Impulse Noise Denoising Using Total Variation with Overlapping Group Sparsity and Lp-Pseudo-Norm Shrinkage
Round 1
Reviewer 1 Report
It is an interesting research paper. There are some suggestions for revision.
1. There are some grammar errors, such as line 80 "noise are classified into" -> "noises not noise", line 94 "Because the regularization term" -> "Since not because". Please correct them.
2. Please discuss the following paper as reference for spare representation method.
Z.Zhu, H.Yin, Y.Chai, Y.Li, and G.Qi, "A novel multi-modality image fusion method based on image decomposition and sparse representation", Information Sciences, 432: 516-529 (2018)
3. In introduction, the motivation of this paper is not clear. Please discuss staircase effect and compare the shortcomings of existing solutions. It is better to list the comparison in table.
4. Please highlight the contributions of this paper in introduction.
5. Please modify the notations in equation 9-14, 25-36. It is difficult to read the notations.
6. It does not show clearly how to set parameter theta and mu in equations.
7. Paragraph 3 discusses OGS-L1 and OSG_Lp. Please discuss the relationship between proposed method and OGS-L1/OSG_Lp.
8. Paragraph 4 shows the solving process of OSG_Lp and OSG-Lp_Fast. Please highlight your contributions. Explain whether existing solutions can solve OSG_Lp and OSG-Lp_Fast. If yes, what are the shortcomings.
9. The relationship between OSG_Lp and OSG-Lp_Fast is not clear. Please disucss which method is better in different scenarios.
10. In experiment section, please list where you get the original images.
Author Response
Thanks for giving us an opportunity to revise our manuscript. We appreciate the Editors and Reviewers for your profound comments and suggestions on the manuscript entitled “Impulse Noise Denoising Using Total variation with Overlapping Group Sparsity and Lp Pseudo-Norm Shrinkage” (Manuscript ID：applsci-381172). Those comments are valuable and helpful for improving the quality of our manuscript. We have studied the comments carefully and enclosed a revised manuscript. All the corrections are marked in red color in the current manuscript.
Point 1: There are some grammar errors, such as line 80 "noise are classified into" -> "noises not noise" ,line 94 "Because the regularization term" -> "Since not because". Please correct them.
Response 1: Thank you for your careful review. We have revised our manuscript. In this version , Lines 105. “noise is classified into Gaussian noise” has been modified as “noises are classified into Gaussian noise”. We have carefully reviewed our manuscript and employed “noise” instead of “noises”.
In line 119, "Because the regularization term" has been modified as "Since the regularization term"
Point 2: Please discuss the following paper as reference for spare representation method.
Z.Zhu, H.Yin, Y.Chai, Y.Li, and G.Qi, "A novel multi-modality image fusion method based on image decomposition and sparse representation", Information Sciences, 432: 516-529 (2018)
Response 2: Thank you for your suggestions. We have studied the article carefully, which is instructive to us. We quoted it in the 38 line of the article. The modification is as follows:
[10] Z. Zhu, H. Yin, Y. Chai, Y. Li, and G. Qi, "A novel multi-modality image fusion method based on image decomposition and sparse representation," Information Sciences, vol. 432, pp. 516-529, 2018.
Point 3: In introduction, the motivation of this paper is not clear. Please discuss staircase effect and compare the shortcomings of existing solutions. It is better to list the comparison in table.
Response 3: Thank you for your suggestion. We have revised our manuscript to further discuss the staircase effect and compare the shortcomings of existing solutions. In the current version,
Ø In Lines 41-43, we elaborate the reason of the staircase effect caused by TV model.
Ø In Lines 44-48, the advantages and drawbacks of high-order TV norm are discussed.
Ø In Lines 51-53, the advantages and drawbacks of fractional-order TV norm are discussed.
Although these improved methods can alleviate the staircase artifacts, they might lead to “spots” effects on the processed image.
The motivation of this paper is (1) To choose a good regularization functional that balance the staircase artifacts and “spots” effects. (2) To select a better fidelity term to describe sparsity features of impulse noise. (3) To propose an algorithm to solve our model effectively. The presentation can be found in Lines 55-57, Lines 63-84.
Point 4: Please highlight the contributions of this paper in introduction.
Response 4: Thank you for your suggestion. We have revised our manuscript, highlight the contributions of this paper. The presentation can be found in Lines 85-97
Point 5: Please modify the notations in equation 9-14, 25-36. It is difficult to read the notations
Response 5: Thank you for your careful review. In this edition, we have carefully reviewed our manuscript, revised the notations in the equations 9-14, 25-36. We also add a detailed description of some equations which make them more readable.
Point 6: It does not show clearly how to set parameter theta and mu in equations.
Response 6: Thank you for your suggestion, We have revised our manuscript to show how to set parameter and Related modifications can be found in 341-353. The optimal parameters for different images with the noise level form 20% to %50 are given in Table 1.
