## 5 Multiplicative Iterative Algorithms
Briefly, under certain regular conditions, MI converges monotonically to a local maxima satisfying the KKT conditions. For the mean functions given in Equation (4), we have \(\
abla_{j}\mu_{i}(\mathbf{x})=a_{ij}\) for emission and \(\
abla_{j}\mu_{i}(\mathbf{x})=-b_{i}e^{-A_{i}\mathbf{x}}a_{ij}\) for transmission tomography; the corresponding updating formula (53) becomes:  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]1  
for emission tomography, and  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]2  
for transmission tomography. The derivative \(\
abla l_{i}(\mu_{i})\) in the above formulae depends on the log-density \(l_{i}(\mu_{i})\). Some examples are presented below. **Example 5.1** (**MI for emission scans with Poisson noise**). : For emission tomography with Poisson noise, we have the log-density function for \(y_{i}\):  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]3where \(\mu_{i}=A_{i}\mathbf{x}+r_{i}\). Thus \(\
abla l_{i}(\mu_{i})=-1+y_{i}/\mu_{i}\), which gives \(\
abla l_{i}(\mu_{i})^{+}=y_{i}/\mu_{i}\) and \(\
abla l_{i}(\mu_{i})^{-}=-1\). The updating formula (59) becomes, for \(j=1,\ldots,p\),  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]4  
Note that when \(h=0\) (_i.e._, maximum likelihood reconstruction), \(r_{i}=0\) and \(\epsilon=0\), this algorithm coincides with the EM algorithm for emission tomography. After line search, the estimate of \(\mathbf{x}\) at iteration \(k+1\) is given by Equation (57). In this algorithm, there is only one back-projection (for the numerator of Equation (62)) and one forward-projection in each iteration; its computational burden is the same as EM. **Example 5.2** (MI for randoms-precorrected PET emission scans). : __  
Some PET scans produce measurements that have already been corrected for randoms [44] and their measurements no longer follow Poisson distributions. We consider in this example the model weighted least squares which is also used in [11] but under a different context, _i.e._, we reconstruct from randoms-precorrected measurements \(y_{i}\) by maximizing the objective Equation (2) where  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]5  
Here \(\mu_{i}\) is used to denote \(A_{i}\mathbf{x}\), and for this \(\mu_{i}\) formula (59) still applies. Now since  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]6  
we have \(\
abla l_{i}(\mu_{i})^{+}=[(y_{i}+2r_{i})/(\mu_{i}+2r_{i})]^{2}\) and \(\
abla l_{i}(\mu_{i})^{-}=-1\). The MI algorithm updates \(\mathbf{x}\) first according to  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]7  
and then, after the line search step, computes \(\mathbf{x}^{(k+1)}\) according to Equation (57). **Example 5.3** (MI for polyenergetic transmission scans with Poisson noise). : __  
Application of the MI algorithm to polyenergetic X-ray CT is again extremely easy. Under the assumption of Poisson noise, the log-density for measurement \(y_{i}\) is identical to Equation (61) but now with \(\mu_{i}=\sum_{m=1}^{M}b_{im}e^{-\sum_{j}a_{ij}\sum_{r}u_{mr}z_{rj}}+r_{i}\); see Equation (17). In Example 5.1 we have already derived \(\
abla l_{i}(\mu_{i})^{+}\) and \(\
abla l_{i}(\mu_{i})^{-}\) for the Poisson noise log-density. On the other hand, the derivative of \(\mu_{i}\) with respect to \(z_{rj}\) (denoted by \(\
abla_{rj}\mu_{i}\)) is  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]8  
Thus, the updating formula for polyenergetic transmission is  
\[\
abla_{j}\Psi(\mathbf{x}) \leq 0\text{ if }x_{j}=0 \tag{50}\]9for \(r=1,\ldots,a\) and \(j=1,\ldots,p\). After the line search step specified in Equation (57), \(\mathbf{z}^{(k+1)}\) is obtained. This iterative formula involves one forward- and two back-projections in each iteration, and therefore it demands similar amount of computations when compared with the alternative minimization algorithm in [34].