Noiselet

Noiselets are a family of functions which are related to wavelets, analogously to the way that the Fourier basis is related to a time-domain signal. In other words, if a signal is compact in the wavelet domain, then it will be spread out in the noiselet domain, and conversely.[1]

Applications

The complementarity of wavelets and noiselets means that noiselets can be used in compressed sensing to reconstruct a signal (such as an image) which has a compact representation in wavelets.[2] MRI data can be acquired in noiselet domain, and, subsequently, images can be reconstructed from undersampled data using compressive-sensing reconstruction.[3]

References

R. Coifman, F. Geshwind, and Y. Meyer, Noiselets, Applied and Computational Harmonic Analysis, 10 (2001), pp. 27–44. doi:10.1006/acha.2000.0313.
E. Candes and J. Romberg, Sparsity and incoherence in compressive sampling, 23 (2007), pp. 969–985. doi:10.1088/0266-5611/23/3/008.
K. Pawar, G. Egan, and Z. Zhang, Multichannel Compressive Sensing MRI Using Noiselet Encoding, 05 (2015), doi:10.1371/journal.pone.0126386.

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[coifman01-1] R. Coifman, F. Geshwind, and Y. Meyer, Noiselets, Applied and Computational Harmonic Analysis, 10 (2001), pp. 27–44. doi:10.1006/acha.2000.0313.

[candes07-2] E. Candes and J. Romberg, Sparsity and incoherence in compressive sampling, 23 (2007), pp. 969–985. doi:10.1088/0266-5611/23/3/008.

[pawar01-3] K. Pawar, G. Egan, and Z. Zhang, Multichannel Compressive Sensing MRI Using Noiselet Encoding, 05 (2015), doi:10.1371/journal.pone.0126386.