Matrix-valued images appear in many applications, ranging from polarimetric remote sensing to medical imaging. Such images can be represented as 4th-order tensors, where the first two dimensions correspond to spatial variables and the last two encode the matrix feature in each pixel. To efficiently analyze, decompose, and process these images, this paper considers the block-block terms decomposition (2BTD), a versatile low-rank tensor decomposition model that extends bilinear matrix factorization to 4th-order tensors by representing the latter as the sum of outer products of low-rank matrix blocks. Low-rank assumptions allow for a significantly reduced number of parameters to be estimated and enable the enforcement of key physical constraints on matrix sources. We establish both necessary and sufficient conditions for the uniqueness of the 2BTD model. To enable the use of 2BTD in covariance matrix-valued imaging, we develop an optimization framework that allows efficient handling of non-negativity and symmetry constraints together with low-rank assumptions on matrix blocks. Numerical experiments on synthetic and real data from Diffusion Tensor Imaging (DTI) illustrate the potential of the 2BTD model in matrix-valued imaging, as well as its effectiveness in practical settings.

In this article, a probability mass function (PMF) estimation method called partial coupled tensor factorization of 3D marginals or PCTF3D is proposed. To tame the inherent PMF estimation curse of dimensionality, PCTF3D’s principle is to couple 3-dimensional data projections – seen as order-3 tensors – to obtain a low-rank tensor approximation of the PMF. The contribution of PCTF3D relies on partial coupling which consists in choosing a limited subset of 3D marginals. While PMF estimation is possible with all marginals, coupling only a subset of marginals like in PCTF3D permits to reduce the computational burden without losing significant estimation performance. A key concept of PCTF3D is the choice of marginals to be coupled: this problem is formulated and studied with hypergraphs. This Part I paper introduces the algorithmic framework of PCTF3D: optimization problem, coupling strategies, numerical experiments and a real data application of PCTF3D. On the other hand, the Part II paper studies coupled tensor uniqueness properties of the model introduced by PCTF3D.

In this paper, uniqueness properties of a coupled factorization of 3D marginal tensors (or PCTF3D) are studied. The PCTF3D method (detailed in the Part I article) performs estimation of probability mass functions (PMFs) by coupling 3D marginals, seen as order-3 tensors. The core novelty of PCTF3D’s approach relies on the partial coupling which consists in choosing a limited set of 3D marginals to be coupled. PCTF3D uniqueness is examined through the prism of polynomial mappings and their recoverability. A numerical algorithm is proposed for finding the maximal rank for which recoverability is guaranteed. This approach properly accounts for the coupling strategy and simplex constraints. Using the proposed algorithm, the different coupling strategies from Part I are examined with respect to their uniqueness properties. Finally, a new identifiability bound is given for a so-called Cartesian coupling which improves existing sufficient bounds available in the literature.

This paper introduces a complete approach for the recovery of polarimetric images from experimental intensity measurements. In many applications, such images collect, at each pixel, a Stokes vector encoding the polarization state of light. By representing a Stokes vector image as a third-order tensor, we propose a new physically-constrained block-term tensor decomposition called Stokes-BTD. The proposed model is flexible and comes with broad identifiability guarantees. Moreover, physical constraints ensure meaningful interpretation of low-rank terms as Stokes vectors. In practice, Stokes images must be recovered from indirect, intensity measurements. To this aim, we implement two recovery algorithms for StokesBTD based on constrained alternated optimization and highlight constraints related to Stokes vectors. Numerical experiments on synthetic and real data illustrate the potential of the approach.

Abstract. This work introduces polarimetric Fourier phase retrieval (PPR), a physically inspired model to leverage polarization of light information in Fourier phase retrieval problems. We provide a complete characterization of its uniqueness properties by unraveling equivalencies with two related problems, namely, bivariate phase retrieval and a polynomial autocorrelation factorization problem. In particular, we show that the problem admits a unique solution, which can be formulated as a greatest common divisor (GCD) of measurement polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties of Sylvester matrices. Alternatively, existing iterative algorithms for phase retrieval, semidefinite positive relaxation and Wirtinger flow, are carefully adapted to solve the PPR problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. They further demonstrate the fruitful combination of algebraic and iterative approaches toward a scalable, computationally efficient, and robust to noise reconstruction strategy for PPR.

