# Optimizing Parameters for Enhanced Iterative Image Reconstruction Using Extended Power Divergence
## 5 Discussion  
First, we discuss the results of the numerical experiments that enable quantitative evaluation using the true values. The algorithm applied to the PDEM system, as shown in Figures 3 and 7, demonstrates that it operates appropriately as expected, regardless of whether the distance function compares images or projections or the type of function that was used. The parameters \(\lambda(T)\) obtained using the true image significantly reduce the evaluation function compared to \(\lambda(0)\), as seen in Figure 4. Additionally, visual assessment from Figure 5 and the density profiles in Figure 6 indicate that the reconstructed images exhibit less variability in density values, which should be flat, providing high-quality results. In practical applications, using distance functions based on forward and measured projections is necessary. It would be desirable if projections yield the same results as functions based on images. Although Euclidean distance and KL-divergence are commonly used as distance functions related to projections, effective parameters have not been obtained, as shown by the changes in distance functions (Figure 8a,b). No improvement in image quality was observed from MLEM (Figure 9). In contrast, distance functions using EPD and WEPD significantly reduce the evaluation function compared to MLEM (Figure 8c,d), and visual assessments (Figure 9) demonstrate that the reconstructed images are high quality. Table 1 summarizes the results of the numerical experiments for the disc phantom. Focusing on the parameter values, previous experimental studies [14] have shown that to improve the image reconstruction quality with PDEM, it is preferable to set a large value for \(\alpha\) to quickly reduce the evaluation function related to the true image with fewer iterations, and to set a small value for \(\gamma\) to prevent the evaluation function from increasing in subsequent iterations. When the distance functions related to projections were selected as EPD (Equation (15)) and WEPD (Equation (17)), the values of the parameter sets \(\lambda(T)\) were \((0.139,9.21)\) and \((0.143,9.17)\), respectively. These values, like the parameter set \((0.166,3.45)\) obtained when using the Euclidean distance for images (Equation (12)), resulted in a sufficiently small \(\alpha\) and a sufficiently large \(\gamma\), which clearly differs from the case when the Euclidean distance (Equation (13)) or KL-divergence (Equation (14)) was chosen for projections. Due to these parameter estimates, the quantitative evaluation of the reconstructed images also shows that, for projection-based EPD and WEPD distance functions, the results are of the same or even higher quality than those obtained with the image-based distance function. This is similarly observed in the results for the Shepp-Logan phantom in Table 2. However, in the experiment with the phantom, as shown in Figures 5b and 6b, the edges of structures where the density values change sharply are blurred. The PDEM shows favorable results in the objective function and SSIM because the noise suppression effect outweighs the smoothing performance. Equation (16) indicates that the parameter sets \((\gamma^{0},\alpha^{0})\) corresponding to \((1,0)\) and \((1,1)\) align with the square of the Euclidean distance and KL-divergence, respectively. In this study, the parameter set \((0.1,0.1)\) chosen for the distance functions EPD and WEPD is considered effective because it makes the reconstruction more robust to noise. The effectiveness of the distance function related to EPD may be due to the fact that PDEM is a reconstruction method based on the optimization of EPD. However, it is of significance that we demonstrated the use of a non-Euclidean distance function, rather than the commonly used Euclidean distance, for evaluating reconstruction methods, including MLEM, as a function of forward and measured projections.