Nonlinear approximation of functions in two dimensions by sums of wave packets

We consider the problem of approximating functions that arise in wave-equation imaging by sums of wave packets. Our objective is to find sparse decompositions of image functions, over a finite range of scales. We also address the naturally connected task of numerically approximating the wavefront set. We present an approximation where we use the dyadic parabolic decomposition, but the approach is not limited to only this type. The approach makes use of expansions in terms of exponentials, while developing an algebraic structure associated with the decomposition of functions into wave packets.