We address the problem of detecting a multidimensional subspace signal among colored noise with unknown covariance matrix using a second-order model, i.e. assuming that the presence of the signal of interest results in a rank-R modification of the noise covariance matrix. The problem is tackled through the prism of invariance. We derive the group of transformations that leave the problem invariant and subsequently derive the maximal invariant statistic (MIS) along with the induced maximal invariant (IMI). A statistical representation of the MIS is obtained which is quite different from that of a first-order model. Hints at possible test statistics based on the MIS are also provided.

We address the problem of covariance matrix estimation with a view to obtain a regularized version of the sample covariance matrix. Towards this goal, we propose a Bayesian model and we consider the case where the covariance matrix follows an F (beta type II) distribution. The latter offers greater flexibility compared to the usual inverse Wishart distribution and therefore it is of interest to investigate this alternative. We first obtain the corresponding posterior distribution but show that it does not belong to a known class of distributions that could lead to the minimum mean-square error estimator. As an alternative we investigate the maximum a posteriori (MAP) estimator and show that it belongs to the class of orthogonally invariant estimators, i.e., it retains the eigenvectors of the sample covariance matrix but modify, in a non-linear way, its eigenvalues. Two versions of the MAP estimator are derived, based on the Gaussian and Student models for the conditional distribution of the data. Simulations are provided to compare the proposed approach with state-of-the-art techniques based on shrinkage of the sample covariance matrix eigenvalues.

We consider adaptive detection of multidimensional second-order subspace signals in colored noise with a view to provide simpler alternatives to the generalized likelihood ratio test which requires alternate maximization over two sets of matrices. Since the main problem comes from maximum likelihood estimation of the unknown parameters under the alternative hypothesis we consider detection schemes that do not need it. Towards this end we derive closed-form expressions of Rao and Durbin tests for the problem at hand. These detectors are computationally simple and provide a constant false alarm rate. Through numerical simulations we illustrate the trade-off between computational complexity and detection performance.

We consider estimating the population covariance matrix when the number of available samples is less than the size of the observations. The sample covariance matrix (SCM) being singular, regularization is mandatory in this case. For this purpose we consider minimizing Stein's loss function and we investigate a method based on augmenting the partial Cholesky decomposition of the SCM. We first derive the finite sample optimum estimator which minimizes the loss for each data realization, then the Oracle estimator which minimizes the risk, i.e., the average value of the loss. Finally a practical scheme is presented where the missing part of the Cholesky decomposition is filled. We conduct a numerical performance study of the proposed method and compare it with available related methods. In particular we investigate the influence of the condition number of the covariance matrix as well as of the shape of its spectrum.

We consider adaptive radar detection of Rician targets which, when present, result in a shift of the mean of the observations as well as a rank-one modification of the covariance matrix. In the paper, we first derive the Fisher information matrix for the statistical model under consideration. This enables us to obtain the expressions of Rao, Wald, Durbin and gradient tests statistics whose distributions are parameter-free under the null hypothesis. Numerical simulations illustrate their probability of detection in both matched and mismatched cases, i.e., when the signature under test coincides or not with the actual target signature. They indicate that under some scenarios Rao, Durbin and gradient tests may be valuable alternatives to the GLRT.

In this paper we address estimation and detection problems using multiple multichannel observations drawn from a matrix-variate Student distribution, also referred to as uncorrelated t distribution. In a first part we derive the Fisher information matrix when the distribution depends on an unknown parameter vector. Next, we consider adaptive detection of a subspace signal using independent test and training data following a matrix-variate Student distribution. We derive the maximum likelihood estimates of the signal amplitude and of the noise covariance matrix and subsequently the generalized likelihood ratio test. Additionally, the score function and the Fisher information matrix derived in the first part of the paper are used to compute Rao, Wald and gradient tests. The detection performance of these methods is illustrated via numerical simulations.

We consider a modification of the minimum variance distortionless response (MVDR) filter using Stein unbiased risk estimation (SURE). The starting point of this modification lies in the observation that the component of the MVDR filter in the subspace orthogonal to the signal of interest is the maximum likelihood estimate (MLE) of the location of a certain multivariate distribution. This draws us to consider James-Stein type estimates which have been shown to outperform MLE for minimization of some risks. In this letter we propose two kinds of modifications inspired by Stein's approach. A natural risk is defined and we derive a loss function which results in an unbiased estimate of this risk, then proceed to its minimization. Numerical simulations compare the so-modified MVDR filter to its original version.

