Section: Scientific Foundations

Optimal wavelets

The notion of optimal wavelet has been introduced by O. Pont (who is currently holding a post-doc position in GEOSTAT)  [53] ,[4] . Optimal wavelets have the fascinating potential of unlocking the signal's microcanonical cascading properties through simple wavelet decomposition. In other words, the turbulent properties of the signal become apparent from the optimal wavelet decomposition. Nevertheless the proper determination of an optimal wavelet associated to a given signal is a very difficult problem, and GEOSTAT is developping important research in this direction. The subject has a strong potential in speech analysis and oceanic motion determination (for instance: descending motion information through an optimal wavelet across the scales to obtain oceanic motion at high spatial resolution from knowledge at lower spatial resolutions). Optimal wavelets are associated to the notion of microcanonical cascade : they try to catch the cascading properties in the microcanonical sense, around any point in the spatial domain of the signal. They raise research problem that encompass both theoretical and empirical approaches. From a theoretical standpoint, we need to determine wich signals are describable as microcanonical cascades or, more precisely, whether they always have an optimal wavelet. It seems that natural systems, which are compliant with Parisi-Frisch statistical-geometrical duality do have this property. In the practical realm, they open new approaches in the forecasting of temporal series.