Near-optimal compressed sensing guarantees for anisotropic and isotropic total variation minimization

Consider the problem of reconstructing a multidimensional signal from partial information, 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. Recently, guarantees for two-dimensional images x ∈ CN2 were established. This paper extends these theoretical results to signals x ∈ CNd of arbitrary dimension d ≥ 2 and to both the anisotropic and isotropic total variation problems. To be precise, we show that a multidimensional signal x ∈ CNd can be reconstructed from O(sd log(N)) linear measurements y = Ax using total variation minimization to within 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 and a logarithmic factor in the signal dimension N. The proof relies on bounds in approximation theory concerning the compressibility of wavelet expansions of bounded-variation functions.