Algorithm Based on Morphological Operators for Shortness Path Planning
Abstract
:1. Introduction
2. Morphological Search Framework
2.1. Preliminary Concepts
2.2. Prior Definitions
2.3. Search Criterion
Algorithm 1 Morphological search with the constraint of and |
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Algorithm 2 Morphological search with the constraint of |
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2.4. Implementation Issues
3. Experimental Analysis and Results
3.1. Experimental Process
- Testing the location of the best trajectory from a grid topology. This experiment uses the city map encoded as a single binary image that represents a grid space. The starting and final path trajectories are selected according to the most representative topologies. The graph belongs to a part of Querétaro City. This graph is the result of automatic graph generation from a satellite map via avenue segmentation. The scale and resolution refer to the real dimensions of maps expressed in terms of pixels.
- The location of the best trajectory from a graph space. In this stage, the city is encoded as a graph after computing the avenue intersections as nodes. The tested scenario becomes the same as that used in the last point. The weighting considers the distance in a metric space to weigh the graph. The coding process uses the vertex detection since the detection of the maximum values of the distance transformation is given by the segmentation of the road into a connected map. All maxima detected represent the nodes, and the connection arcs result from the connected avenues. Each vertex represents an avenue intersection.
- A comparison with other well-accepted approaches. This process evaluates another search method as a reference. The comparison is considered in terms of their computational complexity.
3.2. Results
4. Discussion
5. Conclusions
Further Works
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Minimum Path Trajectory Step Demonstration
- Base case: Consider that D becomes reachable in one step forward by the dilation-induced lattice. Algorithm 2 reduces to the one-time application of the dilation operator. Let , and ; this is true. If the intersection is the empty set, D is not reachable in one step. Consequently, this is the minimum path step forward in the lattice. Considering that the solution might not be unique, one or more nodes in one step forward reach the destination.
- Generalized case: Consider the situation in which D becomes reachable in k step forward. Algorithm 2 reduces to the application of the dilation operator k times. Let , and ; this is true. Consider, by contradiction, that there is no optimal path. The algorithm reconstructs the path trajectory by the one-step-forward navigation of the natural order generated by the successive dilation operation. A better step trajectory must travel by another path that advances in the lattice, resulting in more than one step. Consequently, this becomes false because it does not follow the one-step-forward travel condition. Similarly, as in the base case, the path trajectory might not be unique; however, the steps to reach the destination are the minimum in all possible paths.
- Whenever G becomes a weighted graph. In the case of a weighted graph, consider the same demonstration considering the weighted step forward as the coding of the weighted connection via the introduction of a function , defined as follows:
- Whenever . The structural element represents an isomorphism with a connection order smaller than the maximum connection order of the graph. Then, for a given reference as , a successive is equivalent, i.e., requires, in the worst case, , where is the round operator. The number of extra steps required has, as the lower bound, one (the optimal case), and, as the upper bound, the factor , which denotes the worst case of dilating each note times.
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Approach | Workspace | Complexity | Weighted | Remarks |
---|---|---|---|---|
Dijkstra with referenced graph [10] | Graph | Yes | An algorithm based on reducing the search space using the information of the most common route taken by users. | |
Node mixing approach [40] | Grid | Yes | The approach uses the information of a simplified matrix, starting from previously visited nodes, weighting those nodes that are most representative. | |
Dijkstra enhancement [41] | Graph | Yes | Dijkstra enhancement algorithm setting an upper bound connection criterion () in a huge spare net. | |
Dijkstra enhancement [4] | Graph | Yes | Local search into adjacent node sorting under connectivity criteria and weight cost, reducing the Dijkstra search space. The sorting criterion introduces the priority and computability feasibility of the possible path. | |
Local search with Levenshtein distance [42] | Graph | Yes | An approach based on a local search aided by the Levenshtein distance to quantify the dissimilarity among graphs. The dissimilarity is used to prioritize in terms of cost the possible path equivalences. | |
Contextual heuristic based on local constraints [43] | Graph | No | Introduces a framework to locate the best path trajectories in a net via the introduction of local heuristics and constraints based on the process functionality. | |
Dijkstra enhancement [44] | Graph&Grid | No | Improves the Dijkstra algorithm via the enhancement of the data structure for the storage and searching of the sorted nodes. The matrix node combination is expressed as a single list for better indexing. | |
Morphological search (proposed) | Graph | Yes | It defines a general framework based on mathematical morphology operators and introduces two new operators, which help to build a natural relation for the most reliable path. |
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Perez-Ramos, J.L.; Ramirez-Rosales, S.; Canton-Enriquez, D.; Diaz-Jimenez, L.A.; Xicotencatl-Ramirez, G.; Herrera-Navarro, A.M.; Jimenez-Hernandez, H. Algorithm Based on Morphological Operators for Shortness Path Planning. Algorithms 2024, 17, 184. https://doi.org/10.3390/a17050184
Perez-Ramos JL, Ramirez-Rosales S, Canton-Enriquez D, Diaz-Jimenez LA, Xicotencatl-Ramirez G, Herrera-Navarro AM, Jimenez-Hernandez H. Algorithm Based on Morphological Operators for Shortness Path Planning. Algorithms. 2024; 17(5):184. https://doi.org/10.3390/a17050184
Chicago/Turabian StylePerez-Ramos, Jorge L., Selene Ramirez-Rosales, Daniel Canton-Enriquez, Luis A. Diaz-Jimenez, Gabriela Xicotencatl-Ramirez, Ana M. Herrera-Navarro, and Hugo Jimenez-Hernandez. 2024. "Algorithm Based on Morphological Operators for Shortness Path Planning" Algorithms 17, no. 5: 184. https://doi.org/10.3390/a17050184
APA StylePerez-Ramos, J. L., Ramirez-Rosales, S., Canton-Enriquez, D., Diaz-Jimenez, L. A., Xicotencatl-Ramirez, G., Herrera-Navarro, A. M., & Jimenez-Hernandez, H. (2024). Algorithm Based on Morphological Operators for Shortness Path Planning. Algorithms, 17(5), 184. https://doi.org/10.3390/a17050184