Algorithmes d’Estimation et de Détection en Contexte Hétérogène Rang Faible

One purpose of array processing is the detection and location of a target in a noisy environment. In most cases (as RADAR or active SONAR), statistical properties of the noise, especially its covariance matrix, have to be estimated using i.i.d. samples. Within this context, several hypotheses are usually made: Gaussian distribution, training data containing only noise, perfect hardware. Nevertheless, it is well known that a Gaussian distribution doesn’t provide a good empirical fit to RADAR clutter data. That’s why noise is now modeled by elliptical process, mainly Spherically Invariant Random Vectors (SIRV). In this new context, the use of the SCM (Sample Covariance Matrix), a classical estimate of the covariance matrix, leads to a loss of performances of detectors/estimators. More efficient estimators have been developed, such as the Fixed Point Estimator and M-estimators.If the noise is modeled as a low-rank clutter plus white Gaussian noise, the total covariance matrix is structured as low rank plus identity. This information can be used in the estimation process to reduce the number of samples required to reach acceptable performance. Moreover, it is possible to estimate the basis vectors of the clutter-plus-noise orthogonal subspace rather than the total covariance matrix of the clutter, which requires less data and is more robust to outliers. The orthogonal projection to the clutter plus noise subspace is usually calculated from an estimatd of the covariance matrix. Nevertheless, the state of art does not provide estimators that are both robust to various distributions and low rank structured.In this Thesis, we therefore develop new estimators that are fitting the considered context, to fill this gap. The contributions are following three axes :- We present a precise statistical model : low rank heterogeneous sources embedded in a white Gaussian noise.We express the maximum likelihood estimator for this context.Since this estimator has no closed form, we develop several algorithms to reach it effitiently.- For the considered context, we develop direct clutter subspace estimators that are not requiring an intermediate Covariance Matrix estimate.- We study the performances of the proposed methods on a Space Time Adaptive Processing for airborne radar application. Tests are performed on both synthetic and real data.Une des finalités du traitement d’antenne est la détection et la localisation de cibles en milieu bruité.Dans la plupart des cas pratiques, comme par exemple pour les traitements adaptatifs RADAR, il fautestimer dans un premier temps les propriétés statistiques du bruit, plus précisément sa matrice de covariance.Dans ce contexte, on formule généralement l’hypothèse de bruit gaussien. Il est toutefois connuque le bruit en RADAR est de nature impulsive et que l’hypothèse gaussienne est parfois mal adaptée.C’est pourquoi, depuis quelques années, le bruit, et en particulier le fouillis de sol, est modélisé pardes processus couvrant un panel plus large de distributions, notamment les Spherically Invariant RandomVectors (SIRVs). Dans ce nouveau cadre théorique, la Sample Covariance Matrix (SCM) estimantclassiquement la matrice de covariance du bruit entraîne des pertes de performances importantes desdétecteurs/estimateurs. Dans ce contexte non-gaussien, d’autres estimateurs (e.g. les M-estimateurs),mieux adaptés à ces statistiques de bruits impulsifs, ont été développés.Parallèlement, il est connu que le bruit RADAR se décompose sous la forme d’une somme d’unfouillis de rang faible (la réponse de l’environnement) et d’un bruit blanc (le bruit thermique). La matricede covariance totale du bruit a donc une structure de type rang faible plus identité. Cette informationpeut être utilisée dans le processus d’estimation afin de réduire le nombre de données nécessaires. Deplus, il aussi est possible de construire des traitements adaptatifs basés sur un estimateur du projecteurorthogonal au sous espace fouillis, à la place d’un estimateur de la matrice de covariance. Les traitementsadaptatifs basés sur cette approximation nécessitent aussi moins de données secondaires pour atteindredes performances satisfaisantes. On estime classiquement ce projecteur à partir de la décomposition envaleurs singulières d’un estimateur de la matrice de covariance.Néanmoins l’état de l’art ne présente pas d’estimateurs à la fois robustes aux distributions impulsives,et rendant compte de la structure rang faible des données. C’est pourquoi nos travaux se focalisentsur le développement de nouveaux estimateurs (de covariance et de sous espace fouillis) directementadaptés au contexte considéré. Les contributions de cette thèse s’orientent donc autour de trois axes :- Nous présenterons le modèle de sources impulsives ayant une matrice de covariance de rang faiblenoyées dans un bruit blanc gaussien. Ce modèle, fortement justifié dans de nombreuses applications, acependant peu été étudié pour la problématique d’estimation de matrice de covariance. Le maximum devraisemblance de la matrice de covariance pour ce contexte n’ayant pas une forme analytique directe,nous développerons différents algorithmes pour l’atteindre efficacement- Nous développerons de plus nouveaux estimateurs directs de projecteur sur le sous espace fouillis, nenécessitant pas un estimé de la matrice de covariance intermédiaire, adaptés au contexte considéré.- Nous étudierons les performances des estimateurs proposés sur une application de Space Time AdaptativeProcessing (STAP) pour radar aéroporté, au travers de simulations et de données réelles

