Compressed Sensing – An Introduction

This week Emmanuel Candès, professor of Mathematics and Statistics at Stanford University has been elected to the American Academy of Arts and Sciences. He is one of the authors (the other being Terence Tao) of the paper which brought about the field of compressed sensing.

If you’ve ever done any signal processing, then you’ll know that according to the Nyquist-Shannon theorem, signals need to be sampled at least greater than twice the highest frequency present in order to accurately reproduce the signal. The field of compressed sensing, which has only been around since about 2004, openly defies this law, by providing near perfect reconstruction of undersampled signals, by employing some prior knowledge of the signal type to take advantage of sparsity.

Unfortunately I haven’t yet found an intuitive explanation for how it works that doesn’t at least require a reasonable knowledge of linear algebra and information theory (of which my own knowledge is more than a little  rusty), but there is some good background in the Wired article. The image in the article appears to be a broken link but according to this article it is the image shown below.

Compressed sensing has most notably been applied to (and indeed was discovered while searching for a way to improve) MRI, and it also has applications in radio astronomy. It may have great applications in audio and this article provides a simple, practical demonstration for how compressed sensing may be used to recover undersampled audio signals.

Demonstration of how Compressed Sensing Works

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