# Reconstructing Nerve Structures from Unorganized Points
## 5 Results and Discussion
(**A**,**A**) Input points; (**B**,**B**\({}^{\prime}\)) reconstructions accomplished using our method; (**C**,**C**\({}^{\prime}\)) reconstructions performed through the HNN-Crust algorithm. The reconstructions performed by HNN-Crust in (**C**) and especially in (**C**\({}^{\prime}\)) exhibit interruptions along the contours and an overall inability to clearly distinguish two adjacent contours. A comparative analysis between our proposed method, HNN-Crust, and the Vicur algorithm is presented in Table 1. Given that we observed minimal variation among the three methods except in instances of fascicle branching, we opted to specifically analyze and present statistics for fascicles exhibiting branching throughout their length. The results represent an average taken from all the cross-sections of such fascicles. This indicates that, while most of the analyzed contours are straightforward, a few within each fascicle are intricate, characterized by sharp corners and notably closed contours. The first column displays the Root-Mean-Squared Error (RMSE) values of each of the methods in comparison to the expert curve. The second column indicates the percentage of points that were omitted during reconstruction, relative to the total number of points in the initial set. The third column presents the count of unintended gaps or interruptions in the reconstructed curve. Table 1 reveals that, when considering the RMSE, Vicur and our method produce closely aligned results. However, in the context of missed points, Vicur excels over both our strategy and the HNN-Crust approach, seamlessly retaining every point from the input set. This is, in fact, the main liability of our method. Conversely, in terms of faithfully reconstructing continuous curves, both HNN-Crust and Vicur demonstrate inconsistencies, especially in the scenarios illustrated in Figures 14 and 15. Emphasizing the importance of the precise reconstruction of neighboring contours, particularly fascicle branches, Table 2 showcases statistical data from realistic nerve models derived from [29] relevant to this study. The data reveal the count of fascicles recognized in the available nerve sections, as well as the overall number of fascicle branchings occurring along their paths. Notably, about a third of all detected fascicles in the examined nerve sections exhibited at least one branching, with some presenting multiple branches. It us noteworthy that a heightened resolution in the count of cross-sectional layers amplifies the chances of encountering such scenarios, as depicted in Figures 14 and 15. While the presented algorithm effectively performs reconstructions for its intended and similar applications, it is important to acknowledge its potential limitations:  
1. Given that our algorithm adopts the valley-seeking clustering method, the primary limitation of our approach lies in its inherent characteristic of omitting certain points deemed non-essential for reconstruction. Hence, it is not particularly tailored for reconstructions reliant on a small number of points. In such cases, there is an increased risk of overlooking or disregarding points that the algorithm determines as not aligning with the intended \"line\" (see Figure 16). However, this shortcoming diminishes with reconstructions based on a larger point set. 2. Considering the sensitivity of the results to the selection of specific parameter values (\(k\), defined in Equation (9), \(\alpha\), from Equation (30), or \(\theta\), from Equation (31)), it is important to reevaluate and adjust these parameter values when dealing with a significantly \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **RMSE** & **\% of Omitted Points** & **No. of Gaps per Fascicle** \\ \hline our method & 0.020 & 1.94 (\%) & 0 \\ HNN-Crust & 0.043 & 0.4 (\%) & 9.5 \\ Vicur & 0.019 & 0 (\%) & 0.2 \\ \hline \hline \end{tabular}  
* Data for branching fascicles.  
\end{table} 
Table 1: Statistical comparison between methods *.