Algorithm Based on Morphological Operators for Shortness Path Planning

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors developed a family of search algorithms for the best trajectory in a graph by means of a mathematical morphology framework. The algorithm builds a lattice over the graph space through morphology operators. The algorithm creates a metric space using the morphological operator to the graph and weighs the cost to travel across the lattice. The cumulative traveling criterion leads to the determination of the optimal path trajectory for cost minimization. The authors employ a test to validate the framework functionality. A sample of the application is used for validating the functionality.

The authors should improve the following issues in the revised version of their manuscript:

The authors’ innovation should be better explained at the end of Introduction.

The paper’s structure should be explained at the end of Introduction section.

Figures 3 to 6: Are these produced in Matlab or other software environment?

The Conclusions section could include some type of quantitative evaluation of the authors’ algorithms performance (convergence etc) against the competition.

Comments on the Quality of English Language

Only minor improvements required

Author Response

Authors Response

The comments and questions about the paper were answered. The strategy to ask each one of the comments were the following: To each comment, we responses individually the modifications and actions performed into the paper to improve the comment done.

Reviewer 1

Comments and Suggestions for Authors

The authors developed a family of search algorithms for the best trajectory in a graph by means of a mathematical morphology framework. The algorithm builds a lattice over the graph space through morphology operators. The algorithm creates a metric space using the morphological operator to the graph and weighs the cost to travel across the lattice. The cumulative traveling criterion leads to the determination of the optimal path trajectory for cost minimization. The authors employ a test to validate the framework functionality. A sample of the application is used for validating the functionality.

The authors should improve the following issues in the revised version of their manuscript:

1. The authors’ innovation should be better explained at the end of Introduction.
2. The paper’s structure should be explained at the end of Introduction section.

Response 1 and 2: Thanks for commenting. To enhance clarity for readers, we rewrote the introduction and the last paragraphs.

3. Figures 3 to 6: Are these produced in Matlab or other software environment?

Response: Thanks for commenting. The software environment used for implementing the algorithm were MATLAB and GNU Octave. All code was written in M language.

4. The Conclusions section could include some type of quantitative evaluation of the authors’ algorithms performance (convergence etc) against the competition.

Response: Thanks for the message. The conclusion section was improved and rewritten. A quantitative criterion was added in this section.

Comments on the Quality of English Language

Only minor improvements required.

Response: The document was submitted for a professional English review to improve and enhance its English quality. The certificate is attached to the system.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper aims to optimize trajectory selection in a graph using mathematical morphology, creating a lattice over the graph space, and determining the minima/maxima cost path through a searching algorithm based on the cumulative traveling criterion. The paper's subject is interesting and in line with the aims and scope of the Journal. However, some issues need to be addressed.

1.     The abstract should be better written. Try to make it as simple and clear. Provide background (and highlight the significance of the problem, i.e. the motivation behind the study), highlight the proposed methodology and what is novel about previous studies, and point out the main results and conclusions.

2.     The authors should separate the literature review from the introduction. In the introduction keep only the background, highlight research questions, aim(s) of the paper, a short explanation of the proposed approach (highlighting the main novelties and contributions), main results, and conclusions. Also, add a short paragraph explaining the remainder of the paper at the end of the Introduction.

3.     The literature review is seriously outdated. Most of the references are more than 10 years old, and there is no reference from the last 4 years. The authors should update the literature review and highlight the research gaps, something they tried to bridge in this study.

4.     Check the equations and notation. Make sure you defined every symbol used in the equations, and that there are no duplications of notation (use of the same symbol for different things).

5.     The authors should explain the limitations of the proposed approach if there are any.

6.     The authors should discuss the theoretical and practical (managerial) implications of their study. Who and for what can use the results of the study? The authors did mention something regarding this issue in the conclusion, but they should elaborate on this.

7.     The conclusion is scarce. The authors should address the main research questions and highlight the main results, conclusions, contributions, and future research directions in the conclusion (they could also move the last one from the previous section).

8.     English writing should be revised. Some syntax and style errors need to be addressed.

9.     Some technical issues should be addressed:

a) The table caption should be placed above the table, not below.

b) The table is not formatted according to the Instructions for authors.

c)     Figure captions should be short and informative. Some figure captions (e.g. for figures 3 and 6) are too extensive. If you need to describe a figure in more detail, do it in the main text.

d)    Values in Figures 3, 4, 5, and 6 are hardly visible. Increase the font size.

e)     Functions in lines 177 and 191 should also be numbered as other equations.

f)     References in the reference list are formatted according to the Instructions for Authors (e.g. journal names are not abbreviated).

