## 3 Alternating Minimization Algorithms for Transmission Tomography  
We have explained in Section 2 that the EM algorithm is not directly suitable for transmission scans as its M-step cannot be computed exactly. In this section, we summarize an alternating minimization algorithm designed to solve the transmission tomographic problem, including X-Ray CT. This algorithm is a generalization to the EM algorithm [33] and its application to transmission tomography can be found in [34]. Following [34], we explain this algorithm using the polyenergetic transmission tomography example. In this context, if assuming transmission scans follow Poisson distributions, the corresponding log-likelihood is  
\[l(\mathbf{z})=\sum_{i=1}^{n}\{y_{i}\log\mu_{i}(\mathbf{z})-\mu_{i}(\mathbf{z})\} \tag{16}\]  
where \(y_{i}\) is the scan count of detector \(i\) and \(\mu_{i}\) (now expressed as a function of vector \(\mathbf{z}\), which will be defined below) is given by Equation (5). Moreover, elements of the attenuation map associated with spectrum \(m\), namely elements of \(\mathbf{x}_{m}\) in Equation (5), are further modeled by  
\[x_{mj}=\sum_{r=1}^{a}u_{mr}z_{rj} \tag{17}\]where \(j\) indexes pixels, \(r\) represents different types of materials, \(u_{mr}\) are known linear attenuation coefficients and \(z_{rj}\) are the unknown partial densities (e.g., [34]) we wish to estimate. In Equation (16), \(\mathbf{z}\) is a vector of size \(pa\times 1\) formed by column-wise stacking the vectors \(\mathbf{z}_{j}=(z_{1j},\ldots,z_{aj})^{T}\). Define set  
\[\mathcal{E}=\{q_{im};i=1,\ldots,n\text{ and }m=0,1,\ldots,M\} \tag{18}\]  
where  
\[q_{im}=b_{im}e^{-\sum_{j=1}^{p}a_{ij}\sum_{r=1}^{a}u_{mr}z_{rj}} \tag{19}\]  
for \(m=1,\ldots,M\) and \(q_{im}\) equals the background noise \(r_{i}\) for \(m=0\). Clearly, \(\mu_{i}\) given in Equation (5) can now be expressed as \(\mu_{i}=\sum_{m=0}^{M}q_{im}\). Define another set  
\[\mathcal{L}=\{p_{im}:p_{im}\geq 0\text{ and }\sum_{m}p_{im}=y_{i};i=1,\ldots,n \text{ and }m=0,1,\ldots,M\} \tag{20}\]  
In [34], \(\mathcal{E}\) is called the exponential family and \(\mathcal{L}\) the linear family. Let \(\mathbf{p}\) and \(\mathbf{q}\) be the vectors created from \(p_{im}\) and \(q_{im}\) respectively. It can be shown that the problem of maximizing the log-likelihood Equation (16) can be re-written as  
\[\max_{\mathbf{z}}l(\mathbf{z})=\min_{\mathbf{q}\in\mathcal{E}}\min_{\mathbf{p}\in\mathcal{L}} \{I(\mathbf{p}\,\|\,\mathbf{q})\} \tag{21}\]  
subject to \(z_{rj}\geq 0\), where \(I(\mathbf{p}\,\|\,\mathbf{q})\) is the \(I\)-divergence [35] given by  
\[I(\mathbf{p}\,\|\,\mathbf{q})=\sum_{i=1}^{n}\sum_{m=0}^{M}\left(p_{im}\log\frac{p_{im }}{q_{im}}-p_{im}+q_{im}\right) \tag{22}\]  
Thus, maximizing the log-likelihood in Equation (16) can be achieved iteratively. Assuming the estimates \(\mathbf{p}^{(k)}\), \(\mathbf{q}^{(k)}\) and \(\mathbf{z}^{(k)}\) are obtained at iteration \(k\), then iteration \(k+1\) contains two steps:  
1. compute \(\mathbf{p}^{(k+1)}\) by minimizing \(I(\mathbf{p}\,\|\,\mathbf{q}^{(k)})\) subject to \(\mathbf{p}\in\mathcal{L}\);
2. compute \(\mathbf{q}^{(k+1)}\) by minimizing \(I(\mathbf{p}^{(k+1)}\,\|\,\mathbf{q})\) subject to \(\mathbf{q}\in\mathcal{E}\). Note that the second step is equivalent to minimizing \(I(\mathbf{p}^{(k+1)}\,\|\,\mathbf{q})\) over \(z_{rj}\geq 0\) with \(q_{im}\) being given by the expression in Equation (19). Minimizing \(I(\mathbf{p}\,\|\,\mathbf{q}^{(k)})\) over \(\mathbf{p}\in\mathcal{L}\) is easily achieved using the Lagrange multiplier, and the result is  
\[p_{im}^{(k+1)}=q_{im}^{(k)}\frac{y_{i}}{\sum_{m^{\prime}=0}^{M}q_{im^{\prime}} ^{(k)}} \tag{23}\]  
On the other hand, direct optimization of \(I(\mathbf{p}^{(k+1)}\,\|\,\mathbf{q})\) over \(z_{rj}\geq 0\) is an unmanageable task as the \(z_{rj}\)'s are mixed (_i.e._, not decoupled or separated from each other) within the objective function. One approach to overcome this problem is by using a decoupled objective function representing an upper bound of the original objective function.