Turbulence power laws and inverse motion modeling in images

Based on scaling laws describing the statistical structure of turbulent motion across scales, we propose a multiscale and non-parametric regularizer for the estimation of velocity fields of  bidimensional or quasi-bidimensional flows from image sequences. The spatial regularization principle used in order to close the ill-posed nature of motion estimation is achieved by constraining motion increments  to behave thro
ugh scales as the most likely self-similar process given some image data. Motion estimation is formulated as a minimization problem where the solution  second order structure function is constrained to behave as a power law. The most likely power law given the image data is jointly inferred by maximization of Bayesian evidence. The motion estimator accuracy is first evaluated on a synthetic image sequence of simulated bidimensional turbulence. Results obtained with the approach based on the physics exceeds the best state of the art results. Then, the method is used to analyze a real meteorological image sequence. Selecting from images the most evident multiscale motion model enables the recovery of physical quantities which are of major interest for turbulence characterization. In particular, from the meteorological images we are able to estimate the energy and enstrophy flux, which are in agreement with previous in situ measurements, and to reconstruct structure functions and the energy spectrum with information of the turbulent cascades.


1-  P. Héas, E. Mémin, D. Heitz, P.D. Mininni. Power laws and inverse motion modeling: application to turbulence measurements from satellite images. Conditionally accepted for publication in Tellus Series A: Dynamic Meteorology and Oceanography.

2- P. Héas, E. Mémin, D. Heitz, P.D. Mininni. Bayesian selection of scaling laws for motion modeling in images. In International Conference on Computer Vision (ICCV'09), Kyoto, Japan, October 2009.

3- P.Héas, E. Mémin, Heitz. Multiscale regularization based on turbulent kinetic energy decay for PIV estimations with high spatial resolution. In Symposium on Particle Image Velocimetry (PIV'09), Melbourne, Australia, July 2009.

4- P. Héas, E. Mémin. Inference on Gibbs optic flow prior : application to atmospheric turbulence characterization, IEEE International geoscience and remote sensing symposium (IGARSS), Cape town, South Africa, 2009.



Bidimensional turbulence image sequence. Simulated particle images (right), true (middle) and estimated (left) velocity field with ~[Baker07] color system for scalar visualization of vector fields.

Power law model selection. Left : Inferred power law (continuous red line) for 2-nd order structure function. True (dash line) and estimated  (crosses) 2-nd order structure functions  in horizontal-vertical (in blue and turquoise) and diagonal (in pink and green) directions . Right : Energy spectra E(k) of first order (in turquoise), div-curl (in blue or pink) and self-similar (in green) regularizers compared to the true (in red)  spectrum.

Motion estimation accuracy. Velocity field scalar representation (1-st line), Barron's angular error (2-nd line) and end point errors (3-rd line) comparisons with state of the art estimators. RMSE and average Barron angular error are displayed above the figures.


Input meteorological images. Sparse pressure difference maps of layers at intermediate altitude. White pixels of image are areas with no observations. The images characterize the layers evolution in a time interval of 15 minutes.

Horizontal winds & energy flux. Vector (left) and color (right) motion representations for increasing energy flux (including the most likely flux (in blue) selected by Bayesian inference superimposed on the sparse image data.