Many real-life signal-based applications use the Tucker decomposition of a high dimensional/order tensor. A well-known problem with the Tucker model is that its number of entries increases exponentially with its order, a phenomenon known as the “curse of the dimensionality”. The Higher-Order Orthogonal Iteration (HOOI) and Higher-Order Singular Value Decomposition (HOSVD) are known as the gold standard for computing the range span of the factor matrices of a Tucker Decomposition but also suffer from the curse. In this paper, we propose a new methodology with a similar estimation accuracy as the HOSVD with non-exploding computational and storage costs. If the noise-free data follows a Tucker decomposition, the corresponding Tensor Train (TT) decomposition takes a remarkable specific structure. More precisely, we prove that for a$Q$-order Tucker tensor, the corresponding TT decomposition is constituted by$Q-3$3-order TT-core tensors that follow a Constrained Canonical Polyadic Decomposition. Using this new formulation and the coupling property between neighboring TT-cores, we propose a JIRAFE-type scheme for the Tucker decomposition, called TRIDENT. Our numerical simulations show that the proposed method offers a drastically reduced complexity compared to the HOSVD and HOOI while outperforming the Fast Multilinear Projection (FMP) method in terms of estimation accuracy.

Hyperspectral super-resolution based on coupled Tucker decomposition has been recently considered in the remote sensing community. The state-of-the-art approaches did not fully exploit the coupling of information contained in hyperspectral and multispectral images of the same scene. This paper proposes a new algorithm that overcomes this limitation. It accounts for both the high-resolution and the low-resolution information in the model by solving a set of least-squares problems. In addition, we provide exact recovery conditions for the super-resolution image in the noiseless case. Our simulations show that the proposed algorithm achieves very good reconstruction quality with a very low computational complexity.

Supervised learning is a major task to classify datasets. In our context, we are interested into classification from highorder tensors datasets. The "curse of dimensionality" states that the complexities in terms of storage and computation grow exponentially with the order. As a consequence, the method from the state-of-art based on the Higher-Order SVD (HOSVD) works well but suffers from severe limitation in terms of complexities. In this work, we propose a fast Grassmannian kernel-based method for high-order tensor learning based on the equivalence between the Tucker and the tensortrain decompositions. Our solution is linked to the tensor network, where the aim is to break the initial high-order tensor into a collection of low-order tensors (at most 3-order). We show on several real datasets that the proposed method reaches a similar accuracy classification rate as the Grassmannian kernel-based method based on the HOSVD but for a much lower complexity.

Over the last two decades, tensor‐based methods have received growing attention in the signal processing community. In this work, the authors proposed a comprehensive overview of tensor‐based models and methods for multisensor signal processing. They presented for instance the Tucker decomposition, the canonical polyadic decomposition, the tensor‐train decomposition (TTD), the structured TTD, including nested Tucker train, as well as the associated optimisation strategies. More precisely, they gave synthetic descriptions of state‐of‐the‐art estimators as the alternating least square (ALS) algorithm, the high‐order singular value decomposition (HOSVD), and of more advanced algorithms as the rectified ALS, the TT‐SVD/TT‐HSVD and the Joint dImensionally Reduction and Factor retrieval Estimator scheme. They illustrated the efficiency of the introduced methodological and algorithmic concepts in the context of three important and timely signal processing‐based applications: the direction‐of‐arrival estimation based on sensor arrays, multidimensional harmonic retrieval and multiple‐input–multiple‐output wireless communication systems.

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.

Assume that a N-dimensional noisy measurement vector is available via a N × R linear random sensing operation of a R-dimensional Gaussian signal of interest, denoted by s. The problem statement being addressed here is the study of the minimal Bayes' error probability for the detection of s where N → ∞ with N/R → β ∈ (1, ∞). When the exact derivation of this probability is intractable, statistical similarity metrics, nourishing their roots in the information geometry theory, are useful to characterize the exponential rate of the error probability. More precisely, the Chernoff information is asymptotically given by the minimum over s ∈ (0, 1) of the s-divergence. In many applications, it is hard to evaluate the s-divergence. Worse, due to the asymmetry of the s-divergence for the considered detection problem, the Bhattacharyya divergence (s = 1/2), cannot circumvent this problem. As a consequence, the derivation of the optimal value of s requires a costly numerical optimization strategy. In this work, we propose two contributions. The first one is to provide a closed-form expression of the asymptotic normalized s-divergence. The second contribution is to provide an analytic expression for the optimal value of s.

