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Compressed sensing has had a profound influence on sampling and signal processing. Based on the idea of sparsity, it provides a theory and techniques for the recovery of signals and images from highly incomplete sets of of measurements. The key ingredients that permit this so-called subsampling are (i) sparsity of the signal in a particular basis and (ii) mutual incoherence between such basis and the sampling system. Provided the corresponding coherence parameter is sufficiently small, one can recover a sparse signal using a number of measurements that is, up to a log factor, on the order of the sparsity. Unfortunately, many problems that one encounters in practice are not incoherent. For example, Fourier sampling, the type of sampling encountered in Magnetic Resonance Imaging (MRI), is typically not incoherent with wavelet bases. To overcome this `coherence barrier' we introduce a new theory of compressed sensing, based on the principles of so-called asymptotic incoherence and asymptotic sparsity. When combined with a multi-level sampling strategy, this allows for significant subsampling in problems for which standard compressed sensing tools are limited by the lack of incoherence. Moreover, we demonstrate how the amount of subsampling possible with this new approach actually increases with resolution. In other words, this technique is ideally suited to higher resolution problems. This is joint work with Anders Hansen and Bogdan Roman (Cambridge).