Wavelet-based motion estimation


Based on a wavelet expansion of the velocity field, we derived novel optical flow algorithms dedicated to the estimation of continuous motion fields such as fluid flows. This scale-space representation, associated to a simple gradient-based optimization algorithm, naturally sets up a well-defined multi-resolution analysis framework for the optical flow estimation problem, thus avoiding the common drawbacks of standard multi-resolution schemes. Moreover, in the context of the analysis of incompressible fluids, divergence-free wavelet series constitute relevant bases for optical flow estimation: incompressible flows are vectorial functions constrained to be divergence-free. Introducing wavelets in optical flow estimation provides several other interesting properties:  the design of high-order polynomial approximations is very simple and can be done with a low computational complexity;  the continuous motion field representation enables the exact analytical calculation of integrals over the spatial domain; by choosing differentiable wavelets, we easily design high-order derivative regularizers, via the calculation of mass and stiffness matrices; it is possible to reconstruct fractional motion fields by constraining the decay of wavelet coefficients. Numerical results on synthetic and real images of turbulent fluids have shown the efficiency of the estimator.


1- P. Derian, P. Heas, C. Herzet, E. Memin. Wavelets to reconstruct turbulence multifractals from experimental image sequences. In 7th Int. Symp. on Turbulence and Shear Flow Phenomena, TSFP-7, Ottawa, Canada, July 2011.

2- P. Derian, P. Heas, C. Herzet, E. Memin Wavelet-Based Fluid Motion Estimation, Scale Space and Variational Methods (SSVM) in Computer Vision, Israel, 2011.


Estimated motion fields from 2D planar jet image dataset:  Upper figures are the vorticity fields corresponding to motion estimated at two consecutive times and the lower figure represents the flow streamlines. Three different areas are visible: at the output of the jet (top of the field), shear regions begin to oscillate slowly. The middle region clearly shows the development of vortices characteristic of the Kelvin-Helmholtz instability. Finally, in the lower part of the field, structure of vortices collapse due to their tri-dimensionalization.

Estimated motion fields from synthetic turbulence image dataset:  Video above shows input frames, ground truth vorticity and vorticity computed from the velocity field estimated with the 100 first frames of the sequence. Original video (mpeg4) can be found here.