Breaking the coherence barrier: A new theory for compressed sensing

TL;DR

This paper extends compressed sensing theory by introducing asymptotic sparsity, asymptotic incoherence, and multilevel sampling, explaining its practical success in coherent inverse problems and enabling more flexible, efficient sensing strategies.

Contribution

It generalizes the foundational principles of compressed sensing to include asymptotic concepts and multilevel sampling, bridging the gap between theory and real-world applications.

Findings

Compressed sensing is feasible under relaxed conditions.

Asymptotic incoherence suffices for successful reconstruction.

Multilevel sampling improves efficiency and accuracy.

Abstract

This paper provides an extension of compressed sensing which bridges a substantial gap between existing theory and its current use in real-world applications. It introduces a mathematical framework that generalizes the three standard pillars of compressed sensing - namely, sparsity, incoherence and uniform random subsampling - to three new concepts: asymptotic sparsity, asymptotic incoherence and multilevel random sampling. The new theorems show that compressed sensing is also possible, and reveals several advantages, under these substantially relaxed conditions. The importance of this is threefold. First, inverse problems to which compressed sensing is currently applied are typically coherent. The new theory provides the first comprehensive mathematical explanation for a range of empirical usages of compressed sensing in real-world applications, such as medical imaging, microscopy, spectroscopy and others. Second, in showing that compressed sensing does not require incoherence, but instead that asymptotic incoherence is sufficient, the new theory offers markedly greater flexibility in the design of sensing mechanisms. Third, by using asymptotic incoherence and multi-level sampling to exploit not just sparsity, but also structure, i.e. asymptotic sparsity, the new theory shows that substantially improved reconstructions can be obtained from fewer measurements.