## 4 Optimization Transfer Algorithms  
Details of the optimization transfer (OT) algorithm (also called the minorization-maximization (MM) algorithm for maximizations) can be found in, for example, [38]. In this section we present this algorithm briefly and explain its application in emission and transmission tomography. The fundamental idea of the OT algorithm is that it employs a surrogate function to minorize (see the definition below) the objective function \(\Psi(\mathbf{x})\) in each iteration, and then update the parameter estimate by maximizing this surrogate function. More specifically, a function \(\Phi(\mathbf{x};\mathbf{x}^{(k)})\) is said to minorize \(\Psi(\mathbf{x})\) at \(\mathbf{x}^{(k)}\) if it satisfies the following \"minorization\" conditions:  
(i) \(\Psi(\mathbf{x}^{(k)})=\Phi(\mathbf{x}^{(k)};\mathbf{x}^{(k)})\), and  
(ii) \(\Psi(\mathbf{x})\geq\Phi(\mathbf{x};\mathbf{x}^{(k)})\) for all \(\mathbf{x}\). Then at iteration \(k+1\), \(\mathbf{x}\) is estimated by maximizing \(\Phi(\mathbf{x};\mathbf{x}^{(k)})\), _i.e._,  
\[\mathbf{x}^{(k+1)}=\arg\max_{\mathbf{x}\geq 0}\Phi(\mathbf{x};\mathbf{x}^{(k)}) \tag{26}\]If the exact maximum is not easy to obtain, we can find an \(\mathbf{x}^{(k+1)}\) by simply increasing \(\Phi(\mathbf{x};\mathbf{x}^{(k)})\), as this will also guarantee that the monotonic condition stated below remains for \(\Psi(\mathbf{x})\). An attractive property when using this surrogate function is that \(\mathbf{x}^{(k+1)}\) satisfies the monotonic condition, namely  
\[\Psi(\mathbf{x}^{(k+1)})\geq\Psi(\mathbf{x}^{(k)}) \tag{27}\]  
where equality holds only when the iteration has converged. This monotonic property can be easily verified by the minorization conditions since  
\[\Psi(\mathbf{x}^{(k+1)}) =\Phi(\mathbf{x}^{(k+1)};\mathbf{x}^{(k)})+\Psi(\mathbf{x}^{(k+1)})-\Phi(\mathbf{ x}^{(k+1)};\mathbf{x}^{(k)})\geq\Phi(\mathbf{x}^{(k)};\mathbf{x}^{(k)})+\Psi(\mathbf{x}^{(k)})- \Phi(\mathbf{x}^{(k)};\mathbf{x}^{(k)})\] \[=\[PERSON](\mathbf{x}^{(k)})\]  
For implementation of the OT algorithm to medical imaging, a surrogate function \(\Phi(\mathbf{x};\mathbf{x}^{(k)})\) must be determined. There exist different ways of choosing the surrogate function, such as those listed in [38]. We mainly consider two approaches in this paper: (i) the method based on the inequality on concave functions (called the concave inequality hereafter); and (ii) the method based on quadratic lower bounds (also known as paraboloidal surrogates [39]). These ideas are summarized below. Let \(G(\mathbf{x})=\sum_{i=1}^{n}g_{i}(A_{i}\mathbf{x})\) be the objective function we wish to maximize, where \(A_{i}\) is the \(i\)-th row of matrix \(A_{n\times p}\) and \(\mathbf{x}\) is a \(p\)-vector. For matrix \(A\), we assume its elements \(a_{ij}\) are non-negative and \(\sum_{j}a_{ij}\
eq 0\). We also assume that all \(g_{i}(\cdot)\) are concave functions. Let \(\pi_{ij}\geq 0\) be weights satisfying \(\sum_{j=1}^{p}\pi_{ij}=1\). Then according to the concave inequality we have  
\[g_{i}(A_{i}\mathbf{x})=g_{i}\left(\sum_{j=1}^{p}\pi_{ij}\frac{a_{ij}x_{j}}{\pi_{ij }}\right)\geq\sum_{j=1}^{p}\pi_{ij}g_{i}\left(\frac{a_{ij}x_{j}}{\pi_{ij}}\right) \tag{28}\]  
There are different ways of choosing weights \(\pi_{ij}\). For example, we can use \(\pi_{ij}=a_{ij}x_{j}/A_{i}\mathbf{x}\), which is also adopted in [32]. In this case since each \(\pi_{ij}\) is a function of \(\mathbf{x}\), the surrogate function corresponding to Equation (28) is  
\[\Phi(\mathbf{x};\mathbf{x}^{(k)})=\sum_{j=1}^{p}\sum_{i=1}^{n}\frac{a_{ij}x_{j}^{(k)} }{A_{i}\mathbf{x}^{(k)}}\ g_{i}\Bigg{(}\frac{A_{i}\mathbf{x}^{(k)}}{x_{j}^{(k)}}x_{j} \Bigg{)} \tag{29}\]  
and it is easy to verify that this surrogate satisfies the minorization conditions. The right hand side of Equation (29) is a weighted summation of functions \(g_{i}\), each involving a single \(x_{j}\) only (_i.e._, decoupled), and therefore maximization with respect to \(\mathbf{x}\) of \(\Phi(\mathbf{x};\mathbf{x}^{(k)})\) can be achieved by a sequence of 1-D optimizations.