A matching pursuit approach to solenoidal filtering of three-dimensional velocity measurements

Various measurement techniques have been developed recently which provide threedimensional velocity fields. These methodologies have tremendous potential to improve the understanding of complex flows in both engineering and biomedical applications. Many flows of interest are effectively incompressible which translates into a zero-divergence constraint in the Navier-Stokes equations. Noise and intrinsic errors inevitably affect three-dimensional flow measurements generating a spurious, finite divergence. The development of a filter to extract a divergence-free field of minimum distance (in the sense of the closest approximation) from the measurements is therefore of primary interest. Beside improving accuracy, the resulting divergence-free velocity fields are most suitable to extract physical information, notably the pressure field, by integration of the Navier Stokes equations. We propose an iterative procedure which progressively evaluates the `2-nearest zero divergence field from the available data and is compatible with the measurement resolution. Principles developed in the signal processing community (i.e., Mallat & Zhang 1993) are employed to decompose a velocity field as a linear combination of local solenoidal waveforms. We also show that the proposed algorithm is naturally parallelizable and therefore can greatly benefit from implementation on modern GPU architectures.