This paper proposes an unrolled Expectation Maximization (EM) algorithm tailored for robust radio interferometric imaging in the presence of non-Gaussian radio interferences . We introduce a compound Gaussian model for the observation noise and derive an unrolled neural architecture based on the EM algorithm to tackle the reconstruction problem in a robust manner. This innovative approach aims to enhance image reconstruction by simultaneously incorporating model information and generalization for the case of non-Gaussian heavy-tailed noise distribution, while leveraging the benefits of deep learning . Our experiments demonstrate significant improvements over state-of-the-art methods, highlighting the efficacy of our proposed scheme in handling the complexities of radiofrequency interference and improving image reconstruction accuracy.

Logistic regression is a fundamental and widely used statistical method for modeling binary outcomes based on covariates. However, the presence of missing data, particularly in settings involving hybrid covariates (a mix of discrete and continuous variables), poses significant challenges. In this paper, we propose a novel Expectation-Maximization based algorithm tailored for parameter estimation in logistic regression models with missing hybrid covariates. The proposed method is specifically designed to handle these complexities, delivering efficient parameter estimates. Through comprehensive simulations and real-world application, we demonstrate that our approach consistently outperforms traditional methods, achieving superior accuracy and reliability.

Multi-Temporal Interferometric Synthetic Aperture Radar (MT-InSAR) exploits Synthetic Aperture Radar images time series (SAR-TS) for surface deformation monitoring via phase difference (with respect to a reference image) estimation. Most of the actual state-of-the-art MT-InSAR rely on temporal covariance matrix of the SAR-TS, assuming Gaussian distribution. However, these approaches become computationally expensive when the time series lengthens and new images are added to the data vector. This paper proposes a novel approach to sequentially integrate each newly acquired image using Phase Linking (PL) and Maximum Likelihood Estimation (MLE). The methodology divides the data into blocks, using previous images and estimations as a prior to sequentially estimate the phase of the new image. Actually, this framework allows to consider non Gaussian distributions, such as a mixture of scaled Gaussian distribution, which is particularly important to consider when dealing with urban areas.

Interferences and computational cost pose significant challenges in large-scale interferometric sensing, impacting the accuracy and numerical efficiency of imaging algorithms. In this letter, we introduce an imaging algorithm using antenna switching based on expectation-maximization (EM) to address both challenges. By leveraging the low-rank noise model, our approach effectively captures interferences in interferometric data. Additionally, the proposed switching strategy between different sub-arrays reduces significantly the computational complexity during image restoration. Through extensive experiments on simulated datasets, we demonstrate the superiority of the low-rank noise model over the Gaussian noise model in the presence of interferences. Furthermore, we show that the proposed switching approach yields similar imaging performance with fewer antennas compared to the full array configuration, thereby reducing computational complexity, while outperforming non-switching configurations with the same number of antennas.

Traditional Phase-Linking (PL) algorithms are known for their high cost, especially with the huge volume of Synthetic Aperture Radar (SAR) images generated by Sentinel-1 SAR missions. Recently, a COvariance Fitting Interferometric Phase Linking (COFI-PL) approach has been proposed, which can be seen as a generic framework for existing PL methods. Although this method is less computationally expensive than traditional PL approaches, COFI-PL exploits the entire covariance matrix, which poses a challenge with the increasing time series of SAR images. However, COFI-PL, like traditional PL approaches, cannot accommodate the efficient inclusion of newly acquired SAR images. This paper overcomes this drawback by introducing a sequential integration of a block of newly acquired SAR images. Specifically, we propose a method for effectively addressing optimization problems associated with phase-only complex vectors on the torus based on the Majorization-Minimization framework. The proposed approach demonstrates comparable performance to the offline COFI-PL method while achieving a reduction in computation time , of approximately 15% in both simulations and real data experiments. Moreover, it outperforms state-of-the-art sequential approaches in terms of both accuracy and speed.

This paper introduces a novel approach for sequential estimation of the interferometric phase in the context of long Synthetic Aperture Radar (SAR) image time series. When newly acquired data arrive, the data set expands and can be partitioned into two distinct blocks. One represents the previous SAR images and the other represents the newly acquired data. The proposed approach (S-MLE-PL) exploits sequential maximum likelihood estimation of the covariance matrix of the whole data set, taking the existing data set as prior information. This approach facilitates the continuous interferometric phase estimation by incorporating the new data into the previous context. In addition, it presents the advantage of reduced computation time compared to the traditional approaches, making it a more efficient solution for operational displacement estimation.

