# RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
## 5 Experimental Results
### MRI Image Recovery  
Significant research effort based on the sparse representation has been directed towards finding ways to accelerate MR imaging. At the heart of these sparse reconstructions assumes MR images can be under-sampled in such a way that data collection time can be dramatically reduced while maintaining image quality. Here, we report experimental results that demonstrate our proposed MRI recovery algorithm on data from the publicly  
Figure 3: Comparisons of CS recovery performance (i.e., the probability of sparse vector recovery versus CS ratio) using sparsifying dictionary \(D_{\textit{KSVD}}\). Red and blue curves were, respectively, obtained using the benchmark (12) and our approach (11). Sparse vectors \(x\) were randomly generated and each point on the curve is the average of 2000 probability measurements. The positions of non-zero coefficients of \(x\) are uniformly distributed and the values of the non-zero coefficients of \(x\) are uniformly distributed in \([-1,1]\). In **(a1,a2)**, sparsity level is 10; in **(b1,b2)**, sparsity level is 12; and in **(c1,c2)**, sparsity level is 14.  
available fastMRI dataset [31]. Our aim is not to develop an algorithm to achieve the state-of-the-art performance for MRI image recovery (baseline deep neural network methods such as variational networks and U-nets [33, 34]). The experiment is designed to demonstrate that using the proposed matrix factorization method can be a promising approach for recovering under-sampled MRI images. In this application, one of the factors in our proposed decomposition is adapted to the acceleration mask (restricted by physically prescribed sensing mechanism), which otherwise does not allow for direct RIP-based recovery guarantees in the \(\ell_{1}\)-synthesis approach to CS. A recent survey of applying the compressed sensing technique to the MRI image recovery [35] presents a variety of techniques to improve the quality of recovered images. In this study, we compared the performance of our method to that based on the TV approach, which promotes sparsity of spacial gradients by approximates a solution to (see [36, 37, 38]):  
\[\underset{Z\in\mathbb{R}^{n_{1}\times n_{2}}}{\text{argmin}}\|Y-\mathcal{R}F_{1 }ZF_{2}\|_{F}^{2}+\lambda\sum_{i,j}\|\
abla Z(i,j)\|_{2}. \tag{13}\]  
Figure 4: Comparisons of CS recovery performance (i.e., the probability of sparse vector recovery versus CS ratio) using sparsifying dictionary \(D_{n wavelet}\) (CDF 9/7). Red and blue curves were, respectively, obtained using the benchmark (12) and our approach (11). Sparse vectors \(x\) were randomly generated and each point on the curve is the average of 2000 probability measurements. The positions of non-zero coefficients of \(x\) are uniformly distributed and the values of the non-zero coefficients of \(x\) are uniformly distributed in \([-1,1]\). In **(a1,a2)**, sparsity level is 10; in **(b1,b2)**, sparsity level is 12; and in **(c1,c2)**, sparsity level is 14. Under-sampling masks (restrained by the instrumental conditions) for 4-fold and 8-fold acceleration were applied to full _k_-space data according to (5). The masks sample 25%, respectively 12.5%, of horizontal scan lines including a fully sampled center region--corresponding to low spatial frequencies--containing 8%, respectively 4%, of scan lines, as shown in Figure 5. We selected CDF 9/7 wavelets with three levels as the sparsifying dictionaries. The lowpass wavelet band and the central areas of the acceleration masks may not be perfectly aligned, leading to potential artifacts in the reconstruction as a result of cross-band interference between the lowpass and highpass wavelet bands. In the provided images Figures 6-9, we can see a visual comparison between the reconstruction outcomes generated by our algorithm and the total-variation (TV) method. These comparisons are based on 4-fold and 8-fold under-sampled Fourier measurements of knee and brain scans from the fastMRI dataset.