# Optimizing Parameters for Enhanced Iterative Image Reconstruction Using Extended Power Divergence
## 3 Proposed Method  
The proposed method focuses on iterative reconstruction algorithms with adjustable parameters and provides a way to explore parameter values that yield the most effective performance for a set of projection data sharing common characteristics, such as similar noise levels and internal structures.  
First, given \(K\) projections \(Y\in R_{+}^{I\times K}\) with the same number of elements \(I\), we define an iterative algorithm with the target system \(f\) to determine the corresponding pixel values \(Z\in R_{+}^{I\times K}\) as follows:  
\[Z_{k}(n+1)=f(Z_{k}(n);\lambda) \tag{7}\]  
for \(n=0,1,2,\ldots,N-1\) and \(k=1,2,\ldots,K\). Here, \(\lambda\in R_{+}^{M}\) represents the parameters to be adjusted. For example, when the target \(f\) is the PDEM algorithm used in this paper, \(M=2\) and  
\[\lambda=\left(\begin{array}{c}\gamma\\ \alpha\end{array}\right). \tag{8}\]  
Next, starting from the initial value \(Z_{k}(0)\) (\(k=1,2,\ldots,K\)) according to Equation (7), after \(n\) iterations (\(n=1,2,\ldots,N\)), we define  
\[Z_{k}(n)=:f^{n}(Z_{k}(0);\lambda) \tag{9}\]  
for \(k=1,2,\ldots,K\). We then seek the parameters that minimize the evaluation function  
\[g(\lambda;N):=\frac{1}{K}\sum_{k=1}^{K}d(f^{N}(Z_{k}(0);\lambda)) \tag{10}\]  
where \(d\) is the distance function, defined below, evaluated at the \(N\)th iteration. Our goal is to find the elements of the set  
\[\operatorname*{argmin}_{\lambda}g(\lambda;N). \tag{11}\]  
Note that the function \(g\) depends on \(\lambda\) through the computation process of \(Z(N)\), which is obtained by iterating \(f(\cdot;\lambda)\), where \(f\) includes \(\lambda\) as a parameter.  
In the experiments conducted in this study, the distance function \(d\) is defined as follows. First, the Euclidean distance using the true value \(e\in\mathcal{E}\) of the image is defined as:  
\[d(x):=\|e-x\|. \tag{12}\]Next, for the distance between the forward projection and the measured projection \(Y_{k}\) (\(k=1,2,\ldots,K\)), we consider the following four types:  
1. Euclidean distance: \[d(x):=\|Y_{k}-Ax\|,\] (13)
2. KL-divergence: \[d(x):=\text{KL}(Y_{k},Ax),\] (14)
3. EPD: \[d(x):=\Phi_{0.1,0.1}(Y_{k},Ax)\] (15) where \[\Phi_{\gamma^{0}\alpha^{0}}(p,q)=\sum_{i=1}^{I}\int_{p_{i}}^{q_{i}}\frac{s^{ \gamma^{0}}-p_{i}^{\gamma^{0}}}{s^{\gamma^{0}\alpha^{0}}}ds,\] (16)
4. WEPD: \[d(x):=\Psi_{0.1,0.1}(Y_{k},Ax)\] (17) where \[\Psi_{\gamma^{0}\alpha^{0}}(p,q)=\sum_{i=1}^{I}\int_{p_{i}}^{q_{i}}\frac{s^{ \gamma^{0}}-p_{i}^{\gamma^{0}}}{s^{\gamma^{0}\alpha^{0}}}ds\sum_{j=1}^{J}A_{ ij}.\] (18)  
Equation (12) is used as a distance function to accurately evaluate the quality of the reconstructed image, while Equations (13)-(15), and (17) serve as candidate functions for applying the proposed method to actual projections. The parameters \(\gamma^{0}\) and \(\alpha^{0}\) in Equations (16) and (18) are both set to \(0.1\) in Equations (15) and (17), respectively, based on the results from Ref. [14], which suggest that smaller parameter values are more desirable for achieving robustness against noise, and fixed values ensure fairness without dependency on other conditions. The initial values for optimization associated with the set in (11) are assigned to \(\gamma(0)=1\) and \(\alpha(0)=1\). Note that the iterative rule with these fixed parameters coincides with MLEM.