# Algorithm Based on Morphological Operators for Shortness Path Planning
## 3 Experimental Analysis and Results
### Results
The grid representation becomes practical whenever the dimensions of the maps become smaller because the computational resources have increased. Meanwhile, a graph space representation has a non-uniform connection order. Figure 3: Example of searching algorithm with (**a**) avenue scenario with node origin and destination; (**b**) frequency with which dilation is applied; and (**c**) dilations required to reach the goal. Testing the location of the best trajectory from a graph spaceAn encoding process is required to convert a geographical map into a graph. The encoding process takes as input geographical information such that avenues/crossroads are encoded as a graph. The encoding process takes the intersections as nodes and the avenues as arcs. The detection is performed with the distance transformation and maximum zones detected. This representation is a compact representation of the geographical avenue layer with the advantage of requiring minimal computational resources. The avenue intersections denote the local maxima distances in a distance transformation. The positions of all maxima are the result of applying the extended morphological distance. The process of coding the image in a graph is illustrated in Figure 5, where, starting from (a), a geographic binary map layer and (c) the distance transform, considering the avenue as a connected surface, we can compute the minimum distance between each active pixel and the nearest border. The distance transform is illustrated in pseudo-color, where dark zones correspond to small distances, and white zones represent large distances. Consequently, Figure 5d shows the process of maximum detection. The resolution and the thinness of the avenue representation might affect the maxima detection by merging two close maxima. The maximum detection zones are obtained through morphologically extended maxima transformation. This transformation offers a robust process to detect maxima on discrete surfaces. Finally, Figure 5b illustrates the graph representation once the maximum is detected. The connectivity results from the removal of the dilated maximum zones detected. Each arc detected is dilated and the maximum regions that overlap are tested. In a weighted version, the geographical distance might require the conversion of pixel distances into metric units. Similarly, the graph generated from the avenue information is encoded as a graph space. Figure 6 illustrates an example of the searching path in the graph space resulting from encoding a geographical city map. Figure 6 illustrates two positions that represent the start \(x_{0}\) and destination to reach \(D\). In the following graphs, Figure 6b illustrates the covering process, representing the city as a graph, where each node is accompanied by the frequency by which it is dilated. The frequency defines the natural order relation that reconstructs the shortness trajectory in terms of jumps. The solution is marked in red once the destination reaches \(D\). In this particular case, it takes 17 iterations (dilations) to reach the destination. Note that the proposed framework has the advantage of representing a width-first search, resulting in faster convergence. A comparison with other well-accepted approachesThe proposed method matches with well-accepted approaches from the literature. Table 1 shows the main features used to compute the best trajectory. The best path trajectory, as an open task, has three main features to denote: (1) the search space; (2) local constraints for specific environments; and (3) the path searching approach. Figure 4: Multiple solutions for best path trajectory. (**a**) shows the different solutions located; (**b**) the frequency is illustrated as level curves. The above features are discussed in the following paragraph and allow the clarification of the contributions of our proposal. Case (1) can be divided into two classes (workspace column in Table 1).