Research on Application of Fractional Calculus Operator in Image Underlying Processing

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Fractional calculus is a powerful tool that can be considered in different scenarios where the standard differential operators cannot extract the necessary information. In particular, in the context of the manuscript, which analyzes image processing. In the manuscript, the authors successfully analyzed fractional calculus theory in image enhancement and denoising. They also showed that the fractional order integral image denoising operator could improve the image's signal-to-noise ratio compared to traditional denoising methods and better preserve detailed information such as edges and textures of the image. The results are very interesting and helpful for researchers on this subject. I only suggest that the authors discuss the possibility of considering the fractional operators of distributed order or with the nonsingular kernels if it is possible to improve their achievements.

Author Response

I would like to express my gratitude to the experts for their valuable revision suggestions. I accept all the expert's modification suggestions. Expert proposes that “I only suggest that the authors discuss the possibility of considering the fractional operators of distributed order or with the nonsingular kernels if it is possible to improve their achievements.” In response to the valuable opinions raised by experts, my answer is as follows.

1. The distribution problem of fractional order in image enhancement and denoising, which is the adaptive fractional order image enhancement and denoising operator. Many researchers have proposed to adaptively determine the order size of the fractional order template based on the local information of the image, in order to achieve better image enhancement and denoising processing results.
2. “With the nonsingular kernels "is a new method for constructing fractional calculus mask operators. Many literature has discussed the method of constructing singular kernels by replacing Caputo or Riemann Liouville derivatives with nonsingular (bounded) kernels, and has achieved some good improvement results. In future research, I will use "the authors discuss the possibility of considering the fractional operators of distributed order or with the nonsingular kernels if it is possible to improve their achievements." as the next step in the application of fractional calculus theory in image enhancement and denoising processing.
3. In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

Reviewer 2 Report

Comments and Suggestions for Authors

This article proposed fractional operator based methodology on image enhancement and denoising. There are a few major problems with this studies despite some results shown as evidence of the proposed methods.

The significance of the proposed work.

The authors need to emphasize a lot more on the advantage of the proposed methods compared to integer order/traditional ones. Figure 8 and Table 1 appear to be the main results supporting the proposed method, but they are not well addressed in the text. For example, on Line 315, it is stated that fractional order enhancement has better gray level equalization and overall image contrast. Are methods used by Figure 7 (d), (e) and (f) the only methods for enhancing? What’s the deeper reason why fractional operator enhanced better? Similar for Table 1, the authors need to explain more on how fractional order works better than other methods. The SNR only shows a marginal advantage.

Moreover, describe the advantage of the proposed method in Introduction section. How is the proposed method compared to all the studies listed in the literature review? How is the proposed method different from them? The authors need to clearly state the innovation/significance of the proposed method.

The writing requires improvement.

Similar captions are used for Figure 5, 6, 7 and 11. Specify what the figures show in the caption.

Phrasing and language requires extensive revision; the entire passage is hard to read.

Support the statement “The application of fractional calculus theory in image processing…” on Line 49 with previous studies.

Comments on the Quality of English Language

Avoid using prolonged sentences that are hard to read.

Author Response

I would like to express my gratitude to the experts for their valuable revision suggestions. I accept all the expert's modification suggestions. In response to the valuable opinions raised by experts, my answer is as follows.

