  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.
