Team geostat

Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry

Section: Application Domains

Theoretical developments

Participants : Hussein Yahia, Khalid Daoudi, Oriol Pont, Annick Lesne, Antonio Turiel, Véronique Garçon, Sylvie Rocques, Régine-André Obrecht, Christine Provost, Vahid Khanagha, Joel Sudre, Alex Potamianos, Petros Maragos, Ioanis Klasinas, Reda Jourani.

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: Im15 ${h_\#952 =-0.2}$ , Im16 ${h_\#952 =0.0}$ , and Im17 ${h_\#952 =0.1}$ (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:


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