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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. MRI is a prominent application.

Overview

A common goal of signal processing is to reconstruct a signal from a series of sampling measurements. In general, this task is impossible because there is no way to reconstruct a signal during the times that the signal isn’t measured. Nevertheless, with prior knowledge or assumptions about the signal, it turns out to be possible to perfectly reconstruct a signal from a series of measurements. Over time, mathematicians have improved their understanding of which assumptions are practical and how they can be generalized.

An early breakthrough in signal processing was the Nyquist-Shannon sampling theorem. It proved that a signal can be perfectly reconstructed from sampling if the signal’s highest frequency is half (or less) of the sampling rate. The main idea is that if you have prior knowledge about the signal’s frequencies, you need fewer samples to reconstruct the signal.

Around 2004, Emmanuel Candès, Terence Tao, and David Donoho proved that given knowledge about a signal's sparsity, the signal may be reconstructed with fewer samples than the Nyquist-Shannon theorem requires. This idea is the basis of compressed sensing.

History

Compressed sensing relies on L1 techniques, which several other scientific fields have used historically. In statistics, the least-squares method was complemented by the $L^{1}$-norm, which was introduced by Laplace. Following the introduction of linear programming and Dantzig's simplex algorithm, the $L^{1}$-norm was used in computational statistics. In statistical theory, the $L^{1}$-norm was used by George W. Brown and later writers on median-unbiased estimators. It was used by Peter Huber and others working on robust statistics. The $L^{1}$-norm was also used in signal processing, for example, in the 1970s, when seismologists constructed images of reflective layers within the earth based on data that did not seem to satisfy the Nyquist–Shannon criterion. It was used in matching pursuit in 1993, the LASSO estimator by Robert Tibshirani in 1996 and basis pursuit in 1998. There were theoretical results describing when these algorithms recovered sparse solutions, but the required type and number of measurements were sub-optimal and subsequently greatly improved by compressed sensing.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

At first glance, compressed sensing might seem to violate the sampling theorem, because compressed sensing depends on the sparsity of the signal in question and not its highest frequency. This is a misconception, because the sampling theorem guarantees perfect reconstruction given sufficient, not necessary, conditions. A sampling method different from the classical fixed-rate sampling therefore can not "violate" the sampling theorem. Sparse signals with high frequency components can be highly under-sampled using compressed sensing compared to classical fixed-rate sampling.

Method

Underdetermined linear system

An underdetermined system of linear equations has more unknowns than equations and generally has an infinite number of solutions. In order to choose a solution to such a system, one must impose extra constraints or beliefs (such as smoothness) as appropriate.

In compressed sensing, one adds the constraint of sparsity, allowing only solutions which have a small number of nonzero coefficients. Not all underdetermined systems of linear equations have a sparse solution. However, if there is a unique sparse solution to the underdetermined system, then the Compressed Sensing framework allows the recovery of that solution.

Solution / reconstruction method

Compressed sensing takes advantage of the redundancy in many interesting signals—they are not pure noise. In particular, many signals are sparse, that is, they contain many coefficients close to or equal to zero, when represented in some domain. This is the same insight used in many forms of lossy compression.

Compressed sensing typically starts with taking a weighted linear combination of samples also called compressive measurements in a basis different from the basis in which the signal is known to be sparse. The results found by Emmanuel Candès, Justin Romberg, Terence Tao and David Donoho, showed that the number of these compressive measurements can be small and still contain nearly all the useful information. Therefore, the task of converting the image back into the intended domain involves solving an underdetermined matrix equation since the number of compressive measurements taken is smaller than the number of pixels in the full image. However, adding the constraint that the initial signal is sparse enables one to solve this underdetermined system of linear equations.

The least-squares solution to such problems is to minimize the $L^{2}$ norm—that is, minimize the amount of energy in the system. This is usually simple mathematically (involving only a matrix multiplication by the pseudo-inverse of the basis sampled in). However, this leads to poor results for many practical applications, for which the unknown coefficients have nonzero energy.

To enforce the sparsity constraint when solving for the underdetermined system of linear equations, one can minimize the number of nonzero components of the solution.

The function counting the number of non-zero components of a vector was called the $L^{0}$ "norm" by David Donoho. The quotation marks served two warnings. First, the number-of-nonzeros $L^{0}$-"norm" is not a proper F-norm, because it is not continuous in its scalar argument: nnzs(αx) is constant as α approaches zero. Unfortunately, authors now neglect the quotation marks and abused terminology—clashing with the established use of the $L^{0}$ norm for the space of measurable functions (equipped with an appropriate metric) or for the space of sequences with F–norm $(x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}$.

Candès. et al., proved that for many problems it is probable that the $L^{1}$ norm is equivalent to the $L^{0}$ norm, in a technical sense: This equivalence result allows one to solve the $L^{1}$ problem, which is easier than the $L^{0}$ problem. Finding the candidate with the smallest $L^{1}$ norm can be expressed relatively easily as a linear program, for which efficient solution methods already exist. When measurements may contain a finite amount of noise, basis pursuit denoising is preferred over linear programming, since it preserves sparsity in the face of noise and can be solved faster than an exact linear program.

Applications

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. Implementations of compressive sensing in hardware at different technology readiness level is available.

Photography

Compressed sensing is used in a mobile phone camera sensor. The approach allows a reduction in image acquisition energy per image by as much as a factor of 15 at the cost of complex decompression algorithms; the computation may require an off-device implementation.

Compressed sensing is used in single-pixel cameras from Rice University. Bell Labs employed the technique in a lensless single-pixel camera that takes stills using repeated snapshots of randomly chosen apertures from a grid. Image quality improves with the number of snapshots, and generally requires a small fraction of the data of conventional imaging, while eliminating lens/focus-related aberrations.

Holography

Compressed sensing can be used to improve image reconstruction in holography by increasing the number of voxels one can infer from a single hologram. It is also used for image retrieval from undersampled measurements in optical and millimeter-wave holography.

Shortwave Infrared Cameras

InView Technology Corporation has developed commercially-available shortwave infrared cameras based upon Compressed Sensing. These cameras have light sensitivity from 0.9 µm to 1.7 µm, which are wavelengths invisible to the human eye. InView is building upon the Rice University single-pixel camera results.

Optical System Research

InView Technology Corporation has developed optical Compressive Sensing Workstations that allow optical-system researchers to develop and test novel modulation and reconstruction algorithms. These Workstations include a spatial light modulation subsystem, and a light acquisition sub-system.

Facial recognition

Compressed sensing is being used in facial recognition applications.

MRI

Compressed sensing has been used to shorten MRI scanning sessions on conventional hardware.

See also

• Noiselet
• Sparse approximation

References

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Further reading

• "The Fundamentals of Compressive Sensing" Part 1, Part 2 and Part 3: video tutorial by Mark Davenport, Georgia Tech. at IEEE Signal Processing Society Online Tutorial Library.
• Using Math to Turn Lo-Res Datasets Into Hi-Res Samples Wired Magazine article
• Compressive Sensing Resources at Rice University.
• Compressed Sensing: The Big Picture
• A list of different hardware implementation of Compressive Sensing
• Compressed Sensing 2.0
• Compressed Sensing Makes Every Pixel Count – article in the AMS What's Happening in the Mathematical Sciences series
• Nuit Blanche A blog on Compressive Sensing featuring the most recent information on the subject (preprints, presentations, Q/As)
• Online Talks focused on Compressive Sensing
• Wiki on sparse reconstruction