Document Type

Technical Report

Department

Mathematics (CMC)

Publication Date

2013

Abstract

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 were established. This paper extends these theoretical results to signals of arbitrary dimension and to both the anisotropic and isotropic total variation problems. To be precise, we show that a multidimensional signal can be reconstructed from a small number of linear measurements using total variation minimization to within a factor of the best approximation of its gradient. The reconstruction guarantees we provide are necessarily optimal up to polynomial factors in the spatial dimension and a logarithmic factor in the signal dimension. The proof relies on bounds in approximation theory concerning the compressibility of wavelet expansions of bounded-variation functions.

Comments

Previous versions of this paper can be found at arxiv.org/abs/1210.3098

Rights Information

© 2014 Deanna Needell and Rachel Ward

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.