Image Noise Reduction and Solution of Unconstrained Minimization Problems via New Conjugate Gradient Methods

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors used a quadratic model to derive a weighted CG parameter. They have suggested several choices for the weight parameter as well. Convergence analysis has been carried out and some numerical tests have been conducted on the image denoising problem.

       Although the approach presented in this study seem to be interesting, the manuscript needs further improvements to be qualified as a scientific draft. Below, some comments have been provided:

- I suggest the authors to reorder the keywords based on the subject of this study. Also, MSC should be inserted.

- In the literature “\beta” often signifies the CG parameter. So, I suggest the authors to use another letter instead of “\beta” in (1).

- In the first paragraph Section 1, the authors should discuss that why the CG algorithms are frequently used to address the image processing problems.

- Section 2 needs a short introduction.

- Motivations of the study should be stated with enough details in Section 2.

- I suggest the authors to follow the literature for the notations; for example use “x” instead of “u” in (2) and (4), as done in the Step 3 of the algorithm of page 4. This makes the manuscript more legible.

- The organization of the text should be improved; there exist many short paragraphs that could be merged as well. Generally, the text should be polished.

- It seems that equation (19) needs to be corrected.

- The algorithm presented in page 4 needs some corrections such as “Set d_0=-g_0 and k=0”, and “goto Step 1”.  Also, it needs a proper title.

- I suggest the authors to test their algorithm on a set of unconstrained optimization problems as well.

- It is not clear that how the line search has been performed in the numerical experiments.  

- The manuscript is not well-referenced; there exist some recent relevant studies that could be cited.

Comments on the Quality of English Language

The organization of the text should be improved; there exist many short paragraphs that could be merged as well. Generally, the text should be polished.

Author Response

The authors used a quadratic model to derive a weighted CG parameter. They have suggested several choices for the weight parameter as well. Convergence analysis has been carried out and some numerical tests have been conducted on the image denoising problem. Although the approach presented in this study seems to be interesting, the manuscript needs further improvements to be qualified as a scientific draft. Below, some comments have been provided:

Comment 1: I suggest the authors reorder the keywords based on the subject of this study. Also, MSC should be inserted.

 

Response: The authors appreciate the reviewer for the observation. The keywords have been reordered as suggested and the MSC numbers have also been included.

 

Comment 2: In the literature “\beta” often signifies the CG parameter. So, I suggest the authors use another letter instead of “\beta” in (1).

 

Response: The authors appreciate the reviewer for the suggestion.  in (1) has been replaced with .

 

Comment 3: In the first paragraph Section 1, the authors should discuss that why the CG algorithms are frequently used to address the image processing problems.

Response: The authors appreciate the review for the comment. A paragraph has been added based on this comment.

Comment 4: Section 2 needs a short introduction.

Response: An introduction has been added based on this comment.

Comment 5: Motivations of the study should be stated with enough details in Section 2.

Response: The motivation is highlighted in a more detailed form.

Comment 6:  I suggest the authors follow the literature for the notations; for example, use “x” instead of “u” in (2) and (4), as done in the Step 3 of the algorithm of page 4. This makes the manuscript more legible.

Response: The authors have addressed the issue as suggested.

Comment 7:  The organization of the text should be improved; there exist many short paragraphs that could be merged as well. Generally, the text should be polished.

Response: The authors have proofread the manuscript and made necessary corrections.

Comment 8:  It seems that equation (19) needs to be corrected.

Response: The correction is on (18), because we have

(17)

Substituting  , in (17) yields:

(18)

we mistakenly wrote . Now, it has been corrected.

Comment 9:  The algorithm presented in page 4 needs some corrections such as “Set d_0=-g_0 and k=0”, and “go to Step 1”.  Also, it needs a proper title.

Response: The authors have implemented the necessary corrections as suggested.

Comment 10: I suggest the authors test their algorithm on a set of unconstrained optimization problems as well.

Response: The authors have evaluated the performance of the proposed methods on unconstrained optimization problems.

Comment 11: It is not clear how the line search has been performed in the numerical experiments.  

Response: The numerical results on unconstrained optimization problems have demonstrated the efficiency of the proposed direction based on Wolfe line search.

Comment 12: The manuscript is not well-referenced; there exist some recent relevant studies that could be cited.

Response: Recent studies on this topic have been cited as suggested.

Comment 13: Comments on the Quality of English Language. The organization of the text should be improved; there exist many short paragraphs that could be merged as well. Generally, the text should be polished.

Response: The authors have proofread the manuscript to address this issue.

 

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript presents a new approach to the Conjugate Gradient (CG) method, focusing on its application in image processing. While the authors claim significant improvements over traditional methods, particularly the Fletcher-Reeves (FR) method, the manuscript lacks sufficient detail and robust evidence to support these claims. Given the high standards of "Mathematics", which emphasizes both theoretical rigor and practical applicability, the current state of the manuscript does not meet the necessary criteria for publication. 

1. The manuscript does not provide a thorough theoretical basis for the new CG method. The authors must clearly articulate the mathematical derivations and theoretical justifications for the proposed method, including a detailed explanation of how the new method differs from existing CG techniques.

2. The claims of superior performance are not adequately supported by the experimental results presented.   The proposed method in this paper should be applied to solve unconstrained optimization problems to verify its effectiveness and superiority.