1. Image enhancement methods include frequency domain based image enhancement and spatial domain based image enhancement. The classic methods of image enhancement based on frequency domain include image enhancement based on Fourier space, image enhancement in wavelet domain, etc. Image enhancement methods based on spatial domain are divided into statistical image enhancement and differential image enhancement. Representative image enhancement methods based on statistics include histogram enhancement and limited contrast histogram enhancement, which mainly enhance the local or non-local contrast information of the image. This article mainly studies image enhancement methods based on difference. Representative difference algorithms include first-order sobel operator, prewitt operator, second- order laplacian operator, and the fractional order differential enhancement operator proposed in this article. These methods mainly enhance the edge and texture information of images, with a focus on enhancing the clarity of the images. This article focuses on the unique performance of fractional order calculus operators in image enhancement and denoising compared to integer order calculus operators. Therefore, the experiment in Figure 7 only compares several classic differential image enhancement methods.
2. According to expert opinions, I have added parameter comparison experiments on different image enhancement methods in Figure 8 of the paper.
3. Fractional calculus mainly studies non integer order differentials and integrals. Unlike traditional integer order calculus, its memory and non-local properties are very suitable for describing signals with memory and genetic properties in the real world. This article mainly focuses on the application of fractional calculus theory in image enhancement and denoising processing. The fractional order differential operator not only enhances the high-frequency components of the signal, but also preserves the very low-frequency components of the signal nonlinearly. The fractional order integral operator has a attenuation effect on the high-frequency signal of the signal, and the degree of attenuation increases nonlinearly with the increase of the integration order. Due to the unique properties of fractional order calculus operators, the fractional order calculus operator proposed in this paper has higher processing efficiency compared to the integer order calculus operators listed in the literature review.
4. Due to my poor English proficiency, the revised content of the paper was handed over to a professional scientific and technological paper institution for polishing.
5. According to the opinions of experts, I carefully revised the highly similar Figure titles in the paper.
6. According to expert opinions, the paper has added corresponding reference annotations to support the discussion on "The application of fractional calculus theory in image processing..." on Line 49 in the paper.

Reviewer 3 Report

Comments and Suggestions for Authors

Please see the review report in attachment.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Please see the review report in attachment.

Author Response

I would like to express my gratitude to the experts for their valuable revision suggestions. I accept all the expert's modification suggestions. In response to the valuable opinions raised by experts, my answer is as follows.

1. I have added the keyword "partial differential equation theory" to the paper.
2. the references recommended by the expert have been added to the paper, and the article references have been reorganized and numbered.
3. I will modify the parameter "gauss" in formula 11 in the paper to "Gauss".
4. I reduce ”Conclusions” to be more incisive. The statement you suggested to delete by the expert has been deleted, and the conclusion section of the paper has been further streamlined to ensure the overall quality of the paper.

Reviewer 4 Report

Comments and Suggestions for Authors

In the paper under consideration, the authors have studied the application of fractional calculus theory in image enhancement and denoising, including the basic theory of fractional calculus and its amplitude frequency characteristics, the application of fractional differential operator in image enhancement, and the application of fractional integral operator in image denoising. 

After review this throughout paper I have a doubt as follows :

- The authors have denoisy, I would like to know what type of denoisy to use in this example.

- What size images does the authors use? square yes or not? Could you use non-square size?

- Hopefully in the next work there will be applications that are color images.

Moreover, The authors should check all English grammar through of this paper.

Comments on the Quality of English Language

The authors should check all English grammar through of this paper.

Author Response

I would like to express my gratitude to the experts for their valuable revision suggestions. I accept all the expert's modification suggestions. In response to the valuable opinions raised by experts, my answer is as follows.

1. As shown in Figure 11, this paper represents the image denoised by fractional integral image denoising operators for the Barbara image with Gaussian white noise of meanand variance.
2. The experimental image in this article is a square image with a resolution of 256 * 256. For the sake of beautiful layout, all experimental images are in square rows, and of course, non- square images can also be used. The processing effect of an image is mainly related to the shape and weight of the template, and is not related to whether it is a square image.
3. Due to my poor English proficiency, the revised content of the paper was handed over to a professional scientific and technological paper institution for polishing.

Reviewer 5 Report

Comments and Suggestions for Authors

Report on the article “Research on Application of Fractional Calculus Operator in Image Underlying Processing”

For the following reasons, I believe that the article needs major revisions before it is accepted for publication in fractal and fractional:

1. The authors use lengthy sentences in the entire manuscript which makes it hard to read. For instance, see line 284-290, etc.

2. Can you provide a clear matrix for the different operators you suggest, similar to what is commonly known Sober and Prewitt operators.

3. What are the advantage of these suggested operators in terms of efficiency? Compare the time required to enhance the image using well-known operators and the suggested ones.

4. Compare the suggested operators with contrast stretching (linear and non-linear) and histogram equalization methods.

5. The claim of this sentence and property “This operator has anisotropic rotation invariance and its filtering coefficient is shown in Equation (19).” Needs to be proved by showing examples of images.

