## 4 Results  
### Gray Matter Brain Phantom  
Figure 2 shows a select 2D COSY spectrum from the central voxel of a 4D EP-COSI gray matter phantom scan that was retrospectively under-sampled 8\(\times\) using the phantom mask in Figure 3 and reconstructed by CS, TV, MaxEnt, GS\({}_{1}\), and GS\({}_{2}\). Because of the homogeneity of the phantom spectra, a single representative 2D spectrum was chosen to illustrate the qualitative effectiveness of each reconstruction method. The 1D projections of the \(F_{1}\) and \(F_{2}\) dimensions are shown to the right and above each 2D spectrum, respectively. The spectrum from the fully sampled phantom scan is shown in the top left of the figure and the 8\(\times\) NUS spectrum with zeros in place of missing samples is shown to its right. The contour levels of each spectrum are the same as those used for the fully sampled scan to allow noise levels and peak heights to be compared between reconstructions. \begin{table}
\begin{tabular}{l c c c} \hline  & \(\mu\) & \(\lambda\) & \(\gamma\) \\ \hline
**CS** & 1 & \(\
icefrac{{1}}{{2}}\) & N/A \\
**TV** & 1 & \(\
icefrac{{1}}{{50}}\) & \(\
icefrac{{1}}{{100}}\) \\
**GS\({}_{1}\)** & 1 & \(\
icefrac{{1}}{{(2\cdot 32)}}\) & N/A \\
**GS\({}_{2}\)** & 1 & \(\
icefrac{{1}}{{(2\cdot 32)}}\) & N/A \\
**MaxEnt** & N/A & N/A & N/A \\ \hline \end{tabular}
\end{table} 
Table 1: Lagrange multiplier values used during reconstruction. The GS reconstructions use the CS parameter values scaled by the number of coefficients per group. The MaxEnt reconstruction does not use Lagrange multipliers. Figure 1: Illustration of the coefficient grouping support of a single data point for equal sized non-overlapping and 50% overlapping groups. As can be seen in the 8\(\times\) NUS spectrum, the broad point spread function (PSF) of the under-sampling mask caused the large diagonal peaks of NAA, Cr, Cho, and ml to alias along the \(F_{1}\) dimension and obscure the much smaller NAA, Glc, mI, and Asp cross-peaks, which have also been spread along \(F_{1}\) by convolution with the PSF. Because the \(t_{2}\) dimension was fully sampled, the \(F_{2}\) projections of the fully sampled, 8\(\times\) NUS, and reconstructed spectra have similar peak amplitudes and line widths; however, the \(t_{1}\) under-sampling caused the spectral peaks shown in the fully sampled \(F_{1}\) projection to collapse in  
Figure 3: (Top Left) NUS masks used to under-sample the \(k_{y}-t_{1}\) plane of 6 brain phantom scans 2\(\times\), 4\(\times\), 6\(\times\), 8\(\times\), and 10\(\times\) (**a–j**) Mean metabolite peak RMSEs of under-sampled 4D EP-COSI brain phantom scans with respect to the fully sampled data _versus_ NUS rate for TV, MaxEnt, CS, GS\({}_{1}\), and GS\({}_{2}\) reconstructions. Figure 2: Select 2D COSY spectrum from the central voxel of a (**a**) Fourier transformed, fully sampled 4D EP-COSI brain phantom scan, (**b**) Fourier transformed 8\(\times\) NUS, and the (**c**) 8\(\times\) TV, (**d**) 8\(\times\) MaxEnt, (**e**) 8\(\times\) CS, (**f**) 8\(\times\) GS\({}_{1}\), (**g**) 8\(\times\) GS\({}_{2}\) reconstructions. the 8\(\times\) NUS projection. The \(F_{1}\) projections of each reconstruction method show reasonable agreement with the fully sampled dataset, indicating the peak energy of major diagonals, such as NAA, Cr, Cho, and mI, has been deconvolved from the broad PSF by each of the reconstructions. There are noticeable differences among the reconstructions that should be highlighted. The noise floors of the CS and TV reconstructions are higher than all of the other spectra, which makes identifying cross peaks difficult. The CS and TV reconstructed diagonal peaks have lost their Lorentzian line shape and the cross peaks are present but \"choppy\" in appearance, further indicating issues with proper line shape reconstruction. The MaxEnt and GS reconstructions have noise floors equal to or lower than the fully sampled dataset. Cross peaks and diagonals are not \"choppy\" in appearance, and most of the cross peaks shown in the fully sampled dataset can be identified.