Section: Overall Objectives
Overall objectives

Glossary
 Singularity exponent
A measure of the unpredictability around a point in a complex signal. Based on local reconstruction around a point, singularity exponents can be evaluated in different ways and in different contexts (e.g. nonlocalized, through the consideration of moments and structure fonctions, trough the computation of singularity spectra). In GEOSTAT we study approaches corresponding to far from equilibrium hypothesis (e.g. microcanonical) leading to geometrically localized singularity exponents.
 Singularity spectrum
The mapping from scaling exponents to Hausdorff dimensions. The singularity spectrum quantifies the degree of nonlinearity in a signal or process, and is used to characterize globally the complexity of a signal.
 Most Singular Manifold
The set of most unpredictable points in a signal, identified to the set of strongest transitions as defined by the singularity exponents. From that set the whole signal can be reconstructed.
 Adaptive Optics (AO)
This term refers to a set of methodologies used, notably in Astromical observations, to compensate for the loss of spatial resolution in optical instruments caused by atmospheric turbulence.
 Compact Representation
Reduced representation of a complex signal (dimensionality reduction) from which the whole signal can be reconstructed. The reduced representation can correspond to points randomly chosen, such as in Compressive Sensing, or to geometric localization related to statistical information content (framework of reconstructible systems).
 MMF
 Sparse representation
The representation of a signal as a linear combination of elements taken in a dictionary, with the aim of finding the most sparse possible one.
 Optimal wavelet
(OW). Wavelets whose associated multiresolution analysis optimizes inference along the scales in complex systems.
 Universality class
In theoretical physics, the observation of the coincidence of the critical exponents (behaviour near a second order phase transition) in different phenomena and systems is called universality. Universality is explained by the theory of the renormalization group, allowing for determination of the changes a physical system undergoes under different distance scales. As a consequence, different macroscopic phenomena displaying a multiscale structure (and their acquisition in the form of complex signals) can be grouped into different sets of universality classes.
GEOSTAT is a research project in nonlinear digital 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. Hence we first mention:

how singularity exponents can be related to sparse representations with reconstruction formulae,

comparison with embedding techniques, such as the one provided by the classical theorem of Takens [59] , [52] .

Lyapunov exponents, how they are related to intermittency, large deviations and singularity exponents,

multiresolution analysis, specifically when performed on the singularity exponents,

the determination of subsets where statistical information is maximized, their relation to reconstruction and compact representation,

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 [11] . We presently concentrate our efforts on it, but GEOSTAT is intended to explore other ways [48] . 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 [60] . 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 [55] [56] [53] , 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 timefrequency 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 [57] . 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). The problematics in GEOSTAT can be summarized in the following items:

the accurate determination in any ndimensional complex signal of singularity exponents (also called Local Predictability Exponents or LPEs) at every point in the signal domain [61] [7] .

The geometrical determination and organization of singular manifolds associated to various transition fronts in complex signals, the study of their geometrical arrangement, and the relation of that arrangement with statistical properties or other global quantities associated to the signal, e.g. cascading properties [12] .

The study of the relationships between the dynamics in the signal and the distributions of singularity exponents [62][12] .

Multiresolution analysis and inference along the scales [12] , [2] .

The study of the relationships between the distributions of singularity exponents and other formalisms associated to predictability in complex signals and systems, such as cascading variables, large deviations and Lyapunov exponents.

The ability to compute optimal wavelets and relate such wavelets to the geometric arrangement of singular manifolds and cascading properties[5] .

The translation of recognition, analysis and classification problems in complex signals to simpler and more accurate determinations involving new operators acting on singular manifolds using the framework of reconstructible systems.