Breaking the coherence barrier: a new theory for compressed sensing. (English) Zbl 1410.94030

Summary: 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.


94A20 Sampling theory in information and communication theory
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
65R32 Numerical methods for inverse problems for integral equations
92C55 Biomedical imaging and signal processing
Full Text: DOI arXiv


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