Structure dependent sampling in compressed sensing: theoretical guarantees for tight frames

TL;DR

This paper extends compressed sensing theory to tight frames, showing how level-dependent sampling can be optimized based on local coherence and sparsity structures, improving sampling efficiency.

Contribution

It generalizes previous results to tight frames and provides a theoretical framework for level-dependent sampling based on local coherence and sparsity.

Findings

Derived sampling guarantees for tight frames

Established relation between local coherence and sampling levels

Extended variable density sampling theory

Abstract

Many of the applications of compressed sensing have been based on variable density sampling, where certain sections of the sampling coefficients are sampled more densely. Furthermore, it has been observed that these sampling schemes are dependent not only on sparsity but also on the sparsity structure of the underlying signal. This paper extends the result of (Adcock, Hansen, Poon and Roman, arXiv:1302.0561, 2013) to the case where the sparsifying system forms a tight frame. By dividing the sampling coefficients into levels, our main result will describe how the amount of subsampling in each level is determined by the local coherences between the sampling and sparsifying operators and the localized level sparsities -- the sparsity in each level under the sparsifying operator.