Close, but not quite: compressed sensing really works in a lot of domains. You "just" have to know of a basis set in which your signal is sparse. This basis set can be (and often is) overcomplete; i.e. you don't need a minimal orthogonal set of basis functions. So if you know that "most pixels of an MRI are black" or "music only has a few non-zero Fourier frequencies", you can apply compressed sensing techniques to recover/reconstruct the underlying signal. This is a fairly mild and merely structural form of a priori side information, as opposed to having to know detailed precise prior distributions or etc.

See https://en.wikipedia.org/wiki/Sparse_dictionary_learning and https://arxiv.org/abs/1012.0621 for the gory details.