Julien Flamant

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.

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 measurements polynomials. As a result, we propose algebraic solutions for PPR based on approximate GCD computations using the null-space properties 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 towards a scalable, computationally efficient and robust to noise reconstruction strategy for PPR.

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.

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 HR-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 ocean acoustics, shallow water propagation is conveniently described using normal mode propagation. This article proposes a framework to describe the polarization of normal modes, as measured using a particle velocity sensor in the water column. To do so, the article introduces the Stokes parameters, a set of four real-valued quantities widely used to describe polarization properties in wave physics, notably for light. Stokes parameters of acoustic normal modes are theoretically derived, and a signal processing framework to estimate them is introduced. The concept of polarization spectrogram, which enables the visualization of the Stokes parameters using data from a single vector sensor, is also introduced. The whole framework is illustrated on simulated data, as well as on experimental data collected during the 2017 Seabed Characterization Experiment (SBCEX17). By introducing the Stokes framework used in many other fields, the article opens the door to a large set of methods developed and used in other contexts, but largely ignored in ocean acoustics.