Interferometric phase linking (IPL) has become a prominent technique for processing images of areas containing distributed scatterers in SAR interferometry. Traditionally, IPL consists in estimating consistent phase differences between all pairs of SAR images in a time series from the sample covariance matrix (SCM) of pixel patches on a sliding window. This article reformulates this task as a covariance fitting problem; IPL appears then as a form of projection of an input covariance matrix so that it satisfies the phase closure property. This approach yields a systematic methodology to frame IPL as an optimization problem on the torus of phase-only complex vectors. On the modeling side, the formulation is modular and allows for a flexible choice of covariance matrix estimates, regularization options, and matrix distances. In particular, we demonstrate that most existing IPL algorithms appear as special instances of this framework. In addition, we propose some new options, which were not covered by the state of the art, whose merits are illustrated through simulations and a real-world case study. On the computational side, another contribution of this article is the derivation of generic and computationally efficient algorithms for IPL using majorization-minimization (MM) and Riemannian optimization.

We introduce a new method for Through-the-Wall Radar Imaging (TWRI) that detects the location of stationary targets hidden by a wall. A crucial step is the mitigation of wall returns which obscure the scene and which are characterized by their low-rankedness given the radar measurement setup. Whereas existing methods make use of nuclear norm minimization or Truncated Singular Value Decomposition (TSVD), we propose to leverage Riemannian optimization over the manifold of fixed-rank matrices in order to use robust estimation while keeping the original rank constraint without relaxation. A detection step via sparse recovery is then performed and the overall method is compared with existing methods over simulated scenes. The results show that the proposed method achieves a better performance.

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.

This paper considers the problem of obtaining a low-rank factorization of a given correlation matrix. In order to handle possible spurious correlation coefficients within the input, a robust formulation is proposed with a criterion based on the Huber loss function. Minimizing this fitting criterion under the low-rank correlation structure constraint is then addressed using the block majorization-minimization framework. Several algorithm options are explored and compared in terms of computational complexity. The merits of the proposed correlation fitting method are then validated on simulations, and for the process of dimension reduction of microarray data.

This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a Riemannian manifold (a smooth manifold endowed with a Riemannian metric) and follows a given prior distribution. In this setup, the chosen Riemannian metric induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure. Performance bounds for such error measure were previously obtained in the non-Bayesian case (when the unknown parameter is assumed to deterministic), and referred to as intrinsic Cramér-Rao bound. The presented result then appears either as: a) an extension of the intrinsic Cramér-Rao bound to the Bayesian estimation framework; b) a generalization of the Van-Trees inequality (Bayesian Cramér-Rao bound) that accounts for the aforementioned geometric structures. In a second part, we leverage this formalism to study the problem of covariance matrix estimation when the data follow a Gaussian distribution, and whose covariance matrix is drawn from an inverse Wishart distribution. Performance bounds for this problem are obtained for both the mean squared error (Euclidean metric) and the natural Riemannian distance for Hermitian positive definite matrices (affine invariant metric). Numerical simulation illustrate that assessing the error with the affine invariant metric is revealing of interesting properties of the maximum a posteriori and minimum mean square error estimator, which are not observed when using the Euclidean metric.

This paper addresses the problem of robust and sparse PCA. We consider a formulation combining a M-estimation type robust subspace recovery term and a mixed norm that promotes structured sparsity in the basis vectors, which is especially interesting for joint dimension reduction and variable selection. To solve it, we propose to leverage variable splitting methods, with the crucial step then lying on the Stiefel manifold. The resolution of this subproblem, involving the orthonormality constraint, is achieved through a tailored majorization-minimization (MM) step. Numerical experiments on gene expression measurements illustrate the interest of the proposal.

This paper proposes an algorithm for phase differences estimation in multi-temporal InSAR. The proposed approach is based on covariance fitting estimation and the majorization-minimization algorithm. Experiments with Sentinel-1 images of Mexico City demonstrate that the proposed approach compares favorably to the state-of-the-art phase linking (i.e., maximum likelihood-based approaches) when the sample support is low (i.e., when the number of pixels in the multi-look window cannot scale with the number of SAR images). Hence, the proposed approach can improve the spatial resolution of phase difference estimation in case of large SAR image time series.

Phase linking is a prominent methodology to estimate coherence and phase difference in interferometric synthetic-aperture radar. This method is driven by a maximum likelihood estimation approach, which allows to fully exploit all the possible interferograms from a time series. Its performance is, however, known to be affected by the accuracy of the covariance matrix estimation step, which usually requires to introduce additional prior information on its structure when there is a small sample support (spatial window). Moreover, most phase linking algorithms are built upon the sample covariance matrix, due to the assumption of an underlying Gaussian distribution. In a scenario where SAR data is high resolution, or when the study area is spatially heterogeneous (e.g., urban area), this assumption can also limit the accuracy of the covariance matrix estimation step. Considering the two aforementioned issues, we introduce alternative statistical models, whose maximum likelihood estimators then yield new phase linking algorithms. In order to be robust to non-Gaussian data, we consider the use of a more general model of scaled mixture of Gaussian. To address small sample support issues, we also generalize this approach to a possibly low-rank structured covariance matrix. A unified algorithm to perform phase linking given these models is then derived and validated by simulations and a real data case (Sentinel-1 data).

