On the role of total variation in compressed sensing

This paper considers the problem of recovering a one or two dimensional discrete signal which is approximately sparse in its discrete gradient from an incomplete subset of its discrete Fourier coefficients which have been corrupted with noise. We prove that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order $s$ with high probability, it suffices to draw $\mathcal{O}(s \log N)$ of the available Fourier coefficients uniformly at random. However, we also show that if one draws $\mathcal{O}(s \log N)$ samples in accordance to a particular distribution which concentrates on the low Fourier frequencies, then the stability bounds which can be guaranteed are optimal up to $\log$ factors. Finally, we prove that in the one dimensional case where the underlying signal is gradient sparse and its sparsity pattern satisfies a minimum separation condition, then to guarantee exact recovery with high probability, for some $M<N$, it suffices to draw $\mathcal{O}(s\log M\log s)$ samples uniformly at random from the Fourier coefficients whose frequencies are no greater than $M$.