Compressed sensing, image reconstruction and limited measurements

The following blog entry is based on the paper Breaking the coherence barrier: A new theory for compressed sensing by Ben Adcock, Anders C. Hansen, Clarice Poon and Bogdan Roman. It was published online in February of 2017 in Volume 5 of Forum of Mathematics, Sigma.

Compressed sensing has been one of the most important innovations in applied mathematics in the last several decades. In many applications in science, engineering and industry one is faced with the situation of limited measurements. This is the case notably in image reconstruction, where the number of measurements is limited by time, cost, power or other constraints. For example, in Magnetic Resonance Imaging (MRI), one of imaging modalities in which compressed sensing techniques have been most successful, where the long acquisition time places a substantial constraint on the number of measurements that can be acquired.

With its introduction in 2006, compressed sensing provided a new means to attack such problems. Images have an inherent low-dimensional structure, they are characterized by smooth regions separated by edges, and hence possess sparse representations in, for instance, wavelet transforms. While this observation had long been known, the innovation of compressed sensing was to exploit this property through two crucial elements. First, a sparsity-promoting reconstruction algorithm such ℓ1-minimization, and second, a carefully chosen set of measurements.

The design of effective, structure-promoting sampling strategies is a fundamental aspect of the mathematics of compressed sensing.

By 2012, the standard mathematical theory of compressed sensing was reaching a mature state. However, there was a problem. This theory did not explain why compressed sensing worked so well in practice in image reconstruction, nor, and crucially, how its performance could be optimized. In particular, the standard theory was known to lead to sampling strategies which, while theoretically sound, were easily beaten by heuristic approaches.

Adcock, Hansen, Poon and Roman sought to bridge this gap.

Their key insight was that certain local structure played a fundamental role in image recovery via compressed sensing. The conventional theory of compressed sensing was built on global principles, sparsity, incoherence and random sampling. Hence, they set out to develop a new theory for compressed sensing based on new local principles – sparsity in levels, local coherence in levels and multilevel random sampling. This theory is contained in the resulting paper.

Beyond its theoretical contribution, this work also has signi cant practical impact. It provides new and important insight into how to adapt and optimize the performance of compressed sensing in practical applications. It introduces a new generation of sampling strategies based on multilevel random sampling which are both theoretically optimal and outperform previous state-of-the-art approaches in practice. It has led to new approaches in MRI, NMR, uorescence microscopy and helium atom scattering which offer signi cant performance gains.

Imaging has arguably been the biggest bene ciary of compressed sensing techniques. The US FDA approved compressed sensing methods for commercial use MRI in 2017. This work provides the theoretical underpinnings for its success, and guides its optimal implementation in practice.

Breaking the coherence barrier: A new theory for compressed sensing‘ is available for download and is free to access in perpetuity as an open access publication. Forum of Mathematics, Pi and Forum of Mathematics, Sigma are exciting developments in open access publishing. Together they offer fully OA publication combined with peer-review standards set by an international editorial board of the highest calibre, all backed by Cambridge University Press and our commitment to quality.

Stand by for a book on this topic, Compressive Imaging, coming in 2020 from Cambridge University Press.

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