Wavelet-Based Fluid Motion Estimation

Abstract

Abstract. Based on a wavelet expansion of the velocity field, we present a novel optical flow algorithm 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, wavelet properties enable the design of simple yet efficient high-order regularizers or polynomial approximations associated to a low computational complexity. Accuracy of proposed methods is assessed on challenging sequences of turbulent fluids flows.

Divergence-Free Wavelets and High Order Regularization

“…The main drawback of the method described in [25] lies in its high computational cost, caused by the necessity to explicitly evaluate the functional Hessian. Recently, a new wavelet-based method that overcomes this computational burden was proposed by Dérian et al [4]. The main objective in this section is to briefly introduce this method and the principles that underlies the construction of wavelet bases.…”

“…With the previous shorthand notations, the method of [4] consists in searching, at each scale j, motion field u j represented on a wavelet basis at scale j: where Ψ j is the selected 2D wavelet basis and d j the vector of unknown coefficients that have to be estimated to represent the sought motion components. Incorporating (17) in the (DFD) equation and setting…”

“…The proposed approach falls within the context of wavelet-based multiresolution methods [4] sketched in the previous section. Then, the velocity field u is searched in terms of its divergence-free wavelet projection (33)…”

“…Such estimator which extends the approaches of [4,25] is particularly relevant for the estimation of an incompressible fluid flow from two consecutive images. The divergence free constraint, characterizing the physics of incompressible fluid flows, is imposed by directly estimating coefficients of the optical flow projection onto a divergence-free wavelet basis.…”

“…This technique has however a prohibitive computational cost in O(N 6 ), where N is the number of wavelet coefficients. An extension of this work reducing the algorithm complexity has recently been proposed [4]. Besides, considering the introduction of a mass preserving constraint, divergence-free wavelets define natural bases for the solutions of the incompressible NavierStokes equations.…”