# RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
## 4 Application to Accelerated MRI Imaging  
The goal of fast MRI is the recovery of an image \(Z\in\mathbb{R}^{n_{1}\times n_{2}}\) from under-sampled Fourier (\(k\)-space) measurements. The discrete Fourier transform of \(Z\) is the complex matrix:  
\[\widehat{Z}:=F_{1}ZF_{2}\in\mathbb{C}^{n_{1}\times n_{2}},\]  
where \(F_{1}\in\mathbb{C}^{n_{1}\times n_{1}}\), \(F_{2}\in\mathbb{C}^{n_{2}\times n_{2}}\) are (symmetric and unitary) discrete Fourier transform matrices. Data acquired via accelerated MRI can be modeled as:  
\[Y:=\mathcal{R}\widehat{Z}\in\mathbb{C}^{n_{1}\times n_{2}}, \tag{5}\]  
where \(\mathcal{R}\) is a \(\{0,1\}\)-entry \(n_{1}\times n_{1}\) diagonal matrix that selects rows of \(Y\) corresponding to some under-sampling scheme [31]. The image can be assumed to have a sparse representation:  
\[Z=D_{1}XD_{2}^{\top} \tag{6}\]  
in the transform domain of wavelet dictionaries \(D_{i}\in\mathbb{R}^{n_{i}\times N_{i}}\) (\(N_{i}\geq n_{i}\), \(i=1,2\)) that implement a separable bivariate wavelet transform. The coefficient matrix \(X\in\mathbb{R}^{N_{1}\times N_{2}}\) then consists of low-pass channel approximation coefficients, and sparse high-pass channel coefficients that capture edges and singular points in the target image. In this formulation, the CS-MRI problem is then the inverse problem of approximating sparse coefficients \(X\) that solve:  
\[Y=\mathcal{R}F_{1}D_{1}XD_{2}^{\top}F_{2}, \tag{7}\]  
that is, to reconstruct \(Z\) (via (6)) from the measurements \(Y\) which are in general corrupted by noise.  
Being tied to the measurements (7) by physical principles, we first transform the acquired data. Starting from a sensing factorization \(D_{2}=G_{2}AH_{2}\) according to Proposition 1, we consider the transformed observation:  
\[\widetilde{Y}:=YF_{2}^{*}(G_{2}^{\top})^{-1}=\mathcal{R}F_{1}D_{1}X(A_{2}H_{2} )^{\top}.\]We next construct for \(D_{1}\) two (real) factorizations that are adapted to the real and imaginary parts of \(F_{1}\). First, we let:  
\[G_{R}:=\operatorname*{argmin}_{G\in\mathbb{R}^{n_{1}\times n_{1}} \text{ invertible}}\|\mathcal{R}(\operatorname{Re}(F_{1})G-I)\|_{F}^{2} \tag{8}\]  
and, second, we let:  
\[H_{R}:=\operatorname*{argmin}_{H\in\mathbb{R}^{n_{2}\times n_{2}} \text{ unitary}}\|D_{1}-G_{R}AH\|_{F}^{2}. \tag{9}\]  
Analogously, we define \(G_{I}\) and \(H_{I}\) via solving (8) for the imaginary part of \(F_{1}\), before solving (9) for \(G_{I}\). We then let \(H:=H_{R}+iH_{I}\). For details on numerical solutions of (8) and (9) via the augmented Lagrangian approach we refer to the Appendix A.  
Our motivation for desiring in (8) small differences between \(\mathcal{R}\) and the real (respectively imaginary) parts of \(\mathcal{R}F_{1}G_{R}\) (respectively \(\mathcal{R}F_{1}G_{I}\)) is that then (9) implies that the following difference is small:  
\[\Phi(X):=\tfrac{1}{2}\Big{\|}\tilde{Y}-\mathcal{R}AHX(AH_{2})^{ \top}\Big{\|}_{F}^{2},\]  
such that we can attempt to approximate sparse coefficients \(X\) by solving the \(\ell_{1}\)-regularized problem:  
\[\operatorname*{minimize}_{X\in\mathbb{R}^{N_{1}\times N_{2}}} \Phi(X)+\gamma\|X^{H}\|_{1}, \tag{10}\]  
where \(X^{H}\) denotes the highpass coefficients of \(X\) and \(\gamma>0\).