Divergence Free Polar Wavelets
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We present a Parseval tight wavelet frame for the representation of divergence free vector fields in two and three dimensions. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and, for an appropriate choice of the window functions, vanishing moments and steerability. Our construction also allows for well defined directional selectivity, enabling a vector-valued, divergence free analogue of curvelets that models the behavior of solenoidal vector fields in the vicinity of boundaries. We show that the curvelet-like wavelets attain, up to a logarithmic factor, an optimal approximation rate for piecewise C^2 signals in two dimensions. In contrast to scalar curvelets, two different types of directional, divergence free wavelets exist, which, in the case of an incompressible fluid, can be associated with free-slip and no-slip boundary conditions. We demonstrate the practicality and efficiency of our construction by studying the approximation power of the wavelets for divergence free fluid vector fields.
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