Total variation minimization for stable multidimensional signal recovery

Consider the problem of reconstructing a multidimensional signal from partial information. 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. While reconstruction guarantees and optimal measurement designs have been established for related L1-minimization problems, the theory for total variation minimization has remained elusive until recently, when guarantees for two-dimensional images were established. This paper extends the recent theoretical results to signals of arbitrary dimension d>1. To be precise, we show that a multidimensional signal can be reconstructed from O(sd log(N^d)) linear measurements 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^d. The proof relies on bounds in approximation theory concerning the compressibility of wavelet expansions of bounded-variation functions.