  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 to signals of arbitrary dimension d>1 and to isotropic
total variation problems. To be precise, we show that a multidimensional signal
x 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.
