Section: New Results

Self–similar prior and wavelet bases for hidden turbulent motion

Participant : Patrick Héas.

This is a collaboration with Frédéric Lavancier (université de Nantes) and Souleymane Kadri–Harouna (université de la Rochelle)

This work [14] is concerned with the ill–posed inverse problem of estimating turbulent flows from the observation of an image sequence. From a Bayesian perspective, a divergence–free isotropic fractional Brownian motion (fBm) is chosen as a prior model for instantaneous turbulent velocity fields. This self–similar prior characterizes accurately second–order statistics of velocity fields in incompressible isotropic turbulence. Nevertheless, the associated maximum a posteriori involves a fractional Laplacian operator which is delicate to implement in practice. To deal with this issue, we propose to decompose the divergence–free fBm on well–chosen wavelet bases. As a first alternative, we propose to design wavelets as whitening filters. We show that these filters are fractional Laplacian wavelets composed with the Leray projector. As a second alternative, we use a divergence–free wavelet basis, which takes implicitly into account the incompressibility constraint arising from physics. Although the latter decomposition involves correlated wavelet coefficients, we are able to handle this dependence in practice. Based on these two wavelet decompositions, we finally provide effective and efficient algorithms to approach the maximum a posteriori.