## 4 Optimization Transfer Algorithms
Another trick, due to [PERSON] [32], uses the following concave inequality:  
\[g_{i}(A_{i}\mathbf{x})=g_{i}\left(\sum_{j=1}^{p}\pi_{ij}[\frac{1}{\pi_{ij}}a_{ij} (x_{j}-x_{j}^{(k)})+A_{i}\mathbf{x}^{(k)}]\right)\geq\sum_{j=1}^{p}\pi_{ij}g_{i} \left(\frac{1}{\pi_{ij}}a_{ij}(x_{j}-x_{j}^{(k)})+A_{i}\mathbf{x}^{(k)}\right) \tag{30}\]  
If the weights \(\pi_{ij}\) do not depend on \(x_{j}\), then Equation (30) leads to the surrogate function of  
\[\Phi(\mathbf{x};\mathbf{x}^{(k)})=\sum_{j=1}^{p}\sum_{i=1}^{n}\pi_{ij}g_{i}\left(\frac {a_{ij}}{\pi_{ij}}(x_{j}-x_{j}^{(k)})+A_{i}\mathbf{x}^{(k)}\right) \tag{31}\]  
which clearly also meets the minorization conditions. In Equation (31), the choice of \(\pi_{ij}\) is again flexible, and one popular option is to use \(\pi_{ij}=a_{ij}/\sum_{r}a_{ir}\). The above two surrogates are developed based on the concave inequality. Another useful approach is to employ a quadratic lower bound (e.g., [40]). Assume \(g_{i}\) is twice differentiable with its second derivative denoted by \(\
abla^{2}g_{i}\). Let \(d_{i}^{(k)}\) be a number such that \(d_{i}^{(k)}\leq\
abla^{2}g_{i}(A_{i}\mathbf{x})\) for all \(A_{i}\mathbf{x}>0\), then  
\[g_{i}(A_{i}\mathbf{x})\geq g_{i}(A_{i}\mathbf{x}^{(k)})+(\mathbf{x}-\mathbf{x}^{(k)})^{T}A_{i}^ {T}\
abla g_{i}(A_{i}\mathbf{x}^{(k)})+\frac{1}{2}(\mathbf{x}-\mathbf{x}^{(k)})^{T}A_{i}^ {T}d_{i}^{(k)}A_{i}(\mathbf{x}-\mathbf{x}^{(k)}) \tag{32}\]  
The right hand side of Equation (32) is a parabola surrogate of \(g_{i}\) and the condition on \(d_{i}^{(k)}\) guarantees that this function lies below \(g_{i}\). Unlike the previous surrogate functions, this surrogate is not separable in \(\mathbf{x}\), and therefore its maximization with respect to \(\mathbf{x}\) cannot be reduced to a series of 1-D problems. To overcome this problem we can find another function surrogating the above parabola surrogate but with separable \(\mathbf{x}\). Towards this, we denote the right hand side quadratic function of Equation (32) by \(q_{i}^{(k)}(A_{i}\mathbf{x})\). Since \(q_{i}^{(k)}\) is concave in \(A_{i}\mathbf{x}\), we can use either Equations (29) or (31) to find a surrogate to \(q_{i}^{(k)}\) and the resulting algorithm is called the separable paraboloidal surrogate (SPS) algorithm [39]. For example, corresponding to Equation (31), a separable parabola surrogate of \(q^{(k)}\) is  
\[\Phi(\mathbf{x};\mathbf{x}^{(k)})=\sum_{j=1}^{p}\sum_{i=1}^{n}\pi_{ij}q_{i}^{(k)} \left(\frac{a_{ij}}{\pi_{ij}}(x_{j}-x_{j}^{(k)})+A_{i}\mathbf{x}^{(k)}\right) \tag{33}\]  
A careful selection of the curvature \(b_{i}^{(k)}\) in Equation (32) can lead to fast convergence of the SPS algorithm. [PERSON] and [PERSON] [39] derived the optimal curvature for the SPS algorithm in transmission tomography. Next, we present two examples explaining how to implement the OT algorithm to emission and transmission tomography. **Example 4.1** (OT for emission scans with Poisson noise). : In this example we explain the application of OT for MPL reconstruction in emission tomography, where measurements are assumed to follow Poisson distributions. [PERSON]'s modified EM (MEM) [32] coincides with the method discussed below when \(r_{i}=0\). Firstly, under the Poisson model for emission scans, the penalized log-likelihood function is  
\[\Psi(\mathbf{x}^{(k+1)})\geq\Psi(\mathbf{x}^{(k)}) \tag{27}\]0  
where \(\rho\) is assumed a convex function. Let  
\[\Psi(\mathbf{x}^{(k+1)})\geq\Psi(\mathbf{x}^{(k)}) \tag{27}\]1  
where \(\eta_{i}=A_{i}\mathbf{x}\). It is easy to verify that \(l_{i}\) is concave with respect to \(\eta_{i}\), so we can use Equation (28) to define its surrogate function. On the other hand, for the penalty function in Equation (34), \(-\rho\) is concave, so we can use Equation (31) to construct its surrogate. Combining them together we have the following surrogate for \(\Psi(\mathbf{x})\):  
\[\Psi(\mathbf{x}^{(k+1)})\geq\Psi(\mathbf{x}^{(k)}) \tag{27}\]2where \(\pi_{tj}=c_{tj}/\sum_{r}c_{tr}\).