## 4 Results and Discussion  
We use a dataset out of 1309 STL files collected from ShapeNet (The Princeton Shape Benchmark (PSB), Version 1) [78; 79], Princeton ModelNet 40-Class Subset [80], Purdue Engineering Shape Benchmark (ESB) [81], and internet websites. Every model used has a correct geometry. The extrema amplification process described in Section 3.10 (Method 2) generates the most prominent extrema. The amplification includes summation, which is performed per node for all the shells, looking for the extrema and saddle points, and building the convex hull using those points. We expect that the sum assumes a certain amount of independence of the multiple components (as discussed in Section 3.9 (Method 1)). We ran the method described on the database of 3D objects for 1309 objects. It was found that in 94% of cases, the first attempt (performed by steps A5-A7 of the algorithm) to align the objects using the amplified extrema was immediately successful. Five attempts were enough to achieve 99% success. The maximum number of attempts was set to 30, and it allowed us to find the solution in all the cases. Let us find the rationale for this small number of attempts to find the solution. For each object of the dataset mentioned, we calculate the values  
\[f_{\text{spread}}=\sum_{i=1}^{n}\left|\widetilde{f}_{\text{max}}^{(i)}- \widetilde{f}_{\text{min}}^{(i)}\right|,\qquad\widetilde{f}_{\text{spread}} ^{\sim}=\left|\widetilde{f}_{\text{max}}^{\sim}-\widetilde{f}_{\text{min}}\right|\]  
The first one is the sum of the spreads of functions \(\widetilde{f}^{\sim}\) defined by (1), and the second one is the spread of the final sum \(\widetilde{f}\); see Formula (2). Let us consider the ratio \(\widetilde{f}_{\text{spread}}/f_{\text{spread}}\) for the objects of the dataset. It turned out that the expectation and variance of this ratio are 0.90 and 0.08, respectively. It can be interpreted that in 90 percent of cases, there is at least one extremum on each sphere with the same angular coordinates. The total number of extrema points in the last bins near \(\widetilde{f}_{\text{max}}\) and \(\widetilde{f}_{\text{min}}\) appeared following two trends. If the object does not have symmetrical (repeatable) parts, the number of points in the last bins is less than 30 on average. It corresponds to approximately 0.27% of all the points (recall that we use an icosphere with 10,242 vertices). The second trend manifests itself when the number of points in the last bins is relatively high, and this number can come to above two hundred. This situation appears when some parts of the object are similar or the object has some symmetry (for example, rotational symmetry). These symmetrical/similar parts may contribute a large number of extrema points of the convex hull to the last bins. This trend appears when all right-part probabilities are high in inequality (12). If there is some rotational self-similarity of the object, this number of extrema could be rather large, but all these extrema will define the same transformation of the 3D objects. From this reasoning, in both cases, we have to try matching only a limited number of the extrema to check the object mapping. The results of running the method on a dataset demonstrate that a few attempts to match the objects (steps A5-A7) are enough to conclude whether the object is the same or different. Due to grid sampling of the square-integrable functions \(f\) and \(g\) from (4), the matching could have some errors, and the best matching using metrics \(M_{1}\) and \(M_{2}\) (step A7) will define the best acceptable solution (if found). We propose an algorithm to identify similar objects. It is a non-iterative and non-gradient method of optimization. The current state-of-the-art mapping algorithms are the ICP and Go-ICP.