  Applications such as Magnetic Resonance Tomography acquire imaging data by
point samples of their Fourier transform. This raises the question of balancing
the efficiency of the sampling strategies with the approximation accuracy of an
associated reconstruction procedure. In this paper, we introduce a novel
sampling-reconstruction scheme based on a random anisotropic sampling pattern
and a compressed sensing type reconstruction strategy with a variant of
dualizable shearlet frames as sparsifying representation system. For this
scheme, we prove asymptotic optimality in an approximation theoretic sense for
cartoon-like functions as a model class for the imaging data. Finally, we
present numerical experiments showing the superiority of our scheme over other
approaches.
