Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing

We introduce a mathematical framework that bridges a substantial gap between compressed sensing theory and its current use in real-world applications. Although completely general, one of the principal applications for our framework is the Magnetic Resonance Imaging (MRI) problem. Our theory provides a comprehensive explanation for the abundance of numerical evidence demonstrating the advantage of so-called variable density sampling strategies in compressive MRI. Besides this, another important conclusion of our theory is that the success of compressed sensing is resolution dependent. At low resolutions, there is little advantage over classical linear reconstruction. However, the situation changes dramatically once the resolution is increased, in which case compressed sensing can and will offer substantial benefits.