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.
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 is not 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 less than half of the sampling rate.
Around 2004, Emmanuel Candès, Terence Tao, and David Donoho proved that given knowledge about a signal's sparsity, the signal may be reconstructed with even fewer samples than the sampling theorem requires. This idea is the basis of compressed sensing.
Compressed sensing relies on L1 techniques, which several other scientific fields have used historically. In statistics, the least squares method was complemented by the -norm, which was introduced by Laplace. Following the introduction of linear programming and Dantzig's simplex algorithm, the -norm was used in computational statistics. In statistical theory, the -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 -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.
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 cannot "violate" the sampling theorem. Sparse signals with high frequency components can be highly under-sampled using compressed sensing compared to classical fixed-rate sampling.
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 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 "norm" by David Donoho. The quotation marks served two warnings. First, the number-of-nonzeros -"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 norm for the space of measurable functions (equipped with an appropriate metric) or for the space of sequences with F–norm .
Candès. et al., proved that for many problems it is probable that the norm is equivalent to the norm, in a technical sense: This equivalence result allows one to solve the problem, which is easier than the problem. Finding the candidate with the smallest 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.
The field of compressive sensing is related to other topics in signal processing and computational mathematics, such as 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 levels is available.
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.
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.
Compressed sensing is being used in facial recognition applications.
Magnetic resonance imaging
Compressed sensing has showed outstanding results in the application of network tomography to network management. Network delay estimation and network congestion detection can both be modeled as underdetermined systems of linear equations where the coefficient matrix is the network routing matrix. Moreover, in the Internet, network routing matrices usually satisfy the criterion for using compressed sensing.
- For most large underdetermined systems of linear equations the minimal 𝓁1-norm solution is also the sparsest solution; See Donoho, David L, Communications on pure and applied mathematics, 59, 797 (2006) http://dx.doi.org/10.1002/cpa.20132
- M. Davenport, "The Fundamentals of Compressive Sensing", SigView, April 12, 2013.
- CS: Compressed Genotyping, DNA Sudoku - Harnessing high throughput sequencing for multiplexed specimen analysis
- Sparse MRI: The application of compressed sensing for rapid MR imaging; See Lustig, Michael and Donoho, David and Pauly, John M, Magnetic resonance in medicine, 58(6), 1182-1195 (2007) http://dx.doi.org/10.1002/mrm.21391
- Compressed Sensing MRI; See Lustig, M.; Donoho, D.L.; Santos, J.M. ; Pauly, J.M., Signal Processing Magazine, IEEE, 25(2),72-82 (2008) http://dx.doi.org/10.1109/MSP.2007.914728
- Compressive sampling makes medical imaging safer
- Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence (2006). "Stable signal recovery from incomplete and inaccurate measurements". Communications on Pure and Applied Mathematics 59 (8): 1207. doi:10.1002/cpa.20124.
- Donoho, D.L. (2006). "Compressed sensing". IEEE Transactions on Information Theory 52 (4): 1289. doi:10.1109/TIT.2006.871582.
- List of L1 regularization ideas from Vivek Goyal, Alyson Fletcher, Sundeep Rangan, The Optimistic Bayesian: Replica Method Analysis of Compressed Sensing
- Hayes, Brian (2009). "The Best Bits". American Scientist 97 (4): 276. doi:10.1511/2009.79.276.
- Tibshirani, Robert. The Lasso page "Regression shrinkage and selection via the lasso". Journal of the Royal Statistical Society, Series B 58 (1): 267–288.
- "Atomic decomposition by basis pursuit", by Scott Shaobing Chen, David L. Donoho, Michael, A. Saunders. SIAM Journal on Scientific Computing
- Candès, Emmanuel J.; Romberg, Justin K.; Tao, Terence (2006). "Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Fourier Information". IEEE Trans. Inf. Theory 52 (8): 489–509. doi:10.1109/tit.2005.862083.
- Candès, E.J., & Wakin, M.B., An Introduction To Compressive Sampling, IEEE Signal Processing Magazine, V.21, March 2008 
- Stefan Rolewicz. Metric Linear Spaces.
- L1-MAGIC is a collection of MATLAB routines
- Compressive Sensing Hardware, http://sites.google.com/site/igorcarron2/compressedsensinghardware
- David Schneider (March 2013). "New Camera Chip Captures Only What It Needs". IEEE Spectrum. Retrieved 2013-03-20.
- "Compressive Imaging: A New Single-Pixel Camera | Rice DSP". Dsp.rice.edu. Retrieved 2013-06-04.
- The Physics arXiv Blog June 3, 2013 (2013-05-25). "Bell Labs Invents Lensless Camera | MIT Technology Review". Technologyreview.com. Retrieved 2013-06-04.
- Gang Huang; Hong Jiang; Kim Matthews; Paul Wilford (2393). "Lensless Imaging by Compressive Sensing". IEEE International Conference on Image Processing, ICIP , Paper # 2013. arXiv:1305.7181.
- David Brady, Kerkil Choi, Daniel Marks, Ryoichi Horisaki, and Sehoon Lim. Compressive holography. Optics Express, 17:13040–13049, 2009
- Rivenson, Y., Stern, A., & Javidi, B. (2010). Compressive fresnel holography. Display Technology, Journal of, 6(10), 506-509.
- Loic Denis, Dirk Lorenz, Eric Thibaut, Corinne Fournier, and Dennis Trede. Inline hologram reconstruction with sparsity constraints. Opt. Lett., 34(22):3475–3477, 2009.
- Marim, M., Angelini, E., Olivo-Marin, J. C., & Atlan, M. (2011). Off-axis compressed holographic microscopy in low-light conditions. Optics Letters, 36(1), 79-81. http://arxiv.org/abs/1101.1735
- Marim, M. M., Atlan, M., Angelini, E., & Olivo-Marin, J. C. (2010). Compressed sensing with off-axis frequency-shifting holography. Optics letters, 35(6), 871-873. http://arxiv.org/abs/1004.5305
- Christy Fernandez Cull, David A. Wikner, Joseph N. Mait, Michael Mattheiss, and David J. Brady. Millimeter-wave compressive holography. Appl. Opt., 49(19):E67–E82, 2010.
- Engineers Test Highly Accurate Face Recognition
- By Jordan EllenbergEmail Author (2010-03-04). "Fill in the Blanks: Using Math to Turn Lo-Res Datasets Into Hi-Res Samples | Wired Magazine". Wired.com. Retrieved 2013-06-04.
- Why Compressed Sensing is NOT a CSI "Enhance" technology ... yet !
- Surely You Must Be Joking Mr. Screenwriter
- [Network tomography via compressed sensing|http://www.ee.washington.edu/research/funlab/Publications/2010/CS-Tomo.pdf]
- "InView web site". http://www.inviewcorp.com/products.
- "The Fundamentals of Compressive Sensing" Part 1, Part 2 and Part 3: video tutorial by Mark Davenport, Georgia Tech. at SigView, the IEEE Signal Processing Society 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
- Intuitive Compressive Sensing
- CSS Lablanche & Company