Discovering components that are shared in multiple datasets, next to dataset-specific features, has great potential for studying the relationships between different subjects or tasks in functional Magnetic Resonance Imaging (fMRI) data. Coupled matrix and tensor factorization approaches have been useful for flexible data fusion, or decomposition to extract features that can be used in multiple ways. However, existing methods do not directly recover shared and dataset-specific components, which requires post-processing steps involving additional hyperparameter selection. In this paper, we propose a tensor-based framework for multi-task fMRI data fusion, using a partially constrained canonical polyadic (CP) decomposition model. Differently from previous approaches, the proposed method directly recovers shared and dataset-specific components, leading to results that are directly interpretable. A strategy to select a highly reproducible solution to the decomposition is also proposed. We evaluate the proposed methodology on real fMRI data of three tasks, and show that the proposed method finds meaningful components that clearly identify group differences between patients with schizophrenia and healthy controls.

In this paper, we propose to jointly solve the hyperspectral super-resolution problem and the unmixing problem of the underlying super-resolution image using a coupled LL1 block-tensor decomposition. We consider a spectral variability phenomenon occurring between the observed low-resolution images. Exact recovery conditions for the image and mixing factors are provided. We propose two algorithms, an unconstrained one and another one subject to nonnegativity constraints, to solve the problems at hand. We showcase performance of the proposed approach on synthetic and real images.

This paper addresses the problem of rank tracking in real time hyperspectral image unmixing. Based on the On-line Alternating Direction Method of Multipliers (ADMM), we propose a new hyperspectral unmixing approach that integrates prior information as well as joint sparsity regularization, allowing to select only the active components on each sample of the image. This results in a semi-supervised algorithm, well adapted for on-line rank tracking for pushbroom imager. Experimental results on synthetic and real data sets demonstrate the effectiveness of our method for parameter estimation and rank change detection.

Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurements. Indeed, most of factors impacting the asymptotic estimation performance of the parameters of interest can be taken into account via equality constraints. In this communication, we introduce a new constrained Cramér-Rao- like bound for observations where the probability density function (p.d.f.) parameterized by unknown deterministic parameters results from the marginalization of a joint p.d.f. depending on random variables as well. In this setting, it is now possible to consider random equality constraints, i.e., equality constraints on the unknown deterministic parameters depending on the random parameters, which can not be addressed with the usual constrained Cramér-Rao bound. The usefulness of the proposed bound is illustrated by way of a coupled canonical polyadic model with linear constraints applied to the hyperspectral super-resolution problem.

Hyperspectral image (HSI) deconvolution is an ill-posed problem aiming at recovering sharp images with tens or hundreds of spectral channels from blurred and noisy observations. In order to successfully conduct the deconvolution, proper priors are required to regularize the optimization problem. However, handcrafting a good regularizer may not be trivial and complex regularizers lead to difficulties in solving the optimization problem. In this paper, we use the alternating direction method of multipliers (ADMM) to decompose the optimization problem into iterative subproblems where the prior only appears in a denoising subproblem. Then a 3D denoising convolutional neural network (3DDnCNN) is designed and trained with data for solving this problem. In this way, the hyperspectral image deconvolution is then solved with a framework that integrates the optimization techniques and deep learning. Experimental results demonstrate the superiority of the proposed method with several blurring settings in both quantitative and qualitative comparisons.

