## Section: Application Domains

### Theoretical developments

The previous sections have defined precisely the starting point of GEOSTAT's scientific themes. In GEOSTAT we are primarily interested in the determination and the study of geometric super-structures, accessible through a single realization of a physical system, and not through stationary averages. A valuable approach consists in introducing the exponents inside a specific measure attached to a signal, and this point of view has produced many interesting results already published by researchers involved in GEOSTAT. Moreover, the MMF raises complicated numerical problems.

A quite important point is the possibility of studying multiplicative cascading in the framework of the MMF. Consequently, the relation between a particular and important geometric super-structure, the so-called Most Singular Manifold (MSM) and the signal is deterministic, a result which allows the derivation of a reconstruction formula: a signal can be fully reconstructed from its restriction to the MSM, hence the statistical signifiance of the MSM as the most informative transition front , on top of its dynamical properties. This is the framework of reconstructible systems. This notion of reconstruction is at the core of many GEOSTAT thematics, because it is the starting point for the definition of different reduced signals ; these reduced signals, when compared with appropriate tools (such as, for instance, the Radon-Nykodim derivative in the study of convection) to the original signal, allow the study of the dynamic in complex signals. The reconstruction itself consists in the diffusion, using an universal propagator , of the gradient in Fourier space. In figure 5 , we show some examples of reconstruction on a MétéoSat image.

Figure 5. Reconstruction of a MétéoSat image from geometric super-structures. The various geometric structures are associated to different thresholds in the distribution of singularity exponents: , , and (top to bottom). Images on the left: geometric super-structures; densities: 6.01% , 28.69% and 42.40% . Images on the right: reconstructions; PSNRs= 20.06 dB, 30.01 dB and 34.91 dB.      The theoretical objectives in GEOSTAT are:

• To study, from a theoretical point of view, the notions developped for complex signals about predictibility (extensions of Lyapunov exponents, in particular in finite time, various notions of entropy, large deviations); establish the relationships between these notions and those developped in the framework of reconstructible systems such as singularity exponents, and to the dynamics of the multiscale signals as well.

• To enhance recognition techniques, statistical modeling techniques, classification (reproducing kernels, SVMs) of signals having multiscale properties in the framework of reconstructible systems and the MMF.

• Develop and implement news tools for predictibility in complex multiscale signals.

• Describe the geometric super-structures from a methodological and mathematical point of view, in particular understand how they form and how to compute them efficiently in signals. Understand their relationships with the dynamics. Study of optimal wavelet bases to understand the relationships with the multiplicative cascade.

• geometric super-structures and the dynamics of complex multiscale signals: how do these geometric sets record information about the past in a complex multiscale signal, and how to use the MMF to get temporal information in an acquisition. Study of the relationships with other tools in the analysis of complex dynamics.

• More generally, what properties of a complex system are related to the emergence of geometric structures ? And, in return, how to act on these systems to control their response ?

• Develop algorithms and numerical methods to extract and use the geometrical super-structures.

• Study the apparition of geometrical sets associated to the emergence of complexity in new acquisitions datasets.

• Development of the theory and applications of optimal wavelets .

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