Compressive Sensing

Special Thanks to Lingchen Zhu

Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This takes advantage of the signal's sparseness or compressibility in some domain, allowing the entire signal to be determined from relatively few measurements. Tomography, such as MRI and X-ray CT, is a prominent application.

Around 2004 Emmanuel Candès, Terence Tao and David Donoho discovered important results on the minimum amount of data needed to reconstruct an image even though the amount of data would be deemed insufficient by the Nyquist–Shannon criterion. This work is the basis of compressed sensing as currently studied.

Starting with the single-pixel camera from Rice University, an up-to-date list of the most recent implementations of compressive sensing in hardware at different technology readiness level is available. Some hardware implementations (like the one used in MRI or compressed genotyping) do not require an actual physical change, whereas other hardware require substantial re-engineering to perform this new type of sampling. Similarly, a number of hardware implementations already existed before 2004; however, while they were acquiring signals in a compressed manner, they generally did not use compressive sensing reconstruction techniques to reconstruct the original signal. The result of these reconstruction were suboptimal and have been greatly enhanced thanks to compressive sensing.

The field of compressive sensing is related to other topics in signal processing and computational mathematics, such as to underdetermined linear-systems, group testing, heavy hitters, sparse coding, multiplexing, sparse sampling, and finite rate of innovation. Imaging techniques having a strong affinity with compressive sensing include coded aperture and computational photography.