3. The manuscript mentions a comparison with the FR method but does not provide a detailed comparative analysis.  The authors should expand on this by including more state-of-the-art CG methods in their comparison.

4. The manuscript's presentation could be improved. The authors should reorganize the content to ensure a logical flow and clear presentation of ideas. Additionally, the clarity of the language and the precision of the technical terms need to be enhanced to meet the journal's standards. 

Author Response

Comment 1: The manuscript does not provide a thorough theoretical basis for the new CG method. The authors must clearly articulate the mathematical derivations and theoretical justifications for the proposed method, including a detailed explanation of how the new method differs from existing CG techniques.

Response: Thank you for your insightful feedback. We recognize the importance of a solid theoretical foundation for our new CG method. In response, we revised the manuscript to include comprehensive mathematical derivations and theoretical justifications. We also provide a detailed explanation of how our method differs from existing CG techniques, emphasizing the unique aspects and advantages of our approach.

Comment 2: The claims of superior performance are not adequately supported by the experimental results presented. The proposed method in this paper should be applied to solve unconstrained optimization problems to verify its effectiveness and superiority.

Response:  We appreciate your comments regarding the experimental validation of our method. We will address this by including additional experiments where our proposed method is applied to a wider range of unconstrained optimization problems. This will help to more robustly verify its effectiveness and superiority. We will also provide a more detailed analysis of the results to support our claims.

Comment 3: The manuscript mentions a comparison with the FR method but does not provide a detailed comparative analysis. The authors should expand on this by including more state-of-the-art CG methods in their comparison.

Response: Thank you for pointing out the need for a more comprehensive comparative analysis. We will expand our comparison to include additional state-of-the-art CG methods. This will involve a thorough evaluation and discussion of the performance of our method relative to these techniques, providing a clearer context for its advantages and limitations.

Comment 4: The manuscript's presentation could be improved. The authors should reorganize the content to ensure a logical flow and clear presentation of ideas. Additionally, the clarity of the language and the precision of the technical terms need to be enhanced to meet the journal's standards.

Response: We appreciate your suggestions for improving the manuscript's presentation. We will reorganize the content to ensure a more logical flow and clearer presentation of ideas. Furthermore, we will carefully review the language and technical terms used in the manuscript to enhance clarity and precision, aligning with the journal's standards. Thank you for your constructive feedback.

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript can be considered for publication if the authors can address the above comments. I am interested in looking at the revised version of this manuscript.

Comments for author File: Comments.pdf

Author Response

Summary of the paper This paper proposes a series of new conjugate gradient coefficients for gray image restoration. And the proposed method has global convergence under the Wolfe condition. Numerical results show that the proposed method performs well on the number of iterations and function evaluations.

Comment 1: The manuscript needs to be proofread before submitting the revision.

Response: The authors appreciate the reviewer for the observation. The manuscript has been proofread as suggested.

 

Comment 2: Delete the section “0. How to Use This Template” on page 1.

Response: The authors appreciate the reviewer for the observation. The section has been deleted.

 

Comment 3: All equations should be aligned.

Response: The authors appreciate the reviewer for the observation. All equations have been aligned.

 

Comment 4. Equation numbering is not uniform.

Response: The authors appreciate the reviewer for the observation. The equations have been checked and adjusted.

 

Comment 5. The equation of (1) on image restoration problem is not properly defined. Check and update the equation.

Response: The authors appreciate the reviewer for the observation. The equation has been modified.

 

Comment 6. The first Wolfe Equation in (11) should be on one line.

Response: The authors appreciate the reviewer for the observation. The equation has been adjusted.

 

Comment 7. Citation format is not unform. For instance, check line 80 and 89. Kindly update based on the journal template.

Response: The authors appreciate the reviewer for the observation. All citations have been checked and corrected.

 

Comment 8. I suggest that NI and NF can be demonstrated by curves.

Response: The authors appreciate the reviewer for the observation. The study has added results for unconstrained optimization where the metrics including NOI, NOF, and CPU time are plotted using performance profile curved.

 

Comment 9. I suggest the authors to follow the literature for the notations; for example, use “x” instead of “u” in (2) and (4), as done in the Step 3 of the algorithm of page 4. This makes the manuscript more legible.

Response: The authors appreciate the reviewer for the observation. The authors have addressed this issue.

 

Comment 10. Why is the convergence proof missing?

Response: The authors appreciate the reviewer for the observation. Since the proposed method can be reduced to some classical methods available in literature whose convergence have been established, proving the convergence would be redundant and thus, omitted.

 

Comment 11. In Section 5, there is a lack of text description for Figures 1, 2, 3 and 4.

Response: The authors appreciate the reviewer for the observation. Description about the figures have been updated.

 

Comment 12. I suggest the authors reorder the keywords based on this study. Also, MS should be inserted.

Response: The authors appreciate the reviewer for the observation. The keywords have been reordered as suggested and the MSC numbers have also been included.

The authors are grateful to all reviewers for their constructive comments and suggestions which have greatly improved the manuscript.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

My comments have been well-addressed. 

Comments on the Quality of English Language

My comments have been well-addressed. 

Reviewer 2 Report

Comments and Suggestions for Authors

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