# RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
## 1 Introduction
In CS MRI, sensing mechanisms exploit quantum mechanical principles and existing hardware constraints restrict sensing matrix design beyond non-uniformly subsampling the measured (complex-valued) Fourier spatial frequency coefficients of the target image \(\widetilde{x}=Dx\), which we suppose is synthesized from (real-valued) sparse coefficients \(x\) with respect to a dictionary \(D\). The sensing matrix \(S=\mathcal{R}F\) is thus modeled as the product of a discrete Fourier transform \(F\) and a row subsampling \(\mathcal{R}\). The CS recovery of \(x\) can then be formulated as the \(\ell_{1}\)-_synthesis_ problem:  
\[\underset{x}{\text{minimize}}\,\left\|x\right\|_{1}\quad\text{ subject to }\quad\left\|y-\mathcal{R}FDx\right\|_{2}\leq\epsilon \tag{3}\]  
in which \(\mathcal{R}F\) possesses the RIP [9, 10, 11], while, in general, \(\mathcal{R}FD\) does not, such that the RIP recovery guarantees do not directly apply. Using the dictionary factorization idea, we attempt to neutralize the effect of \(F\) in order to obtain an RIP synthesis problem. We do so by choosing (real) factors \(G\) such that (both real and imaginary parts of) \(\mathcal{R}FG\) optimally match \(\mathcal{R}\), so that the factorizations \(GAH\) that optimally approximate \(D\) allow, by virtue of \(\mathcal{R}FD\sim\mathcal{R}FGAH\sim\mathcal{R}AH\), to replace the constraint in (3) by \(\left\|y-\mathcal{R}AHx\right\|_{2}\leq\epsilon\). Both the real and the imaginary parts of \(\mathcal{R}AH\) are incoherent sampling matrices for sparse signals that possess the RIP. We provide a review of CS application in Section 2 and detail the application of CS factorization method to MRI in Section 4, and report in Section 5 numerical experiments comparing our approach to total-variation based CS MRI [12]. Related workIf the sparsifying dictionary in the general \(\ell_{1}\)-synthesis approach is a tight frame, i.e., \(x=D^{\top}\widetilde{x}\), recovery guarantees can be derived by considering the \(\ell_{1}\)-analysis approach of minimizing over \(\widetilde{x}\) the objective \(\left\|D^{\top}\widetilde{x}\right\|_{1}\) subject to the constraint \(\left\|y-S\widetilde{x}\right\|_{2}\leq\epsilon\). Recovery guarantees for the \(\ell_{1}\)-analysis approach are connected to the notion of D-RIP [13], which requires the sensing matrix to satisfy the RIP inequality for all images of \(k\)-sparse vectors under a tight frame \(D\). Any RIP matrix satisfies the D-RIP when multiplied by a random sign matrix [14]. Unless \(D\) is orthonormal, the geometric structures, properties and empirical performances of the \(\ell_{1}\)-analysis and \(\ell_{1}\)-synthesis approaches in general differ [15]. A weaker condition than the RIP that can facilitate recovery guarantees for \(SD\) with general dictionary \(D\) is the mutual incoherence between the sensing matrix and the dictionary [2; 16]. However, sparsity ranges for signals to be recovered, as well as necessary number of measurements are more restrictive than for RIP guarantees, and overcomplete dictionaries with highly coherent columns in general lead to large coherence of their product with sensing matrices. Finally, the nullspace property of a sensing matrix gives a necessary and sufficient condition for stable recovery of sparse signals using the convex optimization [17].