This article shows a comparison of rigid image registration methods of monomodal and multimodal images. These methods are applied on different images of oak wood pieces. The work presented in this article is a part of a complete vision system which aims to analyze the visual and physicochemical aspect of oak wood piece surface. In this context, a multi-sensor acquisition using a multimodal imagery platform is performed. The acquired images are not superimposable due to the rigid deformations resulting from the use of different sensor scanners and the presence of imperfections during the image acquisition process. This leads to consider image registration as a preprocessing step. The efficiency of the registration method depends on the deformation itself and the type of image to be registered. That is why we propose a comparison of different built-in and extended MATLAB registration methods, based on cross-correlation, phase correlation, mutual information and alignment of geometric primitives. The registration evaluation of the different methods is done by using quantitative measure of image alignment, visual inspection and computational time. Finally, the choice of the most adapted image registration methods, based on the existing differences between the acquired images, in terms of type of deformations and image modality, is discussed.

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.

We propose an algorithm to perform the low-rank Boolean Canonical Polyadic Decomposition (BCPD) of a binary tensor. The proposed approach is based on the AO-ADMM strategy introduced in [1] and uses a post-nonlinear mixture model for binary sources. We show that this new method is better suited for low-rank approximation of binary tensors compared to other similar methods. We also provide an easy-to-check uniqueness condition for the BCPD. This is the first time that such a condition is derived for Boolean decompositions.

We propose a novel approach for hyperspectral super-resolution that is based on low-rank tensor approximation for a coupled low-rank multilinear (Tucker) model. We show that the correct recovery holds for a wide range of multilinear ranks. For coupled tensor approximation, we propose an SVD-based algorithm that is simple and fast, but with a performance comparable to that of the state-of-the-art methods.

This paper proposes a hyperspectral image deconvolution algorithm for the online restoration of hyperspectral images as provided by wiskbroom and pushbroom scanning systems. We introduce a least-mean-squares (LMS)-based framework accounting for the convolution kernel noncausality and including nonquadratic (zero attracting and piecewise constant) regularization terms. This results in the so-called sliding block regularized LMS (SBR-LMS), which maintains a linear complexity compatible with real-time processing in industrial applications. A model for the algorithm mean and mean-squares transient behavior is derived and the stability condition is studied. Experiments are conducted to assess the role of each hyper-parameter. A key feature of the proposed SBR-LMS is that it outperforms standard approaches in low SNR scenarios such as ultra-fast scanning.

This work aims at studying a method to automatically estimate regularization parameters of hyperspectral images deconvolution methods. The deconvolution problem is formulated as a multi-objective optimization problem and the properties of the corresponding response surface are studied. Based on these properties, the minimum distance criterion (MDC) is proposed to estimate regularization parameters. It has good theoretical properties (uniqueness, robustness) from which a grid search based approach is proposed. It results in a fast approach to estimate the regularization parameters.

Zero-attracting least-mean-square (ZA-LMS) algorithm has been widely used for online sparse system identification. It combines the LMS framework and l1-norm regularization to promote sparsity, and relies on subgradient iterations. Despite the significant interest in ZA-LMS, few works analyzed its transient behavior. The main difficulty lies in the nonlinearity of the update rule. In this study, a detailed analysis in the mean and mean-square sense is carried out in order to examine the behavior of the algorithm. Simulation results illustrate the accuracy of the model and highlight its performance through comparisons with an existing model.

This paper aims at studying a method to automatically estimate the regularization parameters of non-negative hyperspectral image deconvolution methods. The deconvolution problem is formulated as a multi-objective optimization problem and the properties of the corresponding response surface are studied. Based on these properties, the minimum distance criterion (MDC) and the maximum curvature criterion (MCC) are proposed to estimate regularization parameters especially for the non-negativity constrained deconvolution problem. MDC has good theoretical properties (convexity and uniqueness) but requires to choose a reference point. On the contrary, MCC does not need to choose any reference point but does not have interesting theoretical properties. A grid-search-based approach to minimize the computational cost of MDC and MCC is proposed. It results in fast approaches to estimate the regularization parameters. Based on simulated 2D images, the proposed approaches are compared with the state-of-the-art methods, confirming the effectiveness of the MDC and MCC for the non-negativity constrained image deconvolution problem. In the case of non-negative hyperpsectral image deconvolution, the fast MDC yields better performances than the fast MCC. An application to real-world hyperspectral fluorescence microscopy images is also provided; it confirms the superiority of MDC.