Clutter Subspace Estimation in Low Rank Heterogeneous Noise Context

International audienceThis paper addresses the problem of the Clutter Subspace Projector (CSP) estimation in the context of a disturbance composed of a Low Rank (LR) heterogeneous clutter , modeled here by a Spherically Invariant Random Vector (SIRV), plus a white Gaussian noise (WGN). In such context, the corresponding LR adaptive filters and detectors require less training vectors than classical methods to reach equivalent performance. Unlike classical adaptive processes, which are based on an estimate of the noise Covariance Matrix (CM), the LR processes are based on a CSP estimate. This CSP estimate is usually derived from a Singular Value Decomposition (SVD) of the CM estimate. However, no Maximum Likelihood Estimator (MLE) of the CM has been derived for the considered disturbance model. In this paper, we introduce the fixed point equation that MLE of the CSP satisfies for a disturbance composed of a LR-SIRV clutter plus a zero mean WGN. A recursive algorithm is proposed to compute this solution. Numerical simulations validate the introduced estimator and illustrate its interest compared to the current state of art

CFAR property and robustness of the lowrank adaptive normalized matched filters detectors in low rank compound gaussian context

International audienceIn the context of an heterogeneous disturbance with a Low Rank (LR) structure (referred to as clutter), one may use the LR approximation for detection process. Indeed, in such context, adaptive LR schemes have been shown to require less secondary data to reach equivalent performances as classical ones. The LR approximation consists on cancelling the clutter rather than whitening the whole noise. The main problem is then the estimation of the clutter subspace instead of the noise covariance matrix itself. Maximum Likelihood estimators (MLE), under different hypothesis [1][2][3], of the clutter subspace have been recently proposed for a noise composed of a LR Compound Gaussian (CG) clutter plus a white Gaussian Noise (WGN). This paper focuses on the performances of the LR Adaptive Normalized Matched Filter (LR-ANMF) detector based on these different clutter subspace estimators. Numerical simulations illustrate its CFAR property and robustness to outliers

Through the Wall Radar Imaging via Kronecker-structured Huber-type RPCA

The detection of multiple targets in an enclosed scene, from its outside, is a challenging topic of research addressed by Through-the-Wall Radar Imaging (TWRI). Traditionally, TWRI methods operate in two steps: first the removal of wall clutter then followed by the recovery of targets positions. Recent approaches manage in parallel the processing of the wall and targets via low rank plus sparse matrix decomposition and obtain better performances. In this paper, we reformulate this precisely via a RPCA-type problem, where the sparse vector appears in a Kronecker product. We extend this approach by adding a robust distance with flexible structure to handle heterogeneous noise and outliers, which may appear in TWRI measurements. The resolution is achieved via the Alternating Direction Method of Multipliers (ADMM) and variable splitting to decouple the constraints. The removal of the front wall is achieved via a closed-form proximal evaluation and the recovery of targets is possible via a tailored Majorization-Minimization (MM) step. The analysis and validation of our method is carried out using Finite-Difference Time-Domain (FDTD) simulated data, which show the advantage of our method in detection performance over complex scenarios

Robust estimation of the clutter subspace for a Low Rank heterogeneous noise under high Clutter to Noise Ratio assumption

