sparse representation for image denoising When I read papers on image denoising, I always encounter sparse representation. For image denoising, we try to separate image signal from noise. It is assumed that signal is correlated and noise is uncorrelated. Sparse representation represents one signal as a linear combination of a small number of dictionary elements. It hope to use as few as non-zero coefficients to represent signal. In my view, it is appropriate for image compression. How can it is related to image denoising?
Here are two interpretations of why sparse representations are useful for denoising:

1) Your clean image is well represented by your dictionary (usually a wavelet or 2D-DCT dictionary), but the noise isn't. This means that when calculating your sparse representation from the noisy signal, you will only capture the clean signal, and not the noise.

2) Enforcing the representation vector to be sparse (i.e. with only a few non-zero elements), means that you will eventually threshold out all the other low-energy noisy components.

As you said sparse representations can be pretty useful for compression. Compression is merely about keeping the relevant information, while discarding the irrelevant one. Well denoising is about keeping the relevant part of your signal (the one that fits your model, or your dictionary), while discarding the irrelevant one (the noise).