# Optimizing Parameters for Enhanced Iterative Image Reconstruction Using Extended Power Divergence
## 4 Experiments
### Experimental Method  
We consider the adjustment of parameters \(\lambda=(\gamma,\alpha)\) for the PDEM iterative algorithm in Equation (6).  
In the numerical experiments, we simulate a first-generation CT scan with 180-degree rotation and reconstruct images (\(J=128\times 128\)) from projections with 184 bins and 90 projection directions (\(I=184\times 90\)). We use the disc or modified Shepp-Logan phantom shown in Figure 1 as the true image \(e\in R_{+}^{J}\). For each \(k\) of \(K=8\) projections with the same noise level but different noise patterns, we create projections using  
\[Y_{k}=Ae+\sigma_{k}\,\in\,R_{+}^{I} \tag{19}\]  
for \(k=1,2,\ldots,K\). Here, the normally distributed random noise \(\sigma_{k}\) is set to have a variance such that the signal-to-noise ratio of the projection is 20 dB and is independently generated for each \(k\). This choice is justified because the noise in X-ray CT is primarily quantum noise and electronic noise, which are well-represented by a normal distribution. The intention is to seek parameters that are common to the average projections, which share the same uncertainty in terms of signal-to-noise ratio, rather than focusing on a specific random variable. The number of iterations \(N\) for optimization is set to 30.  
In the physical experiments, projection data were acquired using an X-ray CT scanner (Canon Medical Systems, Tochigi, Japan) and a phantom simulating the human body [25]. To investigate the noise suppression effect, which is a characteristic of the PDEM reconstruction method, we used an abdominal phantom with relatively simple density variations. The scanner was operated with a tube voltage of 80 kVp, tube current of 10 mA, and scan time of 0.75 s per rotation. This setup corresponds to imaging conditions under relatively high levels of measurement noise. Projections were obtained from 900 directions covering 360 degrees with 957 bins, and data were subsampled to achieve 90 projection directions (\(I=86,130\)) to simulate sparse projections. Optimization was performed using projection data from four consecutive slices (\(K=4\)) and an evaluation function with \(N=100\) iterations. The reconstructed image size was \(674\times 674\) (\(J=454,276\)). Figure 2 shows images created from one of the projection data slices using the filtered back-projection (FBP) method with a Shepp-Logan filter and the iterative reconstruction method using MLEM.  
We solved the problem of finding the elements of the set in (11) using the solver fmincon provided in MATLAB R2023a (MathWorks, Natick, USA) for finding the minimum of constrained nonlinear multivariable functions. The initial parameter values were set to \(\lambda(0)=(1,1)\). The lower bound for parameter variation was set to 0.01 and the upper bound to 10.0. Although a relatively large value is set as the exponent for the upper limit necessary for numerical accuracy, the internal state is also checked in the numerical experiments to ensure that no illegal values or divergence occurs.  
For the numerical calculations, we used a computer equipped with a 12-core Apple M2 Max processor and 64 GB of memory, and the program utilized parallel computing capabilities.