## 2 Theory
### Split Bregman Based Group Sparse Reconstruction  
The GS reconstruction problem is generally considered difficult to solve due to its mixed norm structure and the non-smooth nature of the \(l_{2,1}\) objective function. However, by applying Split Bregman iterative relaxation, we can solve Equation (9) for the optimal MRSI reconstruction to within the standard deviation of noise with ADMM convergence guarantees [19]. In order to apply the Split Bregman algorithm, Equation (9) is formed as an unconstrained problem and both constraints converted to equality:  
\[\min_{u,z}E(u,z)+\lambda H(u,z) \tag{10}\]  
where  
\[ \begin{array}{ll}E(u,z)&=\|z\|_{2,1}+\mu\|R\mathcal{F}u-f\|_{2}^{2}\\ H(u,z)&=\|z-Gu\|_{2}^{2}\end{array}  \tag{11}\]  
Relaxing Equation (9) into Equation (10), forms \(E(u,z)\) as a combination of the objective function and fidelity constraint through the Lagrange multiplier, \(\mu\), which would normally cause the reconstruction result to depend heavily on the initial value for \(\mu\); however, as will be shown, the Bregman iterative framework reduces the dependence on \(\mu\) by treating the sampled data, \(f\), as a Bregman parameter and updating it accordingly throughout the reconstruction [29]. By following the process defined in Equations (5)-(7) and splitting \(E(u,z)\) into its \(l_{2,1}\) and \(l_{2}\) components, we can derive the \(u\) and \(z\) sub-problems and a Bregman parameter update that solves the unconstrained problem in Equation (10):  
\[ \begin{array}{ll}u^{k+1}&=\min_{u}\mu\|R\mathcal{F}u-f\|_{2}^{2}+\lambda\|z ^{k}-Gu-b_{z}^{k}\|_{2}^{2}\\ z^{k+1}&=\min_{z}\|z\|_{2,1}+\lambda\|z-Gu^{k+1}-b_{z}^{k}\|_{2}^{2}\\ b_{z}^{k+1}&=b_{z}^{k}+\left(Gu^{k+1}-z^{k+1}\right)\end{array}  \tag{12}\]  
The \(u^{k+1}\) sub-problem is now a function of \(u\) with \(z^{k}\) treated as a constant, and the \(z^{k+1}\) sub-problem treats \(u^{k+1}\) similarly. The \(z^{k+1}\) sub-problem is non-differentiable; however, its equivalent problem can be solved as:  
\[ \begin{array}{ll}z^{k+1}&=\min_{z}\sum\limits_{i=1}^{L}\left[\|z_{gi}\|_{2}+ \lambda\|z_{gi}-(Gu^{k+1})_{gi}-(b_{z}^{k})_{gi}\|_{2}^{2}\right]\\ &=gshrink(Gu^{k+1}+b_{z}^{k},\frac{1}{\lambda},G)\end{array}  \tag{13}\]where \(gshrink\) is group-wise shrinkage over each group, instead of individual coefficients, and has the closed form solution:  
\[z_{gi}^{k+1}= max\left(0,1-\frac{1}{\lambda\|x_{gi}\|_{2}}\right)\cdot x_{gi} \tag{14}\]  
where \(x_{gi}=(Gu^{k+1})_{gi}+(b_{z}^{k})_{gi}\). The \(u^{k+1}\) sub-problem is differentiable; therefore, optimality conditions can be derived for \(u\) and simplified to:  
\[(\mu\mathcal{F}^{\prime}R^{\prime}R\mathcal{F}+\lambda G^{\prime}G)u^{k+1}=\mu \mathcal{F}^{\prime}R^{\prime}f+\lambda G^{\prime}\left(z^{k}-b_{z}^{k}\right) \tag{15}\]  
where \(G^{\prime}G\) is a diagonal matrix with each \(\left(G^{\prime}G\right)_{ii}\) being the number of groups that contain transform coefficient \(u_{i}^{k+1}\)[33]. If each transform coefficient is in the same number of groups, hence \(G^{\prime}G\) is a multiple of \(I\), the left-hand side of Equation (15) is circulant and can be easily inverted by the Fourier transform [19]:  
\[u^{k+1}=\mathcal{F}^{\prime}K^{-1}\mathcal{F}\left(\mu\mathcal{F}^{\prime}R^ {\prime}f+\lambda G^{\prime}\left(z^{k}-b_{z}^{k}\right)\right) \tag{16}\]  
where \(K=(\mu R^{\prime}R+\lambda G^{\prime}G)\) is diagonal. If each transform coefficient is not in the same number of groups and \(G^{\prime}G\) is not a multiple of \(I\), \(u^{k+1}\) in Equation (15) can be solved for by the Gauss-Seidel method or other iterative solvers for linear systems.