6. Figures captions must be more informative and clear. Revise all captions and their reference in the main text. Give proper and specific reference for every figure.

7. Compare the suggested operator with Cant edge detector to show how the suggested operators are better in the sense of subjective criteria.

8. Is the suggested operators robust against different types oof noise? Try other types of noise such as salt and pepper noise, etc.

Finally, I hope the authors accept my opinion as advice.

Comments on the Quality of English Language

The authors use lengthy sentences in the entire manuscript which makes it hard to read. For instance, see line 284-290, etc.

Author Response

I would like to express my gratitude to the experts for their valuable revision suggestions. I accept all the expert's modification suggestions. In response to the valuable opinions raised by experts, my answer is as follows.

1. Due to my poor English proficiency, the revised content of the paper was handed over to a professional scientific and technological paper institution for polishing.
2. In the paper, based on specific enhancement and denoising experiments, the numerical values of fractional calculus orders can be determined, and the corresponding "a clear matrix" can be obtained by incorporating mask formulas (15) and (19).
3. Compared with first-order and second-order integer order differential image enhancement methods, the     fractional order differential image enhancement method has a time complexity of one order of magnitude, as its basic principle is to use image processing templates and convolution operations on the image to be enhanced. The main factors that determine its time complexity are the size of the template and the resolution of the image.
4. Following the expert's suggestions, I added contrast stretching (linear and non-linear) and histogram equalization methods experimental contents to the image enhancement section of my paper.
5. Fractional order ladder degree itself is a linear operator, and due to the involvement of square and root calculations, the modulus of the fractional ladder degree vector is clearly not linear, so the fractional ladder degree vector does not have rotation invariant isotropy. However, the modulus of the fractional gradient vector is isotropic, as detailed in reference [PU Yi-Fei,WANG Wei-Xing.Fractional Differential Masks of Digital Image and Their Numerical Implementation Algorithm[J].Acta Automatica Sinica,2007(11):1128-1135]. The fractional calculus operator proposed in this article has a high degree of symmetry, that is, the weights of the fractional calculus mask operator are consistent in the eight directions of the digital image, and the fractional gradient modulus value is used to quantify the results of image processing. Therefore, the fractional calculus operator has the property of "anisotropic rotation variance".
6. I have revised the title of the paper and added content to the sub-graph title. I revise all captions and their reference in the paper, and give proper and specific reference for every figure.
7. The main advantage of the fractional order differential enhancement operator proposed in this article compared to the "Cant edge detector" is that it can extract both strong edges and weak edges (texture details) of the image, and then linearly fit the fractional order differential enhanced image with the original image to obtain a new image. Subjectively observed, the enhanced image has richer texture and better contrast.
8. The fractional order integral image denoising algorithm proposed in this paper mainly targets Gaussian white noise, and compared to traditional single theory denoising methods such as Mean denoising, Gaussian denoising and Wiener denoising, it has a relatively good effect of edge preserving denoising. Common image noise includes Gaussian noise, salt and pepper noise, Poisson noise, multiplicative noise, etc. The 'salt and pepper noise' generally differs significantly from the normal pixel values around the image, and the adaptive median denoising method is generally used for better denoising effect. In future research, I will follow the opinions of experts and attempt to use fractional order integration methods to carefully study images containing Poisson noise and multiplicative noise, in order to verify the robustness of fractional order integration denoising operators.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

My comments have been addressed. No more comments and questions.

Comments on the Quality of English Language

Language quality is fine after revising.

Reviewer 5 Report

Comments and Suggestions for Authors

Many thanks to the authors that they considered and took care of my remarks in the first report. However, the authors still use lengthy sentences in the entire manuscript and the paper needs a minor editing of the English language. However, I recommend the article be accepted for publication in fractal and fractional.

Finally, I hope the authors accept my opinion as advice.

Comments on the Quality of English Language

Many thanks to the authors that they considered and took care of my remarks in the first report. However, the authors still use lengthy sentences in the entire manuscript and the paper needs a minor editing of the English language. However, I recommend the article be accepted for publication in fractal and fractional.

Finally, I hope the authors accept my opinion as advice.