  Practical applications of compressed sensing often restrict the choice of its
two main ingredients. They may (i) prescribe using particular redundant
dictionaries for certain classes of signals to become sparsely represented, or
(ii) dictate specific measurement mechanisms which exploit certain physical
principles. On the problem of RIP measurement matrix design in compressed
sensing with redundant dictionaries, we give a simple construction to derive
sensing matrices whose compositions with a prescribed dictionary have a high
probability of the RIP in the $k \log(n/k)$ regime. Our construction thus
provides recovery guarantees usually only attainable for sensing matrices from
random ensembles with sparsifying orthonormal bases. Moreover, we use the
dictionary factorization idea that our construction rests on in the application
of magnetic resonance imaging, in which also the sensing matrix is prescribed
by quantum mechanical principles. We propose a recovery algorithm based on
transforming the acquired measurements such that the compressed sensing theory
for RIP embeddings can be utilized to recover wavelet coefficients of the
target image, and show its performance on examples from the fastMRI dataset.