We consider hyperspectral unmixing problems where the observed images are blurred during the acquisition process, e.g. in micro / spectroscopy. Geometrical spectral unmixing consists in extracting the pure materials contained in the image as the vertices of the minimum-volume simplex (MVS) enclosing the data. In [1], we showed that the blur caused a contraction of the MVS, which implies that a deconvolution step is necessary to correctly unmix the image. In this paper, we study two sequential procedures consisting in deblurring and unmixing the blurred hyperspectral image. Despite its computational appeal, we will show that an unmixing / deconvolution strategy is outperformed by a deconvolution / unmixing approach.

In this paper, we consider hyperspectral unmixing problems where the observed images are blurred during the acquisition process, e.g., in microscopy and spectroscopy. We derive a joint observation and mixing model and show how it affects end-member identifiability within the geometrical unmixing framework. An analysis of the model reveals that nonnegative blurring results in a contraction of both the minimum-volume enclosing and maximum-volume enclosed simplex. We demonstrate this contraction property in the case of a spectrally invariant point-spread function. The benefit of prior deconvolution on the accuracy of the restored sources and abundances is illustrated using simulated and real Raman spectroscopic data.

We consider a hyperspectral image restoration problem in which the solution is known to be nonnegative. The image estimate is obtained as the constrained minimizer of a convex criterion incorporating prior information on its spatial and spectral regularity. We previously proposed a fast algorithm for Tikhonov regularization. Here, we adapt this algorithm to edge-preserving image restoration.

In this paper we propose a generalized non-collocated electromagnetic (EM) vector-sensor configuration allowing the use of the vector cross-product direction finding scheme. The presented work extends the results and the philosophy of [1] to a more general array configuration. We provide a sufficient condition ensuring identifiability of source DOA parameters and propose a novel algorithm allowing the DOA estimation for inter-antenna spacing larger than λ2. The effectiveness of the proposed approach is illustrated by numerical simulations.

We propose a sparse modal estimation approach for analyzing 2-D NMR signals. It consists in decomposing the 2-D problem into two 1-D modal estimations. Each 1-D problem is formulated in terms of simultaneous sparse approximation which is efficiently solved using the Simultaneous Orthogonal Matching Pursuit method associated with a multi-grid dictionary refinement. Then, we propose a new criterion for mode pairing which comes down to solve a sparse approximation problem involving a low dimensional dictionary. The effectiveness of the method is demonstrated on real NMR data.

In this brief, we provide an efficient scheme for performing deconvolution of large hyperspectral images under a positivity constraint, while accounting for spatial and spectral smoothness of the data.

We revisit the positive cone condition given by Efronfor the over-determined least absolute shrinkage and selection operator (LASSO). It is a sufficient condition ensuring that the number of nonzero entries in the solution vector keeps increasing when the penalty parameter decreases, based on which the least angle regression (LARS)and homotopyalgorithms yield the same iterates. We show that the positive cone condition is equivalent to the diagonal dominance of the Gram matrix inverse, leading to a simpler way to check the positive cone condition in practice. Moreover, we elaborate on a connection between the positive cone condition and the mutual coherence condition given by Donoho and Tsaig, ensuring the exact recovery of any$k$-sparse representation using both LARS and homotopy.

In this paper we investigate the uniqueness of the 4-way CANDECOMP/PARAFAC (CP) model in the case where file only possible linear dependencies between the columns of the loading matrices take die form of collinear loadings. For this special configuration we state a necessary and sufficient condition for having full column rank of the Khatri-Rao product of two loading matrices. This allows to derive a sufficient condition for uniqueness of the 4-way CP model with collinear loadings in at most three modes. The result is illustrated by analyzing 4-way fluorescence data.

