Near-optimal compressed sensing guarantees for total variation minimization

TL;DR

This paper extends compressed sensing guarantees for total variation minimization to multidimensional signals, showing near-optimal measurement bounds for accurate reconstruction of natural images and movies.

Contribution

It generalizes recent 2D image guarantees to arbitrary dimensions and isotropic total variation, establishing near-optimal measurement bounds for signal recovery.

Findings

Reconstruction from O(sd*log(N^d)) measurements is possible.

Guarantees are optimal up to polynomial factors in dimension d.

Total variation minimization effectively reconstructs multidimensional signals.

Abstract

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.