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.

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 Canonical Polyadic tensor decomposition (CPD), also known as Candecomp/Parafac, is very useful in numerous scientific disciplines. Structured CPDs, i.e. with Toeplitz, circulant, or Hankel factor matrices, are often encountered in signal processing applications. As subsequently pointed out, specialized algorithms were recently proposed for estimating the deterministic parameters of structured CP decompositions. A closed-form expression of the Cramér-Rao bound (CRB) is derived, related to the problem of estimating CPD parameters, when the observed tensor is corrupted with an additive circular i.i.d. Gaussian noise. This CRB is provided for arbitrary tensor rank and sizes. Finally, the proposed CRB expression is used to asses the statistical efficiency of the existing algorithms by means of simulation results in the cases of third-order tensors having three circulant factors on one hand, and an Hankel factor on the other hand.

The paper derives closed form (nonmatrix) expression for the deterministic Cramér-Rao bound (CRB) for the direction-of-arrival associated with a target embedded in a noisy, multipath channel using the time reversal (TR) MIMO system. By incorporating the TR built-in adaptive waveform processing feature to reshape the MIMO probing signals, we prove that the CRBs for the direction of arrival can be improved in ways not foreseen with the conventional MIMO radars. A second contribution of the paper is the analytical derivation of the Angular Resolution Limit (ARL) defined as the minimal separation between two targets to be separately resolved by the MIMO radar. At high signal-to-noise ratio, we show that the TR ARL inherits the properties of the TR CRB and is superior to its conventional counterpart by a factor proportional to the order of the channel multipath.

Statistical Resolution Limit (SRL), defined as the minimal separation to resolve two closely spaced signals, is one of the important tools to evaluate a given system performance. Based on S.T. Smith's formulation of the SRL, this paper provides a methodology to compute an approximate analytical expression of the resolution limit in the Gaussian model case. As an application, we consider the particular case of two sources located in the near field and consider the resolution limit in terms of minimum angular separation. Discussion and numerical illustrations are then given to get more insights on the proposed derivation and to validate our theoretical results.

Discrete-time Volterra modeling is a central topic in many application areas and a large class of nonlinear systems can be modeled using high-order Volterra series. The problem with Volterra series is that the number of parameters grows very rapidly with the order of the nonlinearity and the memory in the system. In order to efficiently implement this model, kernel eigen-decomposition can be used in the context of a Parallel-Cascade realization of a Volterra system. So, using the multilinear SVD (HOSVD) for de composing high-order Volterra kernels seems natural. In this pa per, we propose to drastically reduce the computational cost of the HOSVD by (1) considering the symmetrized Volterra kernel and (2) exploiting the column-redundancy of the associated mode by using an oblique unfolding of the Volterra kernel. Keeping in mind that the complexity of the full HOSVD for a cubic (I x I x I) un structured Volterra kernel needs 12I4flops, our solution allows reducing the complexity to 2I4flops, which leads to a gain equal to six for a sufficiently large size I.

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.

In this paper, we study the impact of modeling error on the receiver of a MIMO radar. Following other works on classical array processing, we derive closed-form expressions of the Cramér-Rao bounds for an observation model of a widely spaced MIMO radar affected by modeling error. We show that, as the signal-to-noise ratio increases, the Cramér-Rao bound and the mean square error of the maximum likelihood estimator of the angle-of-arrival do not fall to zero (contrary to the classical case without error modeling) and converge to a fixed limit for which we give a closed-form expression. Moreover, we give a simple closed-form expression of the critical value of the signal-to-noise ratio where this limitation of performance appears.

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.

The Spread Transform (ST) robustness facing a Temporal Frame Averaging (TFA) is studied in this paper. Our analysis provides new insights on the effect of the TFA attack in the context of video watermarking when the ST is used. To remove the interferences and the attenuation of the watermark caused by the TFA attack, an adaptation of the Spread Transform is proposed. Our solution is based on the embedding of the watermark thanks to different orthogonal directions and by exploiting the temporal diversity of the video signal, i.e. the message is spread over several video frames. Finally, we show that the robustness and decoding performances are close to the ones before the attack.

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.