This paper addresses the challenge of interference mitigation and reduction of computational cost in the context of radio interferometric imaging. We propose a novel maximum-likelihood-based methodology based on the antenna sub-array switching technique, which strikes a refined balance between imaging accuracy and computational efficiency. In addition, we tackle robustness regarding radio interference by modeling the additive noise as t-distributed. Through simulation results, we demonstrate the superiority of the t-distributed noise model over the conventional Gaussian noise model in scenarios involving interferences. We evidence that our proposed switching approach yields similar imaging performances with far fewer visibilities compared to the full array configuration, thus, diminishing the computational complexity.

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.

In practical applications, non-Gaussianity of the signal at the sensor array is detrimental to the performance of conventional Direction-of-Arrival (DOA) estimators developed under the Gaussian model. In this paper, we propose a novel robust DOA estimator from the data collected at the sensor array under the corruption of non-Gaussian interference and noise. Additionally, the Cramér-Rao bound for DOA parameters under the considered signal model is derived. Simulation results show that the proposed estimator exhibits near-optimal estimation performance under the assumed model while being robust to model mismatch and/or the presence of outliers.

A growing problem in the remote sensing community concerns the estimation of change-points in a time series of Synthetic Aperture Radar (SAR) images. Although the methodologies of change-point estimation have already been investigated in the literature, there are, to the best of our knowledge, no study on the expected performance for the estimation of change-points in a Wishart distributed time series. This is mainly due to the fact that few results exist on change-point estimation performance in the mathematical literature: the classical central limit theorem does not apply and the classical Cramer-Rao Bound does not exist due to the discrete nature of the parameters. To fill this gap, this paper proposes to use a lower-bound on the Mean Square Error (MSE) with fewer regularity conditions. To this end, recent works on hybrid Cramer-Rao/Weiss-Weinstein bound have been adapted to the specific SAR problematic of interest. Since estimation strategies usually rely on a set of parameters which have to be set by the user, we show how the proposed lower bound allows performing an appropriate tuning. Moreover, the proposed bound is computationally efficient which enables an extensive analysis without a high computational cost.

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.

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.

This paper investigates calibration of sensor arrays in the radio astronomy context. Current and future radio telescopes require computationally efficient algorithms to overcome the new technical challenges as large collecting area, wide field of view and huge data volume. Specifically, we study the calibration of radio interferometry stations with significant direction dependent distortions. We propose an iterative robust calibration algorithm based on a relaxed maximum likelihood estimator for a specific context: i) observations are affected by the presence of outliers and ii) parameters of interest have a specific structure depending on frequency. Variation of parameters across frequency is addressed through a distributed procedure, which is consistent with the new radio synthesis arrays where the full observing bandwidth is divided into multiple frequency channels. Numerical simulations reveal that the proposed robust distributed calibration estimator outperforms the conventional non-robust algorithm and/or the mono-frequency case.

This paper derives lower bounds for the mean square errors of parameter estimators in the case of Poisson distributed data subjected to multiple abrupt changes. Since both change locations (discrete parameters) and parameters of the Poisson distribution (continuous parameters) are unknown, it is appropriate to consider a mixed Cramér-Rao/Weiss-Weinstein bound for which we derive closed-form expressions and illustrate its tightness by numerical simulations.

A fairly general class of Bayesian “large-error” lower bounds of the Weiss-Weinstein family, essentially free from regularity conditions on the probability density functions support, and for which a limiting form yields a generalized Bayesian Cramér-Rao bound (BCRB), is introduced. In a large number of cases, the generalized BCRB appears to be the Bobrovsky-Mayer-Wolf-Zakai bound (BMZB). Interestingly enough, a regularized form of the Bobrovsky-Zakai bound (BZB), applicable when the support of the prior is a constrained parameter set, is obtained. Modified Weiss-Weinstein bound and BZB which limiting form is the BMZB are proposed, in expectation of an increased tightness in the threshold region. Some of the proposed results are exemplified with a reference problem in signal processing: the Gaussian observation model with parameterized mean and uniform prior.

This paper investigates parametric direction-of-arrival (DOA) estimation in a particular context: i) each sensor is characterized by an unknown complex gain and ii) the array consists of a collection of subarrays which are substantially separated from each other leading ] to a structured noise covariance matrix. We propose two iterative algorithms based on the maximum likelihood (ML) estimation method adapted to the context of joint array calibration and DOA estimation. Numerical simulations reveal that the two proposed schemes, the iterative ML (IML) and the modified iterative ML (MIML) algorithms for joint array calibration and DOA estimation, outperform the state of the art methods and the MIML algorithm reaches the Cramer-Rao bound for a low number of iterations.

