# RIP Sensing Matrices Construction for Sparsifying Dictionaries with Application to MRI Imaging
## 5 Experimental Results
### MRI Image Recovery
Parameters were chosen for best performance from a set of values as \(\rho=0.02\), \(\
u=0.00016\), \(\mu=0.00024\), as well as \(\lambda=5\times 10^{-5}\) in (13), and \(\gamma=0.0035\) in (10). MRI data: [[https://fastmi.med.nyu.edu/](https://fastmi.med.nyu.edu/)]([https://fastmi.med.nyu.edu/](https://fastmi.med.nyu.edu/)) (accessed date: 1 July 2022). The parameters were established through empirical testing and adjustment (to obtain the source codes for our algorithm, can contact [EMAIL_ADDRESS] or [EMAIL_ADDRESS]). Images (displayed with 90 degree rotation and cut to quadratic center region) taken from coronal proton density knee dataset (image size 372 x 640, _z_-slice 20 of file1000031 (knee _A_) and file1000071 (knee _B_)), and brain dataset (image size 320 x 640, _z_-slice 2 of file_brain_AXT1_201_6002786 (brain _A_) and file_brain_AXT1_201_6002740 (brain _B_)). The parameters were established through empirical testing and adjustment. Table 1 summarizes a comparison of PSNR and SSIM values for the knee and brain images depicted in the figures. Our average PSNR gains for 4-fold under-sampled data are 0.5 dB on the knee and 0.9 dB on the brain images, while our average gains for the 8-fold under-sampled images are about 2 dB. Our SSIM values are superior to that of the TV-method in all experiments. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Proposed**} & \multicolumn{3}{c}{**TV Method**} \\ \cline{2-9}  & \multicolumn{2}{c}{**PSNR**} & \multicolumn{2}{c}{**SSIM**} & \multicolumn{2}{c}{**PSNR**} & \multicolumn{2}{c}{**SSIM**} \\ \cline{2-9}  & **4x** & **8x** & **4x** & **8x** & **4x** & **8x** & **4x** & **8x** \\ \hline Knee \(A\) & 28.5 & 26.4 & 0.657 & 0.570 & 28.2 & 24.5 & 0.615 & 0.538 \\ Knee \(B\) & 28.4 & 26.1 & 0.635 & 0.535 & 27.8 & 24.1 & 0.625 & 0.541 \\ Brain \(A\) & 27.7 & 24.1 & 0.642 & 0.524 & 26.7 & 22.3 & 0.637 & 0.493 \\ Brain \(B\) & 27.7 & 24.6 & 0.668 & 0.494 & 26.8 & 22.7 & 0.644 & 0.474 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison for images in Figures 6–9, for 4-fold and 8-fold acceleration. Figure 5: Scan line locations for 4-fold (**left**) and 8-fold (**right**) under-sampling, including fully sampled low-frequency region. Figure 6: (**a0**) FFT reconstruction of fully sampled MRI \(k\)-space data (file 1000031.h5: the 20 th in a set of 40 datasets, each comprising 372 \(\times\) 640 data points, representing coronal proton density (PD) knee images). (**a1**) Image of the proposed algorithm with 4-fold sampling. (**a2**) Image of the proposed algorithm with 8-fold sampling. (**b1**) Image of the TV algorithm with 4-fold sampling. (**b2**) Image of the TV algorithm with 8-fold sampling. The vertical band patterns observed in the images are artifacts resulting from under-sampling. The artifact in (**a2**) is less noticeable compared to (**b2**). Figure 7: **(a0)** FFT reconstruction of fully sampled MRI \(k\)-space data (file 1000071.h5: the 20 th in a set of 35 datasets, each comprising 372 \(\times\) 640 data points, representing coronal proton density (PD) knee images). **(a1)** Image of the proposed algorithm with 4-fold sampling. **(a2)** Image of the proposed algorithm with 8-fold sampling. **(b1)** Image of the TV algorithm with 4-fold sampling. **(b2)** Image of the TV algorithm with 8-fold sampling. Figure 8: **(a0)** The fully sampled MRI \(k\)-space data (file_brain_AXT1_201_6002786.h5) involves the second set of 16 brain datasets, each containing \(320\times 640\) data points. **(a1)** Image displaying the proposed algorithm with 4-fold sampling. **(a2)** Image displaying the proposed algorithm with 8-fold sampling. **(b1)** Image showing the TV algorithm with 4-fold sampling. **(b2)** Image showing the TV algorithm with 8-fold sampling. The structure is more discernible in **(a2)** and **(a1)** than in **(b2)** and **(b1)**, respectively.