Optimal compressive imaging for Fourier data

One fundamental problem in applied mathematics is the issue of recovery of data from specic samples. Of particular importance is the case of pointwise samples of the associated Fourier transform, which are, for instance, collected in Magnetic Resonance Imaging (MRI). Strategies to reduce the number of samples required for reconstruction with a prescribed accuracy have thus a direct impact on such applications { which in the case of MRI will, for instance, shorten the time a patient is forced to lie in the scanner. In this talk, we will present a sparse subsampling strategy of Fourier samples which can be shown to perform optimally for functions governed by anisotropic features. For this, we will introduce a dualizable shearlet frame for reconstruc- tion, which provides provably optimally sparse approximations of this class of functions { such cartoon-like images are typically regarded as a suitable model for images. The sampling scheme will be based on compressive sensing ideas combined with a coherence-adaptive sampling density considering the coherence between the Fourier basis and the shearlet frame. We nally prove that this general sampling strategy can sparsely approximate a function of the consid- ered model class from a collection of its Fourier samples with optimal sampling rate.