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

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.