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. In particular, we show that from O(slog(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of suitably incoherent matrices.Comment: 25 page

RSP-Based Analysis for Sparsest and Least ℓ1\ell_1-Norm Solutions to Underdetermined Linear Systems

Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does $\ell_1$-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necessary and sufficient condition for the uniqueness of least $\ell_1$-norm solutions to linear systems. From this condition, we deduce that a sparsest solution coincides with the unique least $\ell_1$-norm solution to a linear system if and only if the so-called \emph{range space property} (RSP) holds at this solution. This yields a broad understanding of the relationship between $\ell_0$- and $\ell_1$-minimization problems. Our analysis indicates that the RSP truly lies at the heart of the relationship between these two problems. Through RSP-based analysis, several important questions in this field can be largely addressed. For instance, how to efficiently interpret the gap between the current theory and the actual numerical performance of $\ell_1$-minimization by a deterministic analysis, and if a linear system has multiple sparsest solutions, when does $\ell_1$-minimization guarantee to find one of them? Moreover, new matrix properties (such as the \emph{RSP of order $K$} and the \emph{Weak-RSP of order $K$}) are introduced in this paper, and a new theory for sparse signal recovery based on the RSP of order $K$ is established