (POLSAR) data. The heterogeneous clutter in POLSAR data is described by the spherically invariant random vector model. Three parameters are introduced for the high-resolution POLSAR data clutter: the span, the normalized texture, and the speckle normalized covariance matrix. The asymptotic distribution of the novel span estimator is investigated. A novel heterogeneity test for the POLSAR clutter is also discussed. The proposed method is tested with airborne POLSAR images provided by the Office National d'Études et de Recherches Aerospatiales Radar Aéroporté Multi-spectral d'Etude des Signatures system.

Previously, the well-known Optimum Gaussian Detector (OGD) has been extended to the Multiple-Input Multiple-Output (MIMO) case where all transmit-receive subarrays are considered jointly as a system such that only one detection threshold is used. In this extension, all subarrays have been assumed to be widely separated and the transmitted waveforms are assumed to be orthogonal. However, the necessary separation needed for each subarray to be uncorrelated depends on several factors and it might not be possible to ensure that this condition is always respected, especially in the case of moving platforms. Moreover, perfectly orthogonal waveforms do not exist. Hence, we consider in this paper, a new robust MIMO detector that is able to maintain the same Probability of False Alarm (Pfa) irregardless of the correlation between the subarrays.

Based on the Kennaugh-Huynen decomposition, the Target Scattering Vector Model (TSVM) allows to extract four roll-invariant parameters. Those parameters are necessary for an unambiguous description of the target scattering mechanism. The proposed method consists in applying the TSVM prior to the GLRT-LQ detector for the detection of any oriented target.

Modern SAR systems have high resolution which leads the backscattering clutter to be non-Gaussian. In order to properly classify images from these systems, a non-Gaussian noise model is considered: the SIRV model. A statistical test of equality of covariance matrices is used to classify pixels, taking into account the critical region of the test which rejects the likeliness of a covariance matrix to any of the class centers. This test is applied on experimental data obtained with the ONERA RAMSES system in X-band. The results show a good separation between natural and man-made areas of the image.

This paper presents a new estimation scheme for optimally deriving clutter parameters with high resolution POLSAR data. The heterogeneous clutter in POLSAR data was described by the Spherically Invariant Random Vectors model. Three parameters were introduced for the high resolution POLSAR data clutter: the span, the normalized texture and the speckle normalized covariance matrix. The asymptotic distribution of the novel span estimator is also investigated. The proposed method is tested with airborne POLSAR images provided by the ONERA RAMSES system.

This paper presents an application of the recent advances in the field of spherically invariant random vector (SIRV) modeling for coherency matrix estimation in heterogeneous clutter. The complete description of the polarimetric synthetic aperture radar (POLSAR) data set is achieved by estimating the span and the normalized coherency independently. The normalized coherency describes the polarimetric diversity, while the span indicates the total received power. The main advantages of the proposed fixed-point (FP) estimator are that it does not require anya prioriinformation about the probability density function of the texture (or span) and that it can directly be applied on adaptive neighborhoods. Interesting results are obtained when coupling this FP estimator with an adaptive spatial support based on the scalar span information. Based on the SIRV model, a new maximum-likelihood distance measure is introduced for unsupervised POLSAR classification. The proposed method is tested with both simulated POLSAR data and airborne POLSAR images provided by the Radar Ae¿roporte¿ Multi-Spectral d'Etude des Signatures system. Results of entropy/alpha/anisotropy decomposition, followed by unsupervised classification, allow discussing the use of the normalized coherency and the span as two separate descriptors of POLSAR data sets.

Reducing the number of secondary data used to estimate the Clutter Covariance Matrix (CCM) for Space Time Adaptive Processing (STAP) techniques is still an active research topic. Low rank CCM estimates have already been proposed but only for homogeneous and Gaussian clutter. We propose in this paper to extend the low-rank CCM methods for heterogeneous and/or non-Gaussian clutter. We derive a new detector based on low-rank techniques and exploiting properties of the Normalized Sample Covariance Matrix (NSCM). This detector is shown to exhibit a smaller SNR loss than classical STAP detectors. Moreover, the new detector has a texture-CFAR property with respect to non-Gaussian SIRV model and has more robust behavior when some targets are present in the secondary data. We also give experimental comparison results between the classical STAP detectors and the new one for STAP data.

In this paper, heterogeneous clutter models are introduced to describe Polarimetric Synthetic Aperture Radar (PolSAR) data. Based on the Spherically Invariant Random Vectors (SIRV) estimation scheme, the scalar texture parameter and the normalized covariance matrix are extracted. If the texture parameter is modeled by a Fisher PDF, the observed target scattering vector follows a KummerU PDF. Then, this PDF is implemented in a hierarchical segmentation algorithm. Segmentation results are shown on high resolution PolSAR data at L and X band.

In the context of non-Gaussian polarimetric clutter models, this paper presents an application of the recent advances in the field of Spherically Invariant Random Vectors (SIRV) modelling for coherency matrix estimation in heterogeneous clutter. The complete description of the POLSAR data set is achieved by estimating the span and the normalized coherency independently. The normalized coherency describes the polarimetric diversity, while the span indicates the total received power. Based on the SIRV model, a new maximum likelihood distance measure is introduced for unsupervised POLSAR segmentation. The proposed method is tested with airborne POLSAR images provided by the RAMSES system.

In the general area of radar detection, estimation of the clutter covariance matrix is an important point. This matrix commonly exhibits a persymmetric structure: this is the case for instance for active systems using a symmetrically spaced linear array or pulse train. In this context, this paper provides a new Gaussian adaptive detector called the persymmetric adaptive matched filter (P-AMF). Its theoretical distribution is derived allowing adjustment of the detection threshold for a given probability of false alarm (PFA). Simulations results highlight the improvement in term of probability of detection (PD) of the P-AMF in comparison with the classical adaptive matched filter (AMF).

This paper presents an application of the recent advances in the field of Spherically Invariant Random Vectors modelling. We propose the use of the Fixed Point (FP) estimator for deriving normalized polarimetric coherency matrices in compound Gaussian clutter. The main advantages of the FP estimator are that it does not require any "a priori" information about the probability density function of the texture and it can be directly applied on adaptive neighborhoods. Interesting results are obtained when coupling this FP estimator with an adaptive spatial support driven on the scalar span information. The proposed method is tested with both simulated POLSAR data and high resolution POLSAR data acquired over the French Alps.

Under Gaussian assumptions, the sample covariance matrix (SCM) is encountered in many covariance based processing algorithms. In case of impulsive noise, this estimate is no more appropriate. This is the reason why when the noise is modeled by spherically invariant random vectors (SIRV), a natural extension of the SCM is extensively used in the literature: the well-known normalized sample covariance matrix (NSCM), which estimates the covariance of SIRV. Indeed, this estimate gets rid of a fluctuating noise power and is widely used in radar applications. The aim of this paper is to derive closed-form expressions of the first- and second-order moments of the NSCM

This paper presents a detailed theoretical analysis of a recently introduced covariance matrix estimator, called the fixed point estimate (FPE). It plays a significant role in radar detection applications. This estimate is provided by the maximum likelihood estimation (MLE) theory when the non-Gaussian noise is modelled as a spherically invariant random process (SIRP). We study in details its properties: existence, uniqueness, unbiasedness, consistency and asymptotic distribution. We propose also an algorithm for its computation and prove the convergence of this numerical procedure. These results allow us to study the performance analysis of the adaptive CFAR radar detectors (GLRT-LQ, BORD, ...).