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.

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.

We introduce a versatile approach for Boolean factorization of binary data matrices based on a projected hierarchical alternating least squares method. The general model considered in this work allows for an arbitrary Boolean combination of the binary rank-1 terms. The underlying approximation problem is tackled by relaxing the binary constraints and representing the combining function by a multivariate polynomial. This leads to closed-form and simple to implement updates of the alternating algorithm. Performance comparisons with other methods from the literature are presented for the standard Boolean (‘OR’) mixture model. We also pro-vide results on real data, as well as factorization examples using XOR and 3-term majority logical operators as combining functions.

Quaternion multiway arrays appear naturally as compact representations of 3D or 4D multidimensional signals. However, the non-commutativity of quaternion multiplication prevents a straightforward extension of standard tensor algebra to analyze and process quaternion multiway arrays. After reviewing the theoretical difficulties related to quaternion tensor algebra, we propose the first construction of quaternion tensors as representation of dedicated quaternion multilinear forms. This theoretical construction ensures that usual tensor algebraic properties, such as mode products properties are preserved. This novel framework enables us to generalize Tucker and canonical polyadic tensor decompositions to the quaternion case. For the latter, we carefully design a full quaternion ALS-type algorithm. Its relevance is validated numerically.

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.

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.

This work considers a collaborative wireless sensor network where nodes locally exchange coded informative data before transmitting the combined data towards a remote fusion center equipped with an antenna array. For this communication scenario, a new blind estimation algorithm is developed for jointly recovering network transmitted data and connection topology at the fusion center. The proposed algorithm is based on a two-stage approach. The first stage is concerned with the estimation of the channel gains linking the nodes to the fusion center antennas. The second stage performs a joint estimation of network data and connection topology matrices by exploiting a constrained (PARALIND) tensor model for the collected data at the fusion center. Illustrative simulation results evaluate the performance of the proposed algorithm for some system configurations and network topologies.

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.

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.

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.

This paper presents a version of MUSIC algorithm for linear vector-sensor arrays based on a complexified quaternionic (biquaternionic) modelization of the output three-components vector-signals. A way of computing the eigenvalue decomposition of a biquaternion valued matrix is introduced and the subspace decomposition of the biquaternionic spectral matrix of the observations is used to define the biquaternionic MUSIC estimator (BQ-MUSIC). Performances of the BQ-MUSIC are compared with classical long-vector technique