## 6 Modified [PERSON]'s Method of Scoring Using Jacobi or Gauss-Seidel Over-Relaxations
Let \(r\) be the sub-iteration index for the over-relaxation method and \(\mathbf{x}^{(k,r)}\) the estimate of \(\mathbf{x}\) at the \(r\)-th over-relaxation sub-iteration of the \(k\)-th FS iteration. Let \(f_{jt}^{(k)}\) be the \((j,t)\)-th element of \(F^{(k)}\). Assume \(f_{jj}^{(k)}>0\) for all \(j\). At iteration \(k+1\), first set \(\mathbf{x}^{(k,0)}=\mathbf{x}^{(k)}\). If using JOR to solve Equation (73) we have  
\[x_{j}^{(k,r+1)}=x_{j}^{(k,r)}+\omega\frac{1}{f_{jj}^{(k)}}\left(\
abla_{j}\Psi (\mathbf{x}^{(k)})-\sum_{t=1}^{p}f_{jt}^{(k)}(x_{t}^{(k,r)}-x_{t}^{(k)})\right) \tag{74}\]  
and if using SOR to solve we then have  
\[x_{j}^{(k,r+1)}=x_{j}^{(k,r)}+\omega\frac{1}{f_{jj}^{(k)}}\left(\
abla_{j}\Psi (\mathbf{x}^{(k)})-\sum_{t=1}^{j-1}f_{jt}^{(k)}(x_{t}^{(k,r+1)}-x_{t}^{(k)})-\sum_ {t=j}^{p}f_{jt}^{(k)}(x_{t}^{(k,r)}-x_{t}^{(k)})\right) \tag{75}\]  
where \(r=0,\ldots,m-1\) and \(\omega>0\) is the relaxation parameter. If any \(x_{j}^{(k,r+1)}<0\) then it is reset to zero. This resetting is correct since the only possibility for \(x_{j}^{(k,r+1)}<0\) is that the expressions in the round brackets of Equations (74) and (75) are negative since \(x_{j}^{(k,r)}\geq 0\) and \(f_{jj}^{(k)}>0\). Hence resetting \(x_{j}^{(k,r+1)}\) to zeros assures that the FS-JOR and FS-SOR algorithms converge to, when they converge, the solution satisfying the KKT conditions. At the end of the sub-iterations set \(\mathbf{x}^{(k+1)}=\mathbf{x}^{(k,m)}\). Note that when \(m=1\), the last term in the round brackets of either Equation (74) or (75) becomes zero. Thus 1-step FS-JOR is basically a gradient algorithm and we can therefore replace \(\omega\) by a line search step size \(\omega^{(k)}\), where the search range is fixed at \(0<\omega^{(k)}\leq 1\) as this range will keep the estimate non-negative. The relaxation parameter \(\omega\) is used to achieve convergence of the FS-JOR and FS-SOR algorithms. Results contained in [47] give convergence properties when \(n\rightarrow\infty\) and when the non-negativity constraint is ignored. In fact in this context FS-SOR converges if \(0<\omega<2\) and FS-JOR converges if \(0<\omega<\xi_{\max}\), where \(\xi_{\max}\) is the maximum eigenvalue of \(D_{F}(\hat{\mathbf{x}})^{-1/2}F(\hat{\mathbf{x}})D_{F}(\hat{\mathbf{x}})^{-1/2}\). Here \(\hat{\mathbf{x}}\) is the MPL solution. From the updating formulae given in Equations (74) and (75) we can see that both FS-JOR and FS-SOR involve the gradient \(\
abla\Psi(\mathbf{x})\) and the Fisher information matrix based operation \(F(\mathbf{x})\mathbf{\delta}\). The gradient is standard for most reconstruction algorithms, but the computation of \(F(\mathbf{x})\mathbf{\delta}\) requires more careful consideration. It will become clear in Examples 6.1 and 6.2 that for tomographic reconstructions \(F(\mathbf{x})\) usually exhibits as \(A^{T}W(\mathbf{x})A+\
abla^{2}J(\mathbf{x})\), where \(W(\mathbf{x})=\text{diag}(w_{x}(\mathbf{x}),\ldots,w_{n}(\mathbf{x}))\). It is not wise to compute \(A^{T}W(\mathbf{x})A\) first as this involves multiplications of two huge matrices \(A\) and \(A^{T}\). For FS-JOR, a feasible alternative is to use the forward projection to find \(A\mathbf{\delta}\) first, then to multiply it with the diagonal values of \(W\) to get \(W(\mathbf{x})A\mathbf{\delta}\), and finally to back-project \(W(\mathbf{x})A\mathbf{\delta}\) to obtain \(F(\mathbf{x})\mathbf{\delta}\) (ignoring the penalty term). This approach involves only one forward- and one back-projections in every sub-iteration. The situation for FS-SOR is more complicated since \(\mathbf{\delta}\) changes with the pixel index \(j\). The above approach for FS-JOR cannot be used here as otherwise each FS-SOR sub-iteration will demand infeasible \(p\) pairs of forward- and back-projections. To confront this problem, let  
\[\mathbf{x}_{\succ j}^{(k,r)}=(x_{1}^{(k,r+1)},\ldots,x_{j-1}^{(k,r+1)},x_{j}^{(k,r)},\ldots,x_{p}^{(k,r)})^{T} \tag{76}\]  
The \(F\mathbf{\delta}\) part of Equation (75) involves \(A(\mathbf{x}_{\succ j}^{(k,r)}-\mathbf{x}^{(k)})\).