## 6 Modified [PERSON]'s Method of Scoring Using Jacobi or Gauss-Seidel Over-Relaxations
Note that  
\[A_{i}(\mathbf{x}_{\succ j}^{(k,r)}-\mathbf{x}^{(k)})=A_{i}(\mathbf{x}_{\succ j-1}^{(k,r)} -\mathbf{x}^{(k)})+a_{ij}(x_{j-1}^{(k,r+1)}-x_{j-1}^{(k,r)}) \tag{77}\]  
so we can start with \(A(\mathbf{x}_{\succ 0}^{(k,r)}-\mathbf{x}^{(k)})\equiv A(\mathbf{x}_{\succ p+1}^{(k,r-1)}-\mathbf{ x}^{(k)})\) and obtain \(A(\mathbf{x}_{\succ j}^{(k,r)}-\mathbf{x}^{(k)})\) by applying Equation (77). Although here the number of multiplications for \(A\mathbf{\delta}\) (where vector \(\mathbf{\delta}\) varies with its index \(j\)) becomes the same as \(A\mathbf{x}\), it requires column access to the system matrix \(A\), which can be a problem if \(A\) is generated on-the-fly. We next provide examples of applying FS-JOR and FS-SOR to emission and transmission tomography. **Example 6.1** (**Emission scans with Poisson noise)**. : For emission reconstruction with Poisson noise, the log-density of \(y_{i}\) is given by Equation (61). Thus for the corresponding object function \(\Psi(\mathbf{x})\) of Equation (2), its gradients are  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]0  
and its Fisher information matrix elements are  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]1  
where \(\mu_{i}=A_{i}\mathbf{x}+r_{i}\), \(j\text{ and }t=1,\ldots,p\). Assuming we run only one sub-iteration for FS-JOR or FS-SOR (_i.e._, \(m=1\)), the FS-JOR iterative formula is  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]2  
and the FS-SOR formula is  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]3 \[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]4Then \(x_{j}^{(k+1)}=\max\{0,\tilde{x}_{j}^{(k+1)}\}\). The formula given in Equation (80) is just a gradient algorithm so \(\omega\) can be replaced by a line search step size \(\omega^{(k)}\in(0,1]\). Efficient computation of Equation (81) requires column access to matrix \(A\) as explicated before. [PERSON] _et al_. [PERSON] _et al_. (2014) reported simulation results and a real data application for emission reconstruction. They compared FS-JOR and FS-SOR with EM. The computer time required per iteration for the EM and one-step FS-JOR algorithms were similar. By comparison with the EM algorithm, FS-JOR and FS-SOR accelerated convergence when an appropriate value of \(\omega\) was used. Particularly, FS-SOR had a superior speed of convergence when \(\omega=1\). **Example 6.2** (Transmission scans with Poisson noise). : For transmission reconstructions with Poisson noise, we can easily work out the gradient and Fisher information matrix from its penalized likelihood function. The gradients are  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]5  
and the Fisher information matrix elements are  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]6  
where \(\eta_{i}=A_{i}\mathbf{x}\), \(\mu_{i}=b_{i}e^{-\eta_{i}}+r_{i}\) and \(j\) and \(t=1,\ldots,p\). Note that for this example, the Fisher information matrix is non-negative but the negative Hessian matrix may not be non-negative, making the Newton method non-applicable. Corresponding to \(m=1\), the FS-JOR iterative formula is  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]7 \[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]8  
and the FS-SOR formula is  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Psi^{(k)}(\mathbf{x}) \tag{69}\]9 \[\
abla_{j}\Psi^{(k)}(\mathbf{x})=0\text{ if }x_{j}>0\text{ and } \tag{70}\]0  
Then \(x_{j}^{(k+1)}=\max\{0,\tilde{x}_{j}^{(k+1)}\}\). Again, Equation (84) is a gradient algorithm so that a line search can be used, and efficient implementation of Equation (85) demands unpleasant column access to \(A\). This section explains the Fisher scoring based image reconstruction algorithms using JOR or SOR sub-iterations.