Self-similar prior and wavelet bases for hidden incompressible turbulent motion

TL;DR

This paper introduces wavelet-based methods for estimating incompressible turbulent flows from image sequences using a self-similar prior modeled as divergence-free fractional Brownian motion, improving practical implementation.

Contribution

It proposes novel wavelet decompositions to efficiently implement a self-similar divergence-free prior for turbulent flow estimation, addressing computational challenges.

Findings

Wavelet decompositions effectively approximate the fractional Laplacian operator.

Divergence-free wavelet basis captures incompressibility constraints.

Numerical evaluations demonstrate the method's relevance and efficiency.

Abstract

This work 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 divergent-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. An intensive numerical evaluation proves the relevance of the proposed wavelet-based self-similar priors.