# Enhanced Hybrid Algorithms for Inverse Problem Solutions in Computed Tomography
## 5 Discussion
This applies to both the hybrid with the Nelder-Mead and the Hook-Jeeves methods. For example, comparing Figure 19 (Example 1) and Figure 29 (Example 2) clearly shows that, in the case of the hybrid algorithm, the circles cover the identified shape much more effectively. Figures 31 and 32 present a comparison of the ABC algorithm with the hybrid algorithm in terms of the fitness function values and execution time. The quality of the solutions provided by the hybrid approach ranged from nearly identical to slightly better (in terms of fitness function values), while the time needed to find these solutions was noticeably lower in the case of the hybrid algorithm. In Example 3, the number of polygons was increased to two, which complicates the problem (the search space dimension increases). The goal of this example was to further test the proposed hybrid algorithm. Similar to the examples in Sections 4.1 and 4.2, the results obtained are presented in the form of Figures 34-43. These figures show the fitting of circles to the polygon and the level of energy absorption. Notably, the overall performance of both algorithms was similar, with the Hooke-Jeeves performing slightly better. In particular, the cases using 1, 2, and 3 circles yielded very similar results for both the N-M and H-J methods. Greater differences in the results can be observed in the cases with 4 and 5 circles. For example, Figure 42 shows the arrangement of circles obtained with the two methods. In this case, the hybrid algorithm based on the Hook-Jeeves method proved to be better. In Figures 44 and 45, the fitness function values and computation times for both methods are compared based on the number of circles. Time needed for evaluation remained nearly identical for both algorithms, as well as the fitness function values. By comparing the graphs displaying the fitness function values for the examples in Sections 4.2 and 4.3, it is noticeable that the curve of this graph differs between the scenarios. When identifying a single polygon, the fitness value decreases relatively steadily as the number of circles increases. Additionally, as more circles are used, the original polygon is covered with increasing accuracy. In contrast, with two polygons, the fitness value drops sharply when comparing one and two circles, with the value for two circles being roughly one-fifth that for one circle. However, after this point, the decrease slows significantly, becoming almost stagnant. This is also evident in the figures visualizing the test results--the accuracy of the identification does not noticeably improve with more than two circles, and the recreation with five circles even appears slightly less accurate. After conducting the tests, we decided to increase the number of iterations for the ABC algorithm to 20 and attempt once again to recreate the polygons using five circles, given the initially poor performance. Figure 46 shows the arrangement of circles after extending the runtime of the metaheuristic algorithm. Comparing it to the analogous Figure 42, a significant improvement in the results is easily noticeable. The obtained results show a slight improvement in the fitness function value, but the recreation appears significantly more accurate. However, one of the circles remains distant from the original polygons, suggesting that using five circles may be excessive for identifying these polygons. On the other hand, this circle has a very small radius and seems to have little impact on the overall result. To sum up, after proper tuning, the algorithm successfully handles the problem of identifying two polygons (see Figures 47-49). Figure 36: Recreation of the polygons by hybrid algorithm using 2 circles. Figure 37: Comparison of the energy absorption values of the polygons and their respective recreations. Figure 38: Recreation of the polygons by hybrid algorithm using 3 circles. Figure 41: Comparison of the energy absorption values of the polygons and their respective recreations. Figure 40: Recreation of the polygons by hybrid algorithm using 4 circles.