# Reconstructing Nerve Structures from Unorganized Points
## 2 Related Work  
During the last several decades, numerous methods dedicated to line reconstruction from a set of unstructured points have been presented [11]. In general, these techniques can be classified based on their ability to handle lines, as well as sets of points, of different properties. In this paper, we highlight the most-prominent techniques, differentiated by their responses to the following questions:  
1. Is the point distribution uniform or nonuniform? 2. Are the lines smooth, or do they contain sharp corners? 3. Can the method handle both open and closed curves? Additionally, while some algorithms can reconstruct only a single contour at a time, others are capable of detecting multiple lines. The majority of the techniques focus only on curves without self-intersection. The earlier techniques, such as \(\alpha\)-shapes [12; 13], \(\beta\)-skeletons [14], minimum spanning trees [15], and r-regular shapes [16], were limited to a uniform point distribution and could reconstruct only smooth lines, without sharp corners. The following methods brought improvement in the aspect of working with a nonuniform set of points: Crust [17; 18], Nearest Neighbor Crust (NN-Crust) [19], ConservativeCrust [20], Gathan [21], GathanG [22], Discur [23], and Vicur [24], as well as the algorithms that are based on the Traveling Salesman Problem (TSP) [25; 26]. The Crust algorithm, first introduced in [17], constructs a graph on a planar point set. It is designed for the reconstruction of smooth closed curves, under the condition that the points are dense enough in areas with higher curvature. [PERSON] [18] introduced a simplified version of the original Crust algorithm. Similarly, [PERSON] and [PERSON] [19] proposed their NN-Crust method, which introduces the nearest neighbor technique. The following method Conservative Crust [20] is suitable for the reconstruction of both open and closed lines. However, the biggest difficulty for all the above-mentioned methods is the reconstruction of sharp angles. The problem occurs when two non-adjacent points are nearest neighbors, which leads to their incorrect connection (Figure 1). [PERSON] was the first to offer a solution to this problem [25], developing a reconstruction algorithm that could reconstruct open or closed non-smooth curves. This method, as well as the method proposed by [PERSON] and [PERSON] [26] are based on the Traveling Salesman Problem (TSP). However, due to the way the TSP algorithm works, neither one of those two methods can reconstruct more than a single curve at a time, while the latter method can handle only closed curves. [PERSON] and [PERSON] proposed a method called Gathan [21], which can work with curves that contain sharp corners. The same authors then presented GathanG [22], which guarantees closedcurve reconstruction, with both smooth and corner parts, under specified conditions. [PERSON] and [PERSON] [27] introduced a method able to reconstruct both open and/or closed single or multiple curves with or without sharp corners. However, its limited adoption is often attributed to specific sampling conditions and the lack of open-source code availability. The algorithms Discur [23] and Vicur [24] bring a new perspective, inspired by the way human vision works: nearness (two nearest points are usually connected) and smoothness (the connected points tend to form a smooth curve). Both techniques enable reconstruction of single or multiple open and closed curvesand/or lines with sharp corners. Discur requires a very dense sampling near the sharp corner area, while Vicur brings an advancement in that aspect. The method HNN-Crust, introduced in [28], deals with sampling conditions and improves them. [PERSON] et al. showed that their algorithm requires less-dense sampling than the prior methods of the same type (the Crust family of proximity-based reconstruction algorithms). The main deficiency of this method is it not being able to successfully reconstruct contours with sharp corners or two lines in close proximity.