Point 7: Paragraph 3 discusses OGS-L1 and OSG_Lp. Please discuss the relationship between proposed method and OGS-L1/OSG_Lp.
Response 7: Thank you for your suggestion. We have revised our manuscript to further discuss the relationship between proposed method and OGS-L1/OSG_Lp. L1-norm is commonly used as the fidelity term of impulse noise. However, the L1-norm is only the convex relaxation of L0-norm. The power of Lp-norm (,, for simplicity, we name Lp-pseudo-norm) is another relaxation of L0-norm. In fact, L1-norm constraint is a particular case of Lp-pseudo-norm. The advantages of Lp-quasinorm regularization are listed as follows. (1) The LpS operator may converge to an accurate solution. (2) The Lp-quasinorm is more flexible than L1-norm. This might be useful to adapt the degree of sparsity to the signal being processed. (3) The Lp-quasinorm feasible domain makes the solution robust to noise. Related representation can be found in 140-153, 167-78.
Point 8: Paragraph 4 shows the solving process of OSG_Lp and OSG-Lp_Fast. Please highlight your contributions. Explain whether existing solutions can solve OSG_Lp and OSG-Lp_Fast. If yes, what are the shortcomings.
Response 8: Thank you for your suggestion. The denoising models based on OGS is more time-consuming than the TV-based model. This is mainly because that OGS model considers the gradient information of the neighborhood in a reconstructed image, thus making the computation more complex. Also since the rate of convergence of typical ADMM is , the convergence speed of ADMM is unsatisfactory. So we need to improve the efficiency of OSG_Lp.The paper [12] proposed an accelerated ADMM algorithm with restart that improves the convergence rate of the ADMM algorithm from to . Inspired by that, we adopted this algorithm to improve the OGS_LP model. This modified model is named OGS-Lp-FAST. Related representation can be found in 273-279.
Point 9: The relationship between OSG_Lp and OSG-Lp_Fast is not clear. Please disucss which method is better in different scenarios.
Response 9: Thank you for your suggestion. We have revised our manuscript to further discuss the relationship between OSG_Lp and OSG-Lp_Fast. The related presentation can be found in Line 273-279. Experiments show that OSG_Lp and OSG_Lp_Fast model generate higher PSNR and SSIM values for all the reconstructed images than other methods, indicating its superior denoising effect. Further comparing the values of PSNR and SSIM in the Table2-5, OGS_Lp_Fast and OGS_Lp have the same denoising effect. However, by observing the value of runtime of all testing images, we find that convergence is sped up in the OGS_LP_Fast method by using the accelerated ADMM with a restart. Related modifications can be found in 392-397.
Point 10: In experiment section, please list where you get the original images.
Response 10: Thank you for your careful review. We have revised our manuscript. In the current version, we list the source of the original images, Related modifications can be found in 303-309
Author Response File: 
Author Response.pdf
Reviewer 2 Report
two-dimensional total variation with overlapping group sparsity (OGS-TV) is applied to images with impulse noise. Accelerated alternating direction method of multipliers (ADMM) and Fourier transform is introduced to transform the matrix operation from spatial domain to frequency domain, improving the efficiency.
Lp with p<1 is shown to lead to a more robust model by Fig. 2.
Z need not be redefined in line 150
The indices above Z (0/k) are not clearly displayed
In eq. (9) the subtraction e.g. of Z1-Kh*F isn’t 0? In the way eq. (9) has been expressed this clearly leads to 0 in all Z subtractions. Unless you mean eg. Z(k+1) or something that leads to different value for e.g., Z1 and Kh*F
Eq. (10) is almost unreadable. I cannot understand how it has been derived
In line 180 do you mean the “equality”?
In eq. 22 you estimate v(k+1) from v(k). But you also have v-v0 in the norm. What is v?
Dot product symbol in eq. 28 is not distinguishable from the letter “o”. Better use a different symbol.
In step 7 of Algorithm 1 reference error appears
My main difficulty is to follow the mathematical model of the proposed method. Please justify more clearly how you derive each equation.
The experimental results show that the proposed methods is more time consuming but achieves better PSNR, SSIM especially in low noise levels.
Improve expressions in Lines 79-80 (“noise are classified”) 104: “With”
Author Response
Dear Editors and Reviewers:
Thanks for giving us an opportunity to revise our manuscript. We appreciate the Editors and Reviewers for your profound comments and suggestions on the manuscript entitled “Impulse Noise Denoising Using Total variation with Overlapping Group Sparsity and Lp Pseudo-Norm Shrinkage” (Manuscript ID：applsci-381172). Those comments are valuable and helpful for improving the quality of our manuscript. We have studied the comments carefully and enclosed a revised manuscript. All the corrections are marked in red color in the current manuscript.Please refer to the attachment for detailed reply.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
All my concerns are addressed. It is ready for publication.
Reviewer 2 Report
The authors have explained several points that I had difficulty to understand in the original paper.
They also corrected several points in the notation and symbols used in the mathematical model.