We consider adaptive detection of a Gaussian rank-one component known up to a scaling factor buried in Gaussian noise with unknown statistics, a problem which arises when detecting Swerling I targets in radar systems. From the joint distribution of the samples under test and the training samples, the score function and the Fisher information matrix are derived, which enables us to formulate the Rao, Wald and gradient tests associated with the composite hypotheses problem. The Rao test is shown to bear resemblance with its deterministic counterpart and the Wald test is essentially based on the target power maximum likelihood estimate. The gradient test sort of blends the two other tests. All of them are shown to have a constant false alarm rate. Their performance is evaluated through numerical simulations in matched and mismatched cases.

The fourth-order differential equation describing elastic flexure of the lithosphere is one of the cornerstones of geodynamics that is key to understanding topography, gravity, glacial isostatic rebound, foreland basin evolution, and a host of other phenomena. Despite being fully formulated in the 1940s, a number of significant issues concerning the basic equation have remained overlooked to this day. We first explain the different fundamental forms the equation can take and their difference in meaning and solution procedures. We then show how numerical solutions to flexure problems as they are currently formulated are in general potentially unreliable in an unpredictable manner for cases in which the coefficient of rigidity varies in space due to variations of the elastic thickness parameter. This is due to fundamental issues related to the numerical discretisation scheme employed. We demonstrate an alternative discretisation that is stable and accurate across the broadest conceivable range of conditions and variations of elastic thickness, and we show how such a scheme can simulate conditions up to and including a completely broken lithosphere more usually modelled as an end-loaded, single, continuous plate. Importantly, our scheme will allow breaks in plate interiors, allowing, for instance, the creation of separate blocks of lithosphere which can also share the support of loads. The scheme we use has been known for many years but remains rarely applied or discussed. We show that it is generally the most suitable finite-difference discretisation of fourth-order, elliptic equations of the kind describing many phenomena in elasticity, including the problem of bending of elastic beams. We compare the earlier discretisation scheme to the new one in one-dimensional form and also give the two-dimensional discretisation based on the new scheme. We also describe a general issue concerning the numerical stability of any second-order finite-difference discretisation of a fourth-order differential equation like that describing flexure wherein contrasting magnitudes of coefficients of different summed terms lead to round-off problems, which in turn destroy matrix positivity. We explain the use of 128 bit floating-point storage for variables to mitigate this issue.

We address the problem of detecting a Gaussian rank-one signal using training samples to learn the covariance matrix of the noise present in the samples under test. Towards this end, we propose to use the latter after they have been whitened by the sample covariance matrix of the training samples. As an alternative to the generalized likelihood ratio test, we investigate three simpler alternatives, namely the Rao, gradient and Durbin tests. Closed-form expressions of the corresponding test statistics are derived and the detectors are shown to have a constant false alarm rate. Their performance is assessed via numerical simulations.

In many signal processing applications, conducting a coherent integration over the whole observation duration is not possible due to uncontrolled phase evolutions. In such a case one may refer to Post Detection Integration that consists in wisely combine shorter time integration outputs. In this paper, we propose a new detector that improves the state of art ones. Compared with the GLRT from which it is derived, it consists in a closed-form expression. Moreover, it is based on a simple linear phase assumption and can be used, by the way, in a wide area of applications.

We address the problem of detecting a signal of interest in Gaussian noise with an unknown covariance matrix, when the amplitude of the signal fluctuates along the observations and follows a Rice distribution. This is typical of a target that consists of one large dominant scatterer and a collection of small independent scatterers. We formulate it as a composite hypothesis testing problem, for which we derive the generalized likelihood ratio test, and show that it ensures a constant false alarm rate. Numerical simulations enable to assess its performance for Rician as well as Swerling I and III targets. It is shown that the new detector incurs no loss for Swerling targets but can offer a significant improvement for Rician targets, especially when the number of training samples is small.

Maximum Likelihood (ML) frequency estimation of a single tone in noise is known to be a computationally intensive task that does not cope with many real-time and embedded hardware architectures. Thereby, many sub-optimal techniques, based on approximations, have been proposed in the literature. In this letter, we show that the ML criterion can be solved directly, using an appropriate two-step procedure. The closed-form solution is shown to be asymptotically equivalent to the ML. Moreover, its formulation is very close to the popular Fitz's expression, with a slight correction. Numerical simulations show that the proposed scheme is very close to the ML.

This short note addresses the design of a partially adaptive filter to retrieve a signal of interest in the presence of strong low-rank interference and thermal noise. We consider a generalized sidelobe canceler implementation where the dimension-reducing transformation is build resorting to ideas borrowed from randomized matrix approximations. More precisely, the main subspace of the auxiliary data $Z$ is approximated by $Z\Omega$ where $\Omega$ is a random matrix or a matrix that picks at random columns of $Z$. These transformations do not require eigenvalue decomposition, yet they provide performance similar to those of a principal component filter.

