## 7 Iterative Coordinate Ascent Algorithms  
Another method using SOR is the method of iterative coordinate ascent (ICA) (or iterative coordinate descent (ICD) for minimization problems). ICA was first implemented to tomographic imaging in [51; 52]. The basic idea of ICA is to apply SOR directly to the objective function \(\Psi(\mathbf{x})\), resulting in a sequence of 1-D functions where each \(x_{j}\) is associated with one of these 1-D functions. Then each function is solved exactly or approximately to update the corresponding \(x_{j}\). More specifically, using the SOR principle we can define a function for \(x_{j}\) according to  
\[\psi_{j}^{(k+1)}(x_{j})=\Psi(x_{1}^{(k+1)},\ldots,x_{j-1}^{(k+1)},x_{j},x_{j+1 }^{(k)},\ldots,x_{p}^{(k)}) \tag{86}\]  
This is a function of \(x_{j}\) only and we can update the \(x_{j}\) estimate by  
\[x_{j}^{(k+1)}=\arg\max_{x_{j}\geq 0}\psi_{j}^{(k+1)}(x_{j}) \tag{87}\]  
Since this is a 1-D function, the constraint \(x_{j}\geq 0\) can be easily enforced using, for example, the resetting to zero approach. One computational issue with ICA when applied to tomographic imaging is that it requires repeated calculations of \(\eta_{i}(\mathbf{x})=\sum_{j}a_{it}x_{t}\) for all \(i\) when updating \(x_{j}\). This problem can be rectified by the following approach. Let  
\[\mathbf{x}_{\succ j}^{k}=(x_{1}^{(k+1)},\ldots,x_{j-1}^{(k+1)},x_{j}^{(k)},\ldots, x_{p}^{(k)})^{T} \tag{88}\]  
Consider the evaluation of \(\eta_{i}(\mathbf{x}_{\succ j}^{k})\). Assuming the update of \(x_{j-1}\) is given by \(x_{j-1}^{(k+1)}=x_{j-1}^{(k)}+\delta_{j-1}^{(k)},\) then \(a_{i,j-1}x_{j-1}^{(k+1)}=a_{i,j-1}x_{j-1}^{(k)}+a_{i,j-1}\delta_{j-1}^{(k)},\) and therefore  
\[\eta_{i}(\mathbf{x}_{\succ j}^{k})=\eta_{i}(\mathbf{x}_{\succ j-1}^{k})+a_{i,j-1} \delta_{j-1}^{(k)} \tag{89}\]  
This relationship explains that \(\eta_{i}(\mathbf{x}_{\succ j}^{k})\) can be cheaply computed using the \(\eta_{i}\) value before the \(x_{j}\) update plus a correction term. However, similar to FS-SOR, it necessitates column access to \(A\). This can be a potential issue if \(A\) is generated on-the-fly. Next we use again the emission and transmission examples to elaborate the ICA algorithm. **Example 7.1** (Emission scans with Poisson noise). : Firstly, we define  
\[\mathbf{x}_{(j)}^{(k)}=(x_{1}^{(k+1)},\ldots,x_{j-1}^{(k+1)},x_{j},x_{j+1}^{(k)},\ldots,x_{p}^{(k)})^{T} \tag{90}\]  
From the penalized log-likelihood function of emission measurements \(y_{i}\) (see, for example, Equation (34)), function \(\psi_{j}(x_{j})\) is given by  
\[\psi_{j}(x_{j})=\sum_{i=1}^{n}\left\{-(\eta_{i}(\mathbf{x}_{(j)}^{(k)})+r_{i})+y_ {i}\log(\eta_{i}(\mathbf{x}_{(j)}^{(k)})+r_{i})\right\}-hJ(\mathbf{x}_{(j)}^{(k)}) \tag{91}\]  
Since this is a non-quadratic function of \(x_{j}\), exact maximization is infeasible. We can find its approximate optimization by running a single or multi- step of, for example, the [PERSON] or [PERSON] scoring algorithm. In this example we consider using the [PERSON] scoring algorithm to optimize \(\psi_{j}(x_{j})\) and call the resulting algorithm ICA-FS. After a single step of [PERSON] scoring we have  
\[\tilde{x}_{j}^{(k+1)}=x_{j}^{(k)}+\omega_{j}^{(k)}\frac{1}{\sum_{i}a_{ij}^{2} /\mu_{i}(\mathbf{x}_{\succ j}^{k})+h\
abla_{jj}^{2}J(\mathbf{x}_{\succ j}^{k})}\left( \sum_{i=1}^{n}a_{ij}(y_{i}/\mu_{i}(\mathbf{x}_{\succ j}^{k})-1)-h\
abla_{j}J(\bm {x}_{\succ j}^{k})\right) \tag{92}\]where \(\mu_{i}(\mathbf{x}_{\succ j}^{k})=\eta_{i}(\mathbf{x}_{\succ j}^{k})+r_{i}\) and \(\omega_{j}^{(k)}\) is a line search step size enforcing \(\psi_{j}(x_{j}^{(k+1)})\geq\psi_{j}(x_{j}^{(k)})\), where equality holds only when the algorithm is converged. This monotonic condition eventually leads to \(\Psi(\mathbf{x}^{(k+1)})\geq\Psi(\mathbf{x}^{(k)})\). The update for \(x_{j}\) is then \(x_{j}^{(k+1)}=\max\{0,\tilde{x}_{j}^{(k+1)}\}\). **Example 7.2** (**Transmission scans with Poisson noise)**.