# Algorithm Based on Morphological Operators for Shortness Path Planning
## 3 Experimental Analysis and Results
### Results  
_The location of the best trajectory from a grid topology._ The neighborhood might represent the adjacent local connections of a given node, and it defines the structural element. Thisrepresentation has the advantage that the grid uniformity might refer directly to the uniform sampling of a physical plane space representation. Figure 3 illustrates the grid space that conforms to a graph search space. The uniformity of the structure of the connectivity of each node allows a direct association with the image pixels. The representation of four or eight connections might be applied to images, being useful for illustration purposes. The avenue's directionality is bidirectional. In concrete scenarios, the directionality becomes constrained to the extra layer of information. The first step consists of selecting the initial (the \(x_{0}\) graph) and the destination (the \(D\) graph) positions in the graph (see Figure 3a). In this case, the origin \(x_{0}\) and the destination \(D\) correspond to the same connected zone, and its intersection is the null set. The searching algorithm has two principal subprocesses: (1) the morphological covering of the space until, from a given reference, the destination is reached (see Figure 3b). This process generates the dilation operator several times, where, with each iteration, the covering graph increases. The successive dilations represent the searching criterion, such that it is closely similar to a first-width approach. The approach performs a simultaneous path search, expanding in terms of the topology of the structural element. In our example, the border nodes are illustrated in white and represent the dilating nodes. Each node of the border represents a possible path search to locate the destination. With each successive iteration, the node of the border is incremented one step forward according to the space topology. Hence, these increments mean that the border nodes might be divided, merged, or fused. When the border is divided, the approach introduces one possible path. Whenever two paths are joined, the dilation frequency defines which of the two paths is more reliable. Note that when the space limits are reached, a path solution does not exist. This algorithm's behavior allows the implementation of parallel approaches to benefit the performance. Finally, the first algorithm ends whenever the destination \(D\) is reached, meaning that the covering operators introduce a morphological order expressed by a lattice as all covering subgraphs cover the space. Consequently, the frequency of dilations applied to each node induces an order relation. The infimum corresponds to the destination \(D\) node, denoting the origin. The order in the step-forward natural direction of the frequency dilation nodes represents the possible path trajectories, with the cost associated with reaching the reference \(x_{0}\). The subgraph \(x_{0}\) represents the most dilated nodes, denoting the supremum of the total order. The solution conforms from the subgraph \(D\) to the subgraph \(x_{0}\) direction following the frequency of dilations. Sometimes, the solution might not be unique (see Figure 3), and they might all require the same number of steps to reach the destination from the origin; thus, equivalent minimal path trajectories are provided. For multiple solutions, we focus on the red rectangle marked at the center of the last frame in Figure 3c. This figure augments Figure 4, where (a) shows the different solutions located, and, in (b), the frequency is illustrated as level curves. The red arrows represent the different path trajectories. The solid-arrow solution navigates via the lower rectangle and the dotted-arrow solution via the upper rectangle. Both solutions have an equal cost in traveling from the reference to the destination; it is possible that the cost of traveling via the first and second solutions is the same. Similarly, scenarios with multiple solutions can locate all solutions simultaneously.