Tldr: authors of paper claim their digital signal processing method is superior to those methods imposed by sampling theory; in practice, their method works only on examples where there is high level of a priori knowledge of the signal, i.e., not in any practical environment.
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.
Most of the research in compressed sensing type applications now is focused on more sophisticated prior distributions than sparse/Laplacian. These more general bayesian approaches provide significant performance improvements over LASSO etc.