This paper presents a new classification framework for both first and second order statistics, i.e. mean/location and covariance matrix. In the last decade, several covariance matrix classification algorithms have been proposed. They often leverage the Riemannian geometry of symmetric positive definite matrices (SPD) with its affine invariant metric and have shown strong performance in many applications. However, their underlying statistical model assumes a zero mean hypothesis. In practice, it is often estimated and then removed in a preprocessing step. This is of course damaging for applications where the mean is a discriminative feature. Unfortunately, the distance associated to the affine invariant metric for both mean and covariance matrix remains unknown. Leveraging previous works on geodesic triangles, we propose two affine invariant divergences that use both statistics. Then, we derive an algorithm to compute the associated Riemannian centers of mass. Finally, a divergence based Nearest centroid, applied on the crop classification dataset Breizhcrops, shows the interest of the proposed framework.

This paper presents a new algorithm for improving the estimation of interferometric SAR (InSAR) phases in the context of time series and phase linking approach. Based on maximum likelihood estimator of a multivariate Gaussian model, the estimation of the InSAR phases is solved using a Block Coordinate Descent algorithm. Compared to the state-of-the-art approaches, the main improvement lies on the joint estimation of the covariance matrix and the InSAR phases instead of using a plug-in coherence estimate obtained from the sample covariance of the data or the modeling of the temporal decorrelation of the target under observation. Results of synthetic simulations confirm the improvement brought by the proposed estimator.

We consider the problem of jointly estimating the location and scatter matrix of a Compound Gaussian distribution with unknown deterministic texture parameters. When the location is known, the Maximum Likelihood Estimator (MLE) of the scatter matrix corresponds to Tyler’s M-estimator, which can be computed using fixed point iterations. However, when the location is unknown, the joint estimation problem remains challenging since the associated standard fixed-point procedure to evaluate the solution may often diverge. In this paper, we propose a stable algorithm based on Riemannian optimization for this problem. Finally, numerical simulations show the good performance and usefulness of the proposed algorithm.

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.

We consider the optimal performance of blind separation of Gaussian sources. In practice, this estimation problem is solved by a two-step procedure: estimation of a set of covariance matrices from the observed data and approximate joint diagonalization of this set to find the unmixing matrix. Rather than studying the theoretical performance of a specific method, we are interested in the optimal attainable performance of any estimator. To do so, we consider the so-called intrinsic Cramér-Rao bound, which exploits the geometry of the parameters of the model. Unlike previous works developing a Cramér-Rao bound in this context, our solution does not require any additional hypotheses. To obtain our bound, we define and study a new Riemannian manifold holding the parameters of interest. An original estimation error measure is defined with the help of our Riemannian distance function. The corresponding Fisher information matrix is then obtained from the Fisher information metric and orthonormal bases on the tangent spaces of the manifold. Finally, our theoretical results are validated on simulated data.

This paper considers the problem of detecting changes in multivariate 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.

Testing the similarity of covariance matrices (CMs) from groups of observations has been shown to be a relevant approach for change and/or anomaly detection in synthetic aperture radar images. Although the term “similarity” usually refers to equality or proportionality, we explore the testing of shared properties in the structure of low rank (LR) plus identity CM, which are appropriate for radar processing. Specifically, we derive two new generalized likelihood ratio tests to infer: 1) on the equality of the LR signal component of CMs and 2) on the proportionality of the LR signal component of CMs. The formulation of the second test involves nontrivial optimization problems for which we tailor efficient majorization-minimization algorithms. Eventually, the proposed detection methods enjoy interesting properties that are illustrated on simulations and on an application to real data for change detection.

Scatter matrix and its normalized counterpart, referred to as shape matrix, are key parameters in multivariate statistical signal processing, as they generalize the concept of covariance matrix in the widely used Complex Elliptically Symmetric distributions. Following the framework of [1], intrinsic Cramér-Rao bounds are derived for the problem of scatter and shape matrices estimation with samples following a Complex Elliptically Symmetric distribution. The Fisher Information Metric and its associated Riemannian distance (namely, CES-Fisher) on the manifold of Hermitian positive definite matrices are derived. Based on these results, intrinsic Cramér-Rao bounds on the considered problems are then expressed for three different distances (Euclidean, natural Riemannian, and CES-Fisher). These contributions are therefore a generalization of Theorems 4 and 5 of [1] to a wider class of distributions and metrics for both scatter and shape matrices.

The joint estimation of means and scatter matrices is often a core problem in multivariate analysis. In order to overcome robustness issues, such as outliers from Gaussian assumption,M-estimators are now preferred to the traditional sample mean and sample covariance matrix. These estimators are well established and studied in the real case since the seventies. Their extension to the complex case has drawn recent interest. In this letter, we derive the asymptotic performance of complexM-estimators for multivariate location and scatter matrix estimation.

This paper addresses structured covariance matrix estimation under t -distribution. Covariance matrices frequently reveal a particular structure due to the considered application and taking into account this structure usually improves estimation accuracy. In the framework of robust estimation, the t -distribution is particularly suited to describe heavy-tailed observation. In this context, we propose an efficient estimation procedure for covariance matrices with convex structure under t -distribution. Numerical examples for Hermitian Toeplitz structure corroborate the theoretical analysis.

International audience

Regularized Tyler Estimator's (RTE) have raised attention over the past years due to their attractive performance over a wide range of noise distributions and their natural robustness to outliers. Developing adaptive methods for the selection of the regularisation parameter α is currently an active topic of research. Indeed, the bias-performance compromise of RTEs highly depends on the considered application. Thus, finding a generic rule that is optimal for every criterion and/or data configurations is not straightforward. This issue is addressed in this paper for undersampled configurations (number of samples lower than the dimension of the data). The paper proposes a new regularisation parameter selection based on a subspace reduction approach. The performance of this method is investigated in terms of estimation accuracy and for adaptive detection purposes, both on simulation and real data.

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.