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 vice versa.
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.
References
Noiselet Wikipedia(Text) CC BY-SA