We consider the problem of detecting a signal of interest corrupted by Gaussian noise with unknown mean and covariance matrix when the training samples available have a different mean. We evaluate the robustness of well-known adaptive detectors designed under the assumption of a same mean. More precisely, statistical representations of the generalized likelihood ratio test, the adaptive matched filter and the adaptive coherence estimator are derived for both an additive model with an arbitrary mismatch between the means, and a replacement model which is widely used in hyperspectral imaging. The new representations are given in terms of simple $F$ distributions and are shown to depend in a simple way on the norm of the whitened mean difference and its angle with the whitened signal of interest signature, or on the replacement factor. These new representations allow to identify the key parameters that impact most the probability of false alarm and probability of detection. Numerical simulations illustrate the theoretical results.

Abstract. The 4th order differential equation describing elastic flexure of the lithosphere is one of the cornerstones of geodynamics, key to understanding topography, gravity, glacial isostatic rebound, foreland basin evolution and a host of other phenomena. Despite being fully formulated in the 1940’s, a number of significant issues concerning the basic equation have remained overlooked to this day. We first explain the different fundamental forms the equation can take and their difference in meaning and solution procedures. We then show how numerical solutions to flexure problems in general as they are currently formulated, are potentially unreliable in an unpredictable manner for cases where the coefficient of rigidity varies in space due to variations of the elastic thickness parameter. This is due to fundamental issues related to the numerical discretisation scheme employed. We demonstrate an alternative discretisation that is stable and accurate across the broadest conceivable range of conditions and variations of elastic thickness, and show how such a scheme can simulate conditions up to and including a completely broken lithosphere more usually modelled as an end loaded, single, continuous plate. Importantly, our scheme will allow breaks in plate interiors, allowing for instance, the creation of separate blocks of lithosphere which can also share the support of loads. The scheme we use has been known for many years, but remains rarely applied or discussed. We show that it is generally the most suitable finite difference discretisation of fourth order, elliptic equations of the kind describing many phenomena in elasticity, including the problem of bending of elastic beams. We compare the earlier discretisation scheme to the new one in 1 dimensional form, and also give the 2 dimensional discretisation based on the new scheme.We also describe a general issue concerning the numerical stability of any second order finite difference discretisation of a fourth order differential equation like that describing flexure where contrasting magnitudes of coefficients of different summed terms lead to round off problems which in turn destroy matrix positivity. We explain the use of 128 bit, floating point storage for variables to mitigate this issue.

In this paper, we consider local detection of a target in hyperspectral imaging and we assume that the spectral signature of interest is buried in a background which follows an elliptically contoured distribution with unknown parameters. In order to infer the background parameters, two sets of training samples are available: one set, taken from pixels close to the pixel under test, shares the same mean and covariance while a second set of farther pixels shares the same covariance but has a different mean. When the whole data samples (pixel under test and training samples) follow a matrix-variate t distribution, the one-step generalized likelihood ratio test (GLRT) is derived in closed-form. It is shown that this GLRT coincides with that obtained under a Gaussian assumption and that it guarantees a constant false alarm rate. We also present a two-step GLRT where the mean and covariance of the background are estimated from the training samples only and then plugged in the GLRT based on the pixel under test only. (C) 2020 Elsevier B.V. All rights reserved.

In hyperspectral imaging the replacement model where a target, if present, partly replaces the disturbance is often advocated. In this paper, we consider a somehow more realistic model where only the low-rank background is substituted for the target while a residual noise, which belongs to the orthogonal complement, is unaffected by the presence/absence of the target. A two-step generalized likelkihood ratio test is formulated for such a model. Furthermore we show that the log likelihood can be well approximated by a weighted combination of the log likelihoods of the FTMF and the AMF, and that the dimension of the background subspace is the tuning parameter which enables to balance between these two well-known detectors. A comparison with standard techniques on real hyperspectral data reveals a good performance of the new detectors.

In this paper we analyse the behaviour of adaptive filters or detectors when they are trained with t-distributed samples rather than Gaussian distributed samples. More precisely we investigate the impact on the distribution of some relevant statistics including the signal to noise ratio loss and the Gaussian generalized likelihood ratio test. Some properties of partitioned complex F distributed matrices are derived which enable to obtain statistical representations in terms of independent chi-square distributed random variables. These representations are compared with their Gaussian counterparts and numerical simulations illustrate and quantify the induced degradation.

In this paper we consider Anomaly Detection in the hyperspectral context, and we extend the popular RX detector, initially designed under the standard additive model, to the replacement model case. Indeed, in this more realistic framework, the target, if present, is supposed to replace a part of the background. We show how to estimate this background power variation to improve the standard RX scheme. The obtained Replacement RX (RRX) is shown to be closed-form and outperforms the standard RX on a real data benchmark experiment.