On the role of total variation in compressed sensing
Provides theoretical guarantees for compressed sensing of gradient-sparse signals (e.g., total variation minimization) with Fourier measurements, addressing a known bottleneck in sampling strategies.
The paper proves that for recovering gradient-sparse signals from noisy Fourier measurements, drawing O(s log N) random Fourier coefficients suffices for robust reconstruction, and that a low-frequency sampling distribution yields optimal stability bounds up to log factors. In 1D with a minimum separation condition, O(s log M log s) samples from low frequencies guarantee exact recovery.
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 $M$.