Stable Image Reconstruction Using Total Variation Minimization

This article presents near-optimal guarantees for accurate and robust image recovery from under-sampled noisy measurements using total variation minimization, and our results may be the first of this kind. In particular, we show that from O(s log(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient, up to a logarithmic factor. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of a suitably incoherent matrix.