Infrared Small Target Detection Based on Non-Convex Optimization with Lp-Norm Constraint

Round 1

Reviewer 1 Report

I believe that the authors answered satisfactorily to all my concerns.

Author Response

Thank you very much for your letter and the comments from the referees about our paper submitted to Remote Sensing. If you have any questions about the paper, please do not hesitate to contact us.

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper propose Infrared small target detection based on non convex optimization. One of the most interesting thing in this paper is applying sparisty for detection small Target. I have many concern regrading this paper:

1-line 105:" it requires fewer iteration to converage  " any reference on this claim?

2-line 108: what benefits? any references?

3-I checked the references , authors have to check works that have been done by Micheal Elad and his group .

4- results:  I don't see any comparison between the proposed method and RPCA.... Authers have to include RPCA in the results.

overall, the paper is well written .

Thanks

Author Response

     Thank you very much for your letter and the comments from the referees about our paper submitted to Remote Sensing.

We have learned a lot from the referee’s comments, which is fair, encouraging and constructive. After studying the advices carefully, we have made corresponding changes. Our response shows as below and the words in red are the changes we made in our manuscript.

If you have any questions about the paper, please do not hesitate to contact us.

Point 1: line 105:" it requires fewer iteration to converage  " any reference on this claim?

Response 1:

This conclusion is proved in the reference [44] in this paper, which is specifically referred to as "Chartrand R, Staneva V. Restricted isometry properties and nonconvex compressive sensing [J]. Inverse Problems, 2010, 24(3): 657-682."

In the penultimate paragraph of the numerical experiments in the fourth part of this paper, the following description is included: “For the number of outer iterations chosen, figure 3 shows that when the signal x is recovered, smaller values of p give much smaller reconstruction errors. Thus, using smaller p results in either a more accurate solution, or a solution of specified accuracy obtained more quickly.” The corresponding description can also be seen in the conclusion section: “We also find that when sparse recovery is successful, fewer iterations of our process are required to give very complete convergence when p is small.”

Figure 3 of the paper is as follows:

The situation involved in our paper is to set the iteration termination condition to a fixed value. Thus, as stated in [44], the Lp-norm can get a solution of specified accuracy more quickly. Therefore, the conclusion "it requires fewer iteration to converage" used in this paper has been proved. In order to more clearly illustrate the source of this conclusion, we add a reference to the literature [44] after this sentence(See line 106).

Point 2: line 108: what benefits? any references?

Response 2:

The term "benefits" here refers to the points described in this paragraph of the article:

1. A more sparse solution can be obtained compared to the L1-norm[43-46];

2. When a sparse signal can be recovered, it often requires fewer iterations to converge the equation[44];

3. For the RPCA problem of the Lp norm approximation, the theory has been established a lower bound of nonzero entries in solution of L2-Lp minimization[47].

For the above points, reference has been made to the corresponding position in the paper.

In addition to this we have added another advantage of Lp norm minimization: Lp minimization with 0 < p < 1 recovers sparse signals from fewer linear measurements than does L1 minimization[43] (See line 104-105).

The words in the original text does have a problem that are unclear. Now we have made a change to this sentence, and the correction is: "Although the optimization problem of the Lp-norm is non-convex, it has been studied a lot before, and it has the advantages of being able to obtain a more sparse solution, fewer iterations to converge, and a theoretical basis for the L2-Lp minimization problem." The changes are marked in red(See line 109-111).

Point 3: I checked the references , authors have to check works that have been done by Micheal Elad and his group .

Response 3:

Thank you very much for reminding. In fact, this article refers to Micheal Elad's book "Sparse and Redundant Representations" during the conceptual stage, and I forgot to add it to the citation. The "Promoting Sparse Solutions" in the first chapter of this book has helped me a lot, and we have added references to this part to the article(See line 697).

Point 4: results: I don't see any comparison between the proposed method and RPCA.... Authers have to include RPCA in the results.

Response 4:

Robust principal component analysis(RPCA) describes a type of problem rather than a specific method. RPCA considers the problem of decomposing a matrix into the sum of two matrices, one of which is low rank and the other is sparse. RPCA can be written as an optimization problem as follows:

where A represents a low rank matrix and E represents a sparse matrix.

The “RPCA” mentioned here is a description of such a problem. The matrix represented by the symbols in the formula is not given additional physical meaning, so RPCA is a mathematical concept. The various infrared small target detection methods in [29-41] are extensions of the mathematical problem of RPCA. The rank function and the L0-norm, which are not easy to solve, are approximated, or the penalty factor and the penalty coefficient are added to