## 4 Results
### Gray Matter Brain Phantom
The MaxEnt reconstruction has lower peak amplitudes than the GS reconstructions and possibly TV, but its low noise floor means that low amplitude peaks are not obscured, which was confirmed by adjusting the contour levels down. The peak amplitudes and line shapes of the GS reconstructions are the most similar to the fully sampled dataset from all of the reconstructions, as illustrated by the low amplitude NAA and Glx lower and upper cross peaks (LCP, UCP), whose complex multiplet structures have been properly reconstructed with reasonable line shapes and widths. 4D EP-COSI datasets are self-sparse along the \(Y-F_{1}\) plane [16]. There are large regions between the metabolite peaks that are dominated by noise, therefore Root Mean Square Errors (RMSE) calculated over the entire dataset can be heavily biased by changes to the noise distribution from non-linear reconstruction methods like CS, TV, MaxEnt, and GS. Previous work has demonstrated that constraining the RMSE calculations to the metabolite peaks and ignoring the noise regions is a suitable quantitative metric of reconstruction accuracy that coincides with qualitative observations because it measures amplitude, line width, and line shape simultaneously with minimal influence from the noise distribution [17]. Therefore, the mean metabolite peak reconstruction RMSEs with respect to the fully sampled data of six EP-COSI gray matter brain phantom scans _versus_ NUS rate are shown in Figure 3 along with the 2\(\times\)-10\(\times\)NUS masks used to under-sample the \(k_{y}-t_{1}\) planes of the phantom and _in vivo_ datasets used in these experiments. The RMSEs were calculated for each metabolite peak using the ppm ranges shown in Table 2. The means were calculated from the central 4 \(\times\) 4 VOI of the six phantoms scans used, yielding 96 voxel measurements per metabolite peak. The standard deviations are not shown because they were 2-3 orders of magnitude lower in value than the means. As can be seen in Figure 3, the mean reconstruction RMSEs for the large diagonal peaks, NAA 2.0 ppm, Cr 3.0 ppm, and Cr 3.9 ppm, are shown at the top of the figure, while smaller diagonal peaks and cross peaks are shown in the middle and bottom, respectively. For the large diagonals, all of the reconstruction RMSEs follow similar trends at each NUS rate, which is in agreement with the qualitative results from Figure 2 that indicated each of the reconstruction methods was able to accurately reconstruct the diagonal peaks. However, for the smaller diagonals and cross peaks, the general trend at lower NUS rates is for CS to have the highest RMSE, followed by TV, MaxEnt, GS\({}_{1}\), and GS\({}_{2}\), but at higher NUS rates, MaxEnt has RMSEs equal to or lower than GS\({}_{1}\). The RMSEs of GS\({}_{1}\) for Cho and mI do not follow the general trends of the large diagonal peak RMSEs, nor those of the smaller diagonals and cross peaks; the Cho RMSEs for GS\({}_{1}\) are higher than all other methods for each NUS rate and one of the highest for mI. However, the mean RMSEs for GS\({}_{1}\) of the other metabolites are some of the lowest of the methods tested. These results demonstrate that GS reconstruction has the lowest RMSEs for each metabolite peak and NUS rate, with either GS\({}_{2}\) or GS\({}_{1}\) providing the lowest mean RMSE. The mean RMSEs for GS\({}_{2}\) are consistently lower than GS\({}_{1}\), indicating that overlapping groups may be a more reliable method of metabolite peak reconstruction than non-overlapping groups.