## 3 Results and Discussion of Computational Experiments
### Parametrization of the Demons Algorithm
The values in bold identify the scenarios with the lowest correlation coefficients, which represent the level of performance of each solution. The lowest coefficient obtained with the local optimal parameters (Scenario 5) is greater than almost all of the coefficients in the initial solutions (7 out of 10 scenarios), except from Scenarios 1, 7, and 9. Additionally, for different parameter values, the scenario with the lowest correlation coefficient may vary. However, for the example shown in Table 1, Scenario 5 and 6 obtained the lowest correlation coefficients in both solutions. The improved solution implies an improvement of 0.015 (1.6%) with respect to the initial solution, which is significant for radiotherapy and image analysis. This improvement can be voxels that represent tissue and therefore help make more precise contours of the organs in each scenario, reducing uncertainty in radiation therapy planning, i.e., calculation of delivered dose. When comparing the default parameters provided by Matlab, we see that our observed values generate a better correlation coefficient. Figure 5, with CT scans of patients with liver cancer, shows how significant it is to search for the parameters that best fit the characteristics of the image by maximizing their similarity, in this case the correlation coefficient. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Scenario**} & **Initial Parameters** & **Local Optima** \\ \cline{2-3}  & \(N=200,\rho=3,\sigma=2.5\) & \(N=300,\rho=2,\sigma=2.9\) \\ \hline
1 & 1.0000 & 1.0000 \\
2 & 0.9562 & 0.9585 \\
3 & 0.9538 & 0.9611 \\
4 & 0.9540 & 0.96  
Now, given that each scenario shows a different form of the same organ, and the initial results show that the correlation coefficients not only depend on the three demon parameters but also on the considered scenario, the nonrigid registration can be used to estimate different DVFs in each scenario with different parameter values. Therefore, the local search process is adapted to find a parameter set for each scenario and compare the correlation coefficients and computation time with respect to the previously found parameters. Two main characteristics were taken into account for the adaptation of the Fh algorithm in order to find the appropriate parameters for each scenario. First, the algorithm runs independently for each scenario, therefore it is not necessary to identify the scenario with the lowest correlation coefficient. The second adaptation is related to the computation time of the demon algorithm. In particular, this computation time depends on the chosen parameters. Taking this into account, a set of parameters that allow obtaining a faster solution may be preferable if the difference between the correlation coefficients is very small. Due to the need of establishing a minimal difference between correlation coefficients that merits a given increase in computing time, professional medical judgment is introduced to determine the following analysis. Figure 6 shows the efficient frontiers (correlation coefficients on the horizontal axis and computation time on the vertical axis) where each point represents a determined combination of parameter values for demons. The same scales are used for the different plots per scenario in order to facilitate comparison and analysis. According to Figure 6, as we obtain better (higher) correlation coefficients, a greater computation time is required for executing the demon algorithm. In cases such as Scenarios 5, 6, 8, and 9, the maximum \(S^{corr}\) have a very high demand in computing time (more than 400 seconds). Additionally, the maximum CC for the 9 scenarios is reached in the interval \([0.965,\,0.97]\), which coincides with the highest computation times. A similarity value of 0.9600 is considered good enough in terms of the quality of the registrations and this is obtained in reasonable times in all the scenarios. Figure 5: Comparison between correlation coefficients of the initial parameters and the default parameters in Matlab.