We propose a novel algorithm for the problem of nonorthogonal joint diagonalization of a set of structured matrices based on successive Jacobi-like transformations. Though the elementary transformation matrices we use are not optimal in the sense of the global criterion, they are ensured to be nonsingular, and can be computed in closed form. The algorithm is efficient in virtue of its low computational complexity and fast convergence. The performance of the new algorithm is compared in simulations to the similar algorithms of the recent literature.

This paper presents a methodology to estimate the parameters of two-dimensional damped/undamped exponentials from high complexity noisy signals, which is the case in 2-D nuclear magnetic resonance spectroscopy signals. The proposed approach performs adaptive subband decomposition combined with a classical frequency estimator based on the Prony model. At each node resulting from the decomposition tree, a stopping rule is computed in order to decide whether the decomposition must be continued or not. The rule is a flatness measure applied on residuals resulting from the estimation step. The method is demonstrated using a simulated signal.

By means of the parallel factor (PARAFAC) decomposition, we present a novel method working on a vector-sensor array for blind separation of polarized sources in virtue of their distinct spatial and temporal signatures. Identifiability is studied, and explicit constraints on the sources are derived to ensure the data model identifiable. We show, by numerical simulations, that the estimation performance can approach that of non-blind estimation by optimally designing the source polarizations.

We propose in this paper a new strategy for non-stationary signals denoising based on designing a time-varying filter adapted to the signal short term spectral characteristics. The basic idea leading us to use a new parametric nonlinear weighting of the measured signal short term spectral amplitude (STSA) is exposed. The overall system consists in combining the estimated STSA and the complex exponential of the noisy phase. The proposed technique results in a significant reduction of the noise for a variety of non-stationary signals including speech signals.

This article proposes a new approach to the deconvolution of electron energy loss spectra, used for material characterization. This approach is based on local filters with varying bandwidth, adapted to the local (non-stationary) characteristics of the signal to restore. The local filter synthesis is achieved by means of three parameters determined by optimizing a compound criterion. The effectiveness of this approach is shown on an example

A main issue in source separation is to deal with the indeterminacies. Well known are the ordering and scale ambiguities, but other types of indeterminacies may also occur. In this paper we address these indeterminacies in the case of non-negative sources and non-negative mixing coefficients. On the one hand, we fully develop the case of two sources. On the other hand, in the general case we formulate necessary conditions for the uniqueness of the solution (up to ordering and scale ambiguities).

The paper considers the problem of source separation in the particular case where both the sources and the mixing coefficients are positive. The proposed method addresses the problem in a Bayesian framework. We assume a gamma distribution for the spectra and the mixing coefficients. This prior distribution enforces the non-negativity. This leads to an original method for positive source separation. A simulation example is presented to illustrate the effectiveness of the method.

A new deconvolution method of sparse spike trains is presented. It is based on the coupling of the Hunt filter with a thresholding. We show that a good model for the probability density function of the Hunt filter output is a Gaussian mixture, from which we derive the threshold that minimizes the probability of errors. Based on an interpretation of the method as a maximum a posteriori (MAP) estimator, the hyperparameters are estimated using a joint MAP approach. Simulations show that this method performs well at a very low computation time.

The local Wigner-Ville distribution (LWVD) extends the Cohen's class time-frequency distributions (TFD) by the definition of a kernel for each time-frequency point (local kernel). The subject of the paper is the determination of these local kernels for interference reduction. Starting from the simple idea of the local limitation of the Wigner-Ville TFD integral bounds, a method is presented to estimate these limits and to obtain a reduced interference TFD. The effectiveness for interference reduction of this LWVD, especially when compared to global-kernel methods, is shown using example signals.

Nonstationary gearbox vibration signals are analysed using a time-frequency (TF) representation which is chosen with respect to its interpretability. With its help, a crack transient is examined in detail and decomposed into three physical parts. Following this analysis, a time-domain signal is synthesized. Its good phase fitting proves on one hand, the validity of the analysis and on the other hand, the good accuracy of the TF representation chosen.