Comments on the Quality of English Language

1.     English writing should be revised. Some syntax and style errors need to be addressed.

Author Response

Reviewer 2

Comments and Suggestions for Authors

The paper aims to optimize trajectory selection in a graph using mathematical morphology, creating a lattice over the graph space, and determining the minima/maxima cost path through a searching algorithm based on the cumulative traveling criterion. The paper's subject is interesting and in line with the aims and scope of the Journal. However, some issues need to be addressed.

1. The abstract should be better written. Try to make it as simple and clear. Provide background (and highlight the significance of the problem, i.e. the motivation behind the study), highlight the proposed methodology and what is novel about previous studies, and point out the main results and conclusions.

Response: Thanks, you are right. The abstract has been rewritten to be clearer and highlight the aspect that you suggested.

2. The authors should separate the literature review from the introduction. In the introduction keep only the background, highlight research questions, aim(s) of the paper, a short explanation of the proposed approach (highlighting the main novelties and contributions), main results, and conclusions. Also, add a short paragraph explaining the remainder of the paper at the end of the Introduction.

Response: Thank you for your help. I wanted to let you know that the introduction has been improved. The section now includes a discussion on the current state-of-the-art, and the aim of the research has been rewritten to better emphasize its contribution. Additionally, the article's description and structure have been revised to help readers better understand how the document is organized.

3. The literature review is seriously outdated. Most of the references are more than 10 years old, and there is no reference from the last 4 years. The authors should update the literature review and highlight the research gaps, something they tried to bridge in this study.

Response: Thank you for your comment, certainly true. After conducting a thorough review and update of the literature, we have incorporated recent references and removed outdated ones. Additionally, certain paragraphs have been rewritten in order to better reflect current context.

4. Check the equations and notation. Make sure you defined every symbol used in the equations, and that there are no duplications of notation (use of the same symbol for different things).

Response: Thank you for your message. We have reviewed the mathematical notation and found some minor details that needed to be fixed. These issues have now been addressed.

5. The authors should explain the limitations of the proposed approach if there are any.

Response: Thank you for commenting. We have reviewed the results and experimental process, rewritten some paragraphs, and denotated the aims and limits of the proposal. Some of these limits were emphasized as further works. So, the further works are now located before the final conclusions.

6. The authors should discuss the theoretical and practical (managerial) implications of their study. Who and for what can use the results of the study? The authors did mention something regarding this issue in the conclusion, but they should elaborate on this.
7. The conclusion is scarce. The authors should address the main research questions and highlight the main results, conclusions, contributions, and future research directions in the conclusion (they could also move the last one from the previous section).

Response 6 and 7: Thank you for your message. I agree with you that the document lacked information on possible applications. Therefore, we have revised and restructured the results section to highlight the complexity of our approach, its equivalence to other existing methods, and its potential impact in various fields such as IA, graph theory, logistic, and embedded systems. We have also included a conclusion that demonstrates quantitatively how our proposal has improved.

8. English writing should be revised. Some syntax and style errors need to be addressed.

Response: The document was submitted for a professional English review to improve and enhance its English quality. The certificate is attached to the system.

9. Some technical issues should be addressed:
10. a) The table caption should be placed above the table, not below.
11. b) The table is not formatted according to the Instructions for authors.
12. c)     Figure captions should be short and informative. Some figure captions (e.g. for figures 3 and 6) are too extensive. If you need to describe a figure in more detail, do it in the main text.
13. d)    Values in Figures 3, 4, 5, and 6 are hardly visible. Increase the font size.
14. e)     Functions in lines 177 and 191 should also be numbered as other equations.
15. f)     References in the reference list are formatted according to the Instructions for Authors (e.g. journal names are not abbreviated).

Response: Thanks for commenting. It is true there are some minor edition details. The table captions were relocated, the figure was fixed and resized to be clearer, and the reader should appreciate the samples better.

The math notations were numbered and the references were written in the correct format.

Comments on the Quality of English Language

1. English writing should be revised. Some syntax and style errors need to be addressed.

Response: The document was submitted for a professional English review to improve and enhance its English quality. The certificate is attached to the system.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have successfully addressed all issues identified in the previous review round, thus significantly improving their paper's quality. Accordingly, I suggest an acceptance of the paper in its present form.