International audienceIn 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

Robust Low-rank Change Detection for SAR Image Time Series

International audienceThis paper considers the problem of detecting changes in mul-tivariate Synthetic Aperture Radar image time series. Classical methodologies based on covariance matrix analysis are usually built upon the Gaussian assumption, as well as an unstructured signal model. Both of these hypotheses may be inaccurate for high-dimension/resolution images, where the noise can be heterogeneous (non-Gaussian) and where all channels are not always informative (low-rank structure). In this paper, we tackle these two issues by proposing a new detector assuming a robust low-rank model. Analysis of the proposed method on a UAVSAR dataset shows promising results

Robust Geometric Metric Learning

This paper proposes new algorithms for the metric learning problem. We start by noticing that several classical metric learning formulations from the literature can be viewed as modified covariance matrix estimation problems. Leveraging this point of view, a general approach, called Robust Geometric Metric Learning (RGML), is then studied. This method aims at simultaneously estimating the covariance matrix of each class while shrinking them towards their (unknown) barycenter. We focus on two specific costs functions: one associated with the Gaussian likelihood (RGML Gaussian), and one with Tyler's M -estimator (RGML Tyler). In both, the barycenter is defined with the Riemannian distance, which enjoys nice properties of geodesic convexity and affine invariance. The optimization is performed using the Riemannian geometry of symmetric positive definite matrices and its submanifold of unit determinant. Finally, the performance of RGML is asserted on real datasets. Strong performance is exhibited while being robust to mislabeled data.Comment: Published in EUSIPCO 2022. Best student paper awar

Estimation par Maximum de Vraisemblance du sous espace clutter dans un bruit hétérogène rang faible avec application au STAP

National audienceDans le contexte d'une cible noyée dans un clutter hétérogène de rang faible plus un bruit blanc gaussien, les méthodes de filtrage rang faible requièrent moins de données secondaires que les méthodes classiques pour atteindre des performances équivalentes. Le filtre de rang faible est notamment composé du projecteur sur le sous espace clutter. Celui ci étant en pratique inconnu, il est nécessaire de l'estimer. Nous proposons dans ce papier un estimateur par maximum de vraisemblance du projecteur sur le sous espace clutter pour un bruit composé d'un cluter SIRV (Spherically Invariant Random Vectors) de rang faible et d'un bruit blanc gaussien. Les performances de ce nouvel estimateur sont testées sur des simulations de validation ainsi qu'une application de Space Time Adaptive Processing (STAP) [1]

Riemannian optimization for non-centered mixture of scaled Gaussian distributions

This paper studies the statistical model of the non-centered mixture of scaled Gaussian distributions (NC-MSG). Using the Fisher-Rao information geometry associated to this distribution, we derive a Riemannian gradient descent algorithm. This algorithm is leveraged for two minimization problems. The first one is the minimization of a regularized negative log- likelihood (NLL). The latter makes the trade-off between a white Gaussian distribution and the NC-MSG. Conditions on the regularization are given so that the existence of a minimum to this problem is guaranteed without assumptions on the samples. Then, the Kullback-Leibler (KL) divergence between two NC-MSG is derived. This divergence enables us to define a minimization problem to compute centers of mass of several NC-MSGs. The proposed Riemannian gradient descent algorithm is leveraged to solve this second minimization problem. Numerical experiments show the good performance and the speed of the Riemannian gradient descent on the two problems. Finally, a Nearest centroid classifier is implemented leveraging the KL divergence and its associated center of mass. Applied on the large scale dataset Breizhcrops, this classifier shows good accuracies as well as robustness to rigid transformations of the test set

MAXIMUM LIKELIHOOD ESTIMATION OF CLUTTER SUBSPACE IN NON HOMOGENEOUS NOISE CONTEXT

International audienceIn the context of a disturbance composed of a Low Rank (LR) clutter plus a white Gaussian noise, the corresponding LR filters used to detect a target embedded in this disturbance needs less training vectors than classical methods to reach equivalent performance. Unlike the classical one which is based on covariance matrix of the noise, the LR filter is based on the clutter subspace projector. In this paper, we propose a new estimator of the clutter subspace projector for a disturbance composed of a LR Spherically Invariant Random Vectors (SIRV) plus a zero mean white Gaussian noise that does not require prior information on the SIRV's texture. Numerical simulations validate the introduced estimator, and its performance and robustness are tested on a Space Time Adaptive Processing (STAP) simulation