## 5 Discussion  
In this article, an algorithm to solve the GS reconstruction problem using the Split Bregman framework was derived that allows for arbitrary grouping patterns of transform coefficients and was applied to reconstructing retrospectively under-sampled 4D EP-COSI data. The results for GS reconstruction using overlapping and non-overlapping, equal sized groups were compared against the results from CS, TV, and MaxEnt reconstructions at NUS rates of 2\(\times\), 4\(\times\), 6\(\times\), 8\(\times\), and 10\(\times\) on gray matter phantom and _in vivo_ brain data. Figures 2, 4, and 5 show that spectra reconstructed by GS reconstruction using overlapping coefficient groups (GS\({}_{2}\)) are the most qualitatively similar to the fully sampled spectra with respect to the noise floor, metabolite peak amplitudes, line widths, and spatial localization. The metabolite peak RMSEs calculated over the VOIs of the phantom and _in vivo_ datasets in Figure 3 and Table 3 show that the GS\({}_{2}\) reconstructed spectra also have the lowest values for the majority of peaks evaluated, indicating that they are more accurately reconstructed than the other tested methods. The agreement between these results indicate that GS\({}_{2}\) is the most effective means of reconstructing NUS 4D EP-COSI phantom and _in vivo_ data tested. The higher noise floor and dynamic range from water and fat contamination in the _in vivo_ scans did not prevent GS\({}_{2}\) from successfully reconstructing the highly under-sampled data. Compared with the other non-GS reconstruction methods, GS\({}_{1}\) produced high quality spectra and was able to reconstruct the metabolite peaks to a higher degree of fidelity; however, as indicated by the mean metabolite peak RMSEs in Figure 3 and Table 3, the Cho 3.2 ppm diagonal peak was not reconstructed by GS\({}_{1}\) as accurately as it was by the other reconstruction methods. This discrepancy was caused by the nature of the grouping method used for GS\({}_{1}\), which used equally sized non-overlapping groups of \((F_{2},F_{1})=(8,4)\) coefficients. The Cho 3.2 ppm diagonal peak was located at the intersection of multiple coefficient groups; therefore, the structural block of coefficients that comprised the peak were divided into multiple groups and were not correlated with one another. Instead of increasing the sparsity of the coefficients around the peak and reducing it at the peak, this division had the opposite effect and prevented the Cho 3.2 ppm peak from being properly reconstructed. In contrast, GS\({}_{2}\) was able to reconstruct the Cho 3.2 ppm diagonal peak to a higher degree of fidelity with the same sized  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**CS**} & \multicolumn{2}{c}{**TV**} & \multicolumn{3}{c}{**MaxEnt**} & \multicolumn{3}{c}{**GS\({}_{1}\)**} & \multicolumn{3}{c}{**GS\({}_{2}\)**} \\ \cline{2-13}  & **4\(\times\)** & **6\(\times\)** & **8\(\times\)**coefficient groupings because of their overlap and the increased coverage that it provided. It is clear from this result that the coefficient groupings correlate the reconstructed values of adjacent transform coefficients, but overlapping groups correlate the reconstructed values of adjacent groups and minimize the effects of grouping patterns that do not effectively take advantage of structural sparsity. These results agree with previous work that showed MaxEnt reconstruction outperforming CS reconstruction of NUS 4D EP-COSI simulated and _in vivo_ breast data [17]; however, other work reconstructing NUS 4D EP-JRESI phantom and _in vivo_ data showed TV results at a higher data fidelity than was shown here. The 4D EP-JRESI reconstructions of phantom and _in vivo_ data were completed as part of this work, but were excluded for brevity. In the experiments not shown, TV reconstruction performed equally well or better than some of the other reconstruction methods. The main difference between these experiments is the phase characteristics of 4D EP-COSI _versus_ 4D EP-JRESI data.