In this paper, security evaluation of an important watermarking class based on quantization is given. Theoretical developments and practical simulations are used to measure the security level of watermarking techniques. We give the contribution of each observation available to the attacker on the total gathered information about the watermarking secrecy. By leading on watermarking technique weaknesses, we find that using the Quantization Index Modulation (QIM) with continuous secret key is equivalent to using a secret key with two states. The latter is easier to estimate than a continuous key. Then, we propose a secure version of a classical trellis coded quantization watermarking. The security is guaranteed thanks to the trellis path generated from the discrete key and the message. We show that the spread transform can represent a second or alternative security level for watermarking systems. It allows to increase the watermarking security level and to keep the embedded message hard to read for unauthorise user.

The quantization based embedding systems are usually used in the information hiding domain, thanks to their efficiency and simplicity. In other hand, they are known to be insecure in steganography context (according to Cachins' security definition) because they distort the stego-signal density function. In this paper, we show that an important kind of quantization based systems (those resulting from the combination with the spread transform and classical data hiding schemes based on quantization ) preserve the probability density function of the stego-signal. We prove the undetectability theoretically and show how the spread transform makes the stego-message statistically non-dectectable in the real case.

In the context of the direction-of-arrival (DOA) estimation problem, we can sometimes assume the a priori knowledge of M-S DOA among M. Some authors have propose to incorporate this a priori knowledge to better estimate the DOA of interest (ie., the unknown ones). In a previous work, the authors have proposed two prior MinNorm schemes based on oblique projection which allow the integration of this prior-knowledge. In particular, numerical and theoretical expressions of the variances have been derived. In this work, we go further into the analysis already given. We first focus on the asymptotic (large number of sensors) behavior of the standard, the constrained and the prior versions of the MinNorm algorithm and we show that in this case the exploitation of a prior-knowledge is not beneficial. Next, we derive closed-form approximations of the variance of these algorithms in case of two closely-spaced sources for small/moderate number of sensors and we show that the prior-MinNorm algorithms based on oblique projection is much more insensitive to the proximity of the DOA as compared to the standard and the constrained MinNorm algorithms. Finally, these theoretical analysis are checked against computer simulations by means of Monte-Carlo runs.

Most exiting array signal processing techniques for bearing estimation are strongly relied on the far-field assumption. When the sources are located close to the array, these techniques may no longer perform satisfactorily. In this work, we propose a tensor-based algorithm which is dedicated to the joint estimation of the range and the bearing of multiple narrow-band and near-filed sources in a spatially white Gaussian noise. Automatic paring of the model parameters is achieved for an uniform linear array. By means of numerical simulation, we show that for low signal to noise ratio, the proposed algorithm is more accurate than the higher order statistics (HOS)- based ESPRIT algorithm for small/moderate number of snapshots.

The steganographic security in the Cachin's work is defined as the statistical invisibility between the host signal and its marked version. At contrary, the robustness to an attack is not a prime goal. In robust watermarking, this is exactly the inverse. The scalar costa scheme (SCS) is a typical example of this fact. Indeed, this scheme is robust to additive white Gaussian noise (AWGN) attack but is drastically insecure since its probability density function for Gaussian host signal is severely discontinuous. An improved scheme has been proposed by Guillon et al. which increases the security to the detriment of the robustness. In this paper, we propose a new watermarking scheme, based on the combination of the spread transform (ST) and the trellis coded quantization (TCQ) which is secure and robust to AWGN attack.

Several authors have studied stego-systems based on Costa scheme, but just a few ones gave both theoretical and experimental justifications of these schemes performance in an active warden context. We provide in this paper a steganographic and comparative study of three informed stego-systems in active warden context: scalar Costa scheme, trellis-coded quantization and spread transform scalar sosta Scheme. By leading on analytical formulations and on experimental evaluations, we show the advantages and limits of each scheme in term of statistical undetectability and capacity in the case of active warden. Such as the undetectability is given by the distance between the stego-signal and the cover distance. It is measured by the Kullback-Leibler distance.

Many works has been made in the context of the recursive or sequential MUSIC algorithm for bearing estimation. Indeed, in some difficult scenarios as for closely spaced bearings, correlated sources and at low SNRs, the accuracy of the MUSIC algorithm can be improved by sequentially cancel the previous estimated bearings. In this paper, we propose a new algorithm, called zero-forcing based sequential MUSIC, which efficiently tackles this problem.

Recently, certain solutions have been proposed to solve the common poles estimation problem in a multichannel exponentially damped sinusoidal (EDS) signal. In this work, we tackle the closely related problem consisting of the estimation of the complex amplitudes associated to the common poles of a multichannel EDS signal. Our approach is based on the estimation of an oblique projector on the space of the estimated/common poles along the space of the unknown/non-common poles. We derive three projection/estimation schemes. The first one is based on a sequential (channel by channel) processing of the data while the others are based on joint estimation of the complex amplitudes.

