Section: Overall Objectives

Overall objectives

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GEOSTAT is a research project in nonlinear signal processing, with the fundamental distinction that it considers the signals as the realizations of complex dynamic systems. The driving approach is to understand the relations between complexity (or information content) and the geometric organization of information in a signal. For instance, for signals which are acquisitions of turbulent fluids, the organization of information is related to the effective presence of a multiscale hierarchy, of multifractal nature, which is strongly related to intermittency and multiplicative cascade phenomena; the determination of this geometric organization unlocks key nonlinear parameters and features associated to these signals; it helps understanding their dynamical properties and, as a consequence, their analysis. Specific advances are obtained in GEOSTAT in using this type of statistical/geometric approach to get validated dynamical information of signals acquired in Universe Sciences, e.g. Oceanography or Astronomy. The research in GEOSTAT encompasses nonlinear signal processing and the study of emergence in complex systems, with a strong emphasis on geometric approaches to complexity. Consequently, research in GEOSTAT is oriented towards the determination, in real signals, of quantities or phenomena, usually unattainable through linear methods, that are known to play an important role both in the evolution of dynamical systems whose acquisitions are the signals under study, and in the compact representations of the signals themselves. Research in GEOSTAT is structured in two parts:

• Theoretical and methodological aspects in nonlinear signal processing.

• Applicative aspects which encompass biomedical data (heartbeat signal analysis with IHU LIRYC, biomedical applications in speech signal analysis) and the study of universe science datasets.

The theoretical objectives are:

• multiscale description in terms of multiplicative cascade (essential in the characterization of turbulent systems).

• Excitable systems (cardiac electrophysiology): study of intermittency phenomena. Accessible objective: theoretical derivation of optimal wavelets.

The methodological tools used in reaching these objectives place GEOSTAT in the forefront of nonlinear signal processing and complex systems. We cite: singularity exponents [59] [6] , how these exponents can be related to sparse representations with reconstruction formulae [12] [60] [12]  [12] , [4] , the comparison with embedding techniques, such as the one provided by the classical theorem of Takens [57] , [50] , the use of Lyapunov exponents, how they are related to intermittency, large deviations and singularity exponents, various forms of entropies, persistence along the scales, optimal wavelets [5] , comparison with other approaches such as Compressive Sensing, and, above all, the ways that lead to effective numerical and high precision determination of nonlinear characteristics in real signals. The MMF (Multiscale Microcanonical Formalism) is one of the ways to partly unlock this type of analysis, most notably w.r.t. singularity exponents and reconstructible systems [10] . We presently concentrate our efforts on it, but GEOSTAT is intended to explore other ways [46] . Presently GEOSTAT explores new methods for analyzing and understanding complex signals in different applicative domains through the theoretical advances of the MMF, and the framework of reconstructible systems [58] . Derived from ideas in Statistical Physics, the methods developed in GEOSTAT provide new ways to relate and evaluate quantitatively the local irregularity in complex signals and systems, the statistical concepts of information content and most informative subset. That latter notion is developed through the notion of transition front and Most Singular Manifold. As a result, GEOSTAT is aimed at providing radically new approaches to the study of signals acquired from different complex systems (their analysis, their classification, the study of their dynamical properties etc.). A common characteristic of these signals, which is related to universality classes [53] [54] [51] , being the existence of a multiscale organization of the systems. For instance, the classical notion of edge or border, which is of multiscale nature, and whose importance is well known in Computer Vision and Image Processing, receives profound and rigorous new definitions, in relation with the more physical notion of transition and fits adequately to the case of chaotic data. The description is analogous to the modeling of states far from equilibrium, that is to say, there is no stationarity assumption. From this formalism we derive methods able to determine geometrically the most informative part in a signal, which also defines its global properties and allows for compact representation in the wake of known problematics addressed, for instance, in time-frequency analysis. In this way, the MMF allows the reconstruction, at any prescribed quality threshold, of a signal from its most informative (i. e. most unpredictable) subset, and is able to quantitatively evaluate key features in complex signals (unavailable with classical methods in Image or Signal Processing). It appears that the notion of transition front in a signal is much more complex than previously expected and, most importantly, related to multiscale notions encountered in the study of nonlinearity [55] . For instance, we give new insights to the computation of dynamical properties in complex signals, in particular in signals for which the classical tools for analyzing dynamics give poor results (such as, for example, correlation methods or optical flow for determining motion in turbulent datasets).