The maximum likelihood (ML) and maximum a posteriori (MAP) estimation techniques are widely used to address the direction-of-arrival (DOA) estimation problems, an important topic in sensor array processing. Conventionally the ML estimators in the DOA estimation context assume the sensor noise to follow a Gaussian distribution. In real-life application, however, this assumption is sometimes not valid, and it is often more accurate to model the noise as a non-Gaussian process. In this paper we derive an iterative ML as well as an iterative MAP estimation algorithm for the DOA estimation problem under the spherically invariant random process noise assumption, one of the most popular non-Gaussian models, especially in the radar context. Numerical simulation results are provided to assess our proposed algorithms and to show their advantage in terms of performance over the conventional ML algorithm.

Bayesian hierarchical modelling is a well-established branch of Bayesian inference. In this letter, we derive and study the estimation performance for the Bayesian hierarchical linear model (BHLM). Specifically, we consider a linear model with hierarchical priors for the involved amplitude and noise vectors. We provide closed-form expressions of the Bayesian Cramér-Rao bound (BCRB) for the following settings: (i) an arbitrary prior and hyperprior and (ii) a Gaussian-Y prior for the amplitudes, while the prior of noise is a Gaussian-X in both cases. Gaussian-X means that the conditional prior given the hyperparameter is Gaussian and X is the hyperprior. For the hierarchical distribution associated with spherical invariant random variables, the BCRB has a compact closed-form expression and enjoys several interesting properties that are discussed. Finally, we provide a theoretical analysis of the statistical efficiency of the linear minimum mean square error (MMSE) estimator in the low- and high-noise variance regimes when the hyperparameters are stochastically dominant.

The resolvability of two closely-spaced signals is an important performance measure for parametric estimation problems. In this paper we investigate the so-called resolution limit (RL) in a MIMO radar context, i.e., the minimum angular separation required to resolve two closely-spaced targets. Due to the limited number of elementary scatterers, the Gaussian modeling of the clutter is inappropriate. In our analysis, we consider a K-distributed clutter which is a well-established approximation of the real clutter. Finally, our RL's expression reveals a number of insightful properties that are discussed in detail and, numerical examples are provided to corroborate the theoretical analysis.

This paper is devoted to the analysis of the angular resolution limit (ARL), an important performance measure in the directions-of-arrival estimation theory. The main fruit of our endeavor takes the form of an explicit, analytical expression of this resolution limit, w.r.t. the angular parameters of interest between two closely spaced point sources in the far-field region. As by-products, closed-form expressions of the Cramér-Rao bound have been derived. Finally, with the aid of numerical tools, we confirm the validity of our derivation and provide a detailed discussion on several enlightening properties of the ARL revealed by our expression, with an emphasis on the impact of the signal correlation.

In this paper, we derive the Multidimensional Statistical Resolution Limit (MSRL) to resolve two closely spaced targets using a widely spaced MIMO radar. Toward this end, we perform a hypothesis test formulation using the Generalized Likelihood Ratio Test (GLRT). More precisely, we link the MSRL to the minimum Signal-to-Noise Ratio (SNR) required to resolve two closely spaced targets, for a given probability of false alarm and for a given probability of detection. Finally, theoretical and numerical analysis of the MSRL are given for several scenarios (known/unknown parameters of interest and known/unknown noise variance) including lacunar arrays.

The concept of Statistical Resolution Limit (SRL), which is defined as the minimal separation to resolve two closely spaced signals, is an important tool to quantify performance in parametric estimation problems. This paper generalizes the SRL based on the Cramér-Rao bound to multiple parameters of interest per signal and for multiple signals. We first provide a fresh look at the SRL in the sense of Smith's criterion by using a proper change of variable formula. Second, based on the Minkowski distances, we extend this criterion to the important case of multiple parameters of interest per signal and to multiple signals. The results presented herein can be applied to any estimation problem and are not limited to source localization problems.

Near-field source localization problem by a passive antenna array makes the assumption that the time-varying sources are located near the antenna. In this situation, the far-field assumption (planar wavefront) is no longer valid and we have to consider a more complicated model parameterized by the bearing (as in the far-field case) and by the distance, named range, between the source and a reference sensor. We can find a plethora of estimation schemes in the literature but the ultimate performance has not been fully investigated. In this paper, we derive and analyze the Cramer-Rao Bound (CRB) for a single time-varying source. In this case, we obtain nonmatrix closed-form expressions. Our approach has two advantages: (i) the computational cost for a large number of snapshots of a matrix-based CRB can be high while our approach is cheap and (ii) some useful informations can be deduced from the behavior of the bound. In particular, we show that closer is the source from the array and/or higher is the carrier frequency, better is the estimation of the range.