Near-Optimal Compressed Sensing Guarantees for Total Variation Minimization

Consider the problem of reconstructing a multidimensional signal from an underdetermined set of measurements, as in the setting of compressed sensing. Without any additional assumptions, this problem is ill-posed. However, for signals such as natural images or movies, the minimal total variation estimate consistent with the measurements often produces a good approximation to the underlying signal, even if the number of measurements is far smaller than the ambient dimensionality. This paper extends recent reconstruction guarantees for two-dimensional images x ∈ ℂ^N2 to signals x ∈ ℂ^Nd of arbitrary dimension d ≥ 2 and to isotropic total variation problems. In this paper, we show that a multidimensional signal x ∈ ℂ^Nd can be reconstructed from O(s dlog(N^d)) linear measurements y = Ax using total variation minimization to a factor of the best s-term approximation of its gradient. The reconstruction guarantees we provide are necessarily optimal up to polynomial factors in the spatial dimension d.