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Published online by Cambridge University Press:  15 February 2017

Department of Mathematics, Simon Fraser University, Burnaby BC, V5A 1S6, Canada;
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK;,, Department of Mathematics, University of Oslo, 0316 OSLO, Norway
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK;,,
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK;,,


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This paper presents a framework for compressed sensing that bridges a gap between existing theory and the current use of compressed sensing in many real-world applications. In doing so, it also introduces a new sampling method that yields substantially improved recovery over existing techniques. In many applications of compressed sensing, including medical imaging, the standard principles of incoherence and sparsity are lacking. Whilst compressed sensing is often used successfully in such applications, it is done largely without mathematical explanation. The framework introduced in this paper provides such a justification. It does so by replacing these standard principles with three more general concepts: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. Moreover, not only does this work provide such a theoretical justification, it explains several key phenomena witnessed in practice. In particular, and unlike the standard theory, this work demonstrates the dependence of optimal sampling strategies on both the incoherence structure of the sampling operator and on the structure of the signal to be recovered. Another key consequence of this framework is the introduction of a new structured sampling method that exploits these phenomena to achieve significant improvements over current state-of-the-art techniques.

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