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.

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.

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.

Coupled tensor decompositions (CTDs) perform data fusion by linking factors from different datasets. Although many CTDs have been already proposed, current works do not address important challenges of data fusion, where: 1) the datasets are often heterogeneous, constituting different “views” of a given phenomena (multimodality); and 2) each dataset can contain personalized or dataset-specific information, constituting distinct factors that are not coupled with other datasets. In this work, we introduce a personalized CTD framework tackling these challenges. A flexible model is proposed where each dataset is represented as the sum of two components, one related to a common tensor through a multilinear measurement model, and another specific to each dataset. Both the common and distinct components are assumed to admit a polyadic decomposition. This generalizes several existing CTD models. We provide conditions for specific and generic uniqueness of the decomposition that are easy to interpret. These conditions employ uni-mode uniqueness of different individual datasets and properties of the measurement model. Two algorithms are proposed to compute the common and distinct components: a semi-algebraic one and a coordinate-descent optimization method. Experimental results illustrate the advantage of the proposed framework compared with the state of the art approaches.

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.

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.

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.

This paper introduces a method for background (or baseline) correction in hyperspectral images. The method is based on the optimization of a criterion incorporating a non-quadratic robust loss (data fidelity term) and both spatial and spectral regularization terms to enforce baseline smoothness. Unlike the classical approach based on a pixel-by-pixel baseline correction, the proposed algorithm exploits jointly the spatial and spectral information. The effectiveness of the the proposed algorithm is demonstrated using simulated and geological hyperspectral images.

In this paper, we study three recently proposed probability mass function (PMF) estimation methods for flow cytometry data analysis. By modeling the PMFs as a mixture of simpler distributions, we can reformulate the PMF estimation problem as three different tensor-based approaches: a least squares coupled tensor factorization approach, a least squares partially coupled tensor factorization approach, and a Kullback-Leibler divergence (KLD)-based expectation-maximization (EM) approach. In the coupled methods, the full PMF is estimated from lower-order empirical marginal distributions, while the EM approach estimates the full PMF directly from the observed data. The three approaches are evaluated in the context of simulated and real data experiments.

This article characterizes the rank-one factorization of auto-correlation matrix polynomials. We establish a sufficient and necessary uniqueness condition for uniqueness of the factorization based on the greatest common divisor (GCD) of multiple polynomials. In the unique case, we show that the factorization can be carried out explicitly using GCDs. In the non-unique case, the number of non-trivially different factorizations is given and all solutions are enumerated.

Quaternions are still largely misunderstood and often considered an "exotic" signal representation without much practical utility despite the fact that they have been around the signal and image processing community for more than 30 years now. The main aim of this article is to counter this misconception and to demystify the use of quaternion algebra for solving problems in signal and image processing. To this end, we propose a comprehensive and objective overview of the key aspects of quaternion representations, models, and methods and illustrate our journey through the literature with flagship applications. We conclude this work by an outlook on the remaining challenges and open problems in quaternion signal and image processing.

Résumé – Dans cet article, nous proposons une approche pour la reconstruction d’images à haute résolution par fusion d’images hyperspectrales et multispectrales prenant en compte la variabilité entre les images. L’image haute résolution et la variabilité inter-images sont représentées sous forme de tenseurs et sont supposées admettre une décomposition de Tucker. Deux algorithmes sont proposés, et des conditions pour une reconstruction exacte de l’image à haute résolution sont fournies. Les résultats expérimentaux illustrent les performances des méthodes proposées. Abstract – In this paper we propose a hyperspectral and multispectral image fusion framework accounting for inter-image variability. The images are represented as three dimensional tensors, and both the high-resolution image and inter-image variations are assumed to admit a Tucker decomposition. Two algorithms are proposed, one purely algebraic and computationally fast and another based on an optimization procedure. Exact recovery conditions are provided. Experimental results illustrate the performance of the proposed methods.

In this paper, we propose a new method for automated flow cytometry data analysis. By modeling a multidimensional probability distribution as a mixture of simpler distributions, we can reformulate the problem as a coupled tensor approximation of 3D marginals. In order to reduce the computational load, we use partially coupled strategies. We also propose a grouping of rank-one components together with a new visualization of the results. We demonstrate the usefulness of the proposed methodology on simulated and real data.

This work introduces a novel Fourier phase retrieval model, called polarimetric phase retrieval that enables a systematic use of polarization information in Fourier phase retrieval problems. We provide a complete characterization of uniqueness properties of this new model by unraveling equivalencies with a peculiar polynomial factorization problem. We introduce two different but complementary categories of reconstruction methods. The first one is algebraic and relies on the use of approximate greatest common divisor computations using Sylvester matrices. The second one carefully adapts existing algorithms for Fourier phase retrieval, namely semidefinite positive relaxation and Wirtinger-Flow, to solve the polarimetric phase retrieval problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. We further highlight a reconstruction strategy that combines both approaches for scalable, computationally efficient and asymptotically MSE optimal performance.

We propose a theoretical performance analysis for a class of reconstruction problems, formulated as coupled canonical polyadic decompositions of two low-resolution tensor observations. We study a particular case when all the modes of the tensors are coupled. Unlike the case of a single coupling constraint, a fully-coupled model requires nonlinear constraints in some estimation scenarios. Thus we introduce two probabilistic scenarios. For each scenario, we derive the constrained Cramer-Rao bounds for the parameters and for the mean-squared error of the reconstructed tensor. We show that with a carefully chosen initialization, the maximum likelihood estimators reach the bounds, even in challenging cases (low signal-to-noise ratio or large tensor rank). (c) 2022 Elsevier B.V. All rights reserved.

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 introduces a general framework for solving constrained convex quaternion optimization problems in the quaternion domain. To soundly derive these new results, the proposed approach leverages the recently developed generalized $\mathbb {H}\mathbb {R}$-calculus together with the equivalence between the original quaternion optimization problem and its augmented real-domain counterpart. This new framework simultaneously provides rigorous theoretical foundations as well as elegant, compact quaternion-domain formulations for optimization problems in quaternion variables. Our contributions are threefold: (i) we introduce the general form for convex constrained optimization problems in quaternion variables, (ii) we extend fundamental notions of convex optimization to the quaternion case, namely Lagrangian duality and optimality conditions, (iii) we develop the quaternion alternating direction method of multipliers (Q-ADMM) as a general purpose quaternion optimization algorithm. The relevance of the proposed methodology is demonstrated by solving two typical examples of constrained convex quaternion optimization problems arising in signal processing. Our results open new avenues in the design, analysis and efficient implementation of quaternion-domain optimization procedures.

In this paper, we propose to jointly solve the hyperspectral super-resolution and hyperspectral unmixing problems using a coupled LL1 block-tensor decomposition. We focus on the specific case of spectral variability occurring between the observed low-resolution images. Exact recovery conditions are provided. We propose two algorithms: an unconstrained one and another one subject to non-negativity constraints, to solve the problems at hand. We showcase performance of the proposed approach on a set of synthetic and semi-real images.