# Compressed sensing

**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 is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Shannon-Nyquist sampling theorem. There are two conditions under which recovery is possible.^{[1]} The first one is sparsity which requires the signal to be sparse in some domain. The second one is incoherence which is applied through the isometric property which is sufficient for sparse signals.^{[2]}^{[3]} MRI is a prominent application.^{[4]}^{[5]}^{[6]}^{[7]}

## References

- ↑ CS: Compressed Genotyping, DNA Sudoku - Harnessing high throughput sequencing for multiplexed specimen analysis
- ↑ 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.
- ↑ 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
- ↑ Novel Sampling Strategies for Sparse MR Image Reconstruction; See Wang, Q.; Zenge M., Cetingul H.E., Mueller E., Nadar M.S., International Symposium of Magnetic Resonance in Medicine (2014) http://www.healthcare.siemens.com/siemens_hwem-hwem_ssxa_websites-context-root/wcm/idc/siemens_hwem-hwem_ssxa_websites-context-root/wcm/idc/groups/public/@global/@imaging/@mri/documents/download/MRI-ISMRM-2014-ABSTRACT-WANG-2-01412769.pdf

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