Bivariate signals appear in a broad range of applications where the joint analysis of two real-valued signals is required: polarized waveforms in seismology and optics, eastward and northward current velocities in oceanography, pairs of electrode recordings in EEG or MEG or even gravitational waves emitted by coalescing compact binaries. Simple bivariate signals take the form of an ellipse, whose properties (size, shape, orientation) may evolve with time. This geometric feature of bivariate signals has a natural physical interpretation called polarization. This notion is fundamental to the analysis and understanding of bivariate signals. However, existing approaches do not provide straightforward descriptions of bivariate signals or filtering operations in terms of polarization or ellipse properties. To this purpose, this thesis introduces a new and generic approach for the analysis and filtering of bivariate signals. It essentially relies on two key ingredients: (i) the natural embedding of bivariate signals -- viewed as complex-valued signals -- into the set of quaternions H and (ii) the definition of a dedicated quaternion Fourier transform to enable a meaningful spectral representation of bivariate signals. The proposed approach features the definition of standard signal processing quantities such as spectral densities, linear time-invariant filters or spectrograms that are directly interpretable in terms of polarization attributes. These geometric and physical interpretations are made possible by the use of quaternion algebra. More importantly, the framework does not sacrifice any mathematical guarantee and the newly introduced tools admit computationally fast implementations. By revealing the specificity of bivariate signals, the proposed framework greatly simplifies the design of analysis and filtering operations. Numerical experiment support throughout our theoretical developments. We demonstrate the potential of the approach for the characterization of (polarized) gravitational waves emitted by compact coalescing binaries. A companion Python package called BiSPy implements our findings for the sake of reproducibility.
Directeurs de thèse : M. Pierre CHAINAIS, Professeur des Universités, Centrale Lille, Directeur de thèse M. Nicolas LE BIHAN, Chargé de Recherche CNRS, Université Grenoble Alpes, Directeur de thèse Rapporteurs : Mme Tülay ADALI, Distinguished University Professor, University of Maryland, Baltimore County M. Gabriel PEYRE, Directeur de Recherche CNRS, École Normale Supérieure Membres : Mme Marianne CLAUSEL, Professeure des Universités, Université de Lorraine M. Patrick FLANDRIN, Directeur de Recherche CNRS, École Normale Supérieure de Lyon M. Philippe REFREGIER, Professeur des Universités, Centrale Marseille M. Eric CHASSANDE-MOTTIN, Directeur de Recherche CNRS, Université Paris Diderot, Membre invité
Thesis of the team SIGMA defended on 27/09/2018