The multi-dimensional harmonic model has attracted considerable attention for a variety of applications in signal processing. Stoica and Nehorai have derived the asymptotic (ie., for large analysis duration) Cramer-Rao lower bound (ACRB) which represents the minimal theoretical variance in the estimation of the model parameters for a one-dimensional harmonic model of order M. In this work, we generalize and analyze the ACRB associated to a M-order harmonic model of dimension P with P > 1.

In this work, we deal with multilinear representations of an harmonic process composed by a mixture of complex sinusoids contaminated by a colored Gaussian noise of unknown probability density function. high-order statistics, and in particular fourth-order cumulants (FOC), are popular tools to mitigate the effect of the colored noise and can be viewed as a natural enhancement of the noisy mixture. As the FOC of an harmonic process is a multilinear function, it seems natural to map this quantity onto a structured multi-way array (aha. tensor). However, an harmonic process can be understood as a pure stationary random process or as a deterministic process corrupted by a stationary noise. In this paper, we explore the relation existing between these two models in the context of the multilinear decomposition theory

The higher-order SVD (HOSVD) is a generalization of the SVD to higher-order tensors (ie. arrays with more than two indexes) and plays an important role in various domains. Unfortunately, the computational cost of this decomposition is very high since the basic HOSVD algorithm involves the computation of the SVD of three highly redundant block-Hankel matrices, called modes. In this paper, we present an ultra-fast way of computing the HOSVD of a third-order structured tensor. The key result of this work lies in the fact it is possible to reduce the basic HOSVD algorithm to the computation of the SVD of three non-redundant Hankel matrices whose columns are multiplied by a given weighting function. Next, we exploit an FFT-based implementation of the orthogonal iteration algorithm in an adaptive way. Even though for a square (1 times 1 times 1) tensor the complexity of the basic full-HOSVD is O(I4) and O(rI3) for its r-truncated version, our approach reaches a linear complexity of O(rIlog2(I))

We present a subspace-based scheme for the estimation of the poles (angular-frequencies and damping-factors) of a sum of damped and delayed sinusoids. In our model each component is supported over a different time frame, depending on the delay parameter. Classical subspace based methods are not suited to handle signals with varying time-support. In this contribution, we propose a solution based on the best rank-(R/sub 1/, R/sub 2/, R/sub 3/) approximation of a partially structured Hankel tensor on which the data are mapped. We show, by means of an example, that our approach outperforms the current tensor and matrix-based approaches in terms of the accuracy of the damping parameter estimates.

In this work, we present an evolution of the Damped and Delayed Sinusoidal (DDS) model introduced within the framework of the general signal modeling. This model, named the Partial Damped and Delayed Sinusoidal (PDDS) model, takes into account a single time delay parameter for a set (sum) of damped sinusoids. The proposed modification is more consistent with the transient audio modeling problem. The validity of the approach is shown by its comparison with the well-known Exponentially Damped Sinusoids (EDS) approach. Finally, the performances of the three models based high-resolution parameter estimation algorithms are compared on synthetic fast time-varying signals, and on two typical audio transients.

We present an iterative Fourier-type algorithm which is introduced for DDS (damped and delayed sinusoid) parameter estimation using a sub-band processing approach. This algorithm is shown to improve on existing ones with respect to computational cost and estimation accuracy. Moreover, we derive the Cramer-Rao bound (CRB) expression for the DDS model and we perform a simulation-based performance analysis of a noisy fast time-varying synthetic signal and in the audio transient signal modeling context.

In this paper, we present a novel parametric model based on an important evolution of McAulay & Quatieri sinusoidal models [5]. This model takes into account a damping factor as well as a delay parameter which makes it more efficient for modeling strong audio transients such as castanets onsets without pre-echo effects'. We also develop an original algorithm to estimate the model parameters by means of a high resolution method followed by a sub-band analysis.

An audio signal parametric modeling scheme is proposed that permits higher performance for representing strong sound transients. The exponentially damped sinusoids (EDS) model is considered in association with a high resolution parameter estimation approach. Such a technique is well adapted to almost every audio signal but is unfortunately not efficient when dealing with signals presenting strong temporal variations, such as percussive music signals, and causes pre-echo artifacts and weak onset dynamic reproduction which are prejudicial to listening. A system, based on the EDS model, has been developed with a transient detector and dynamic time segmentation and modeling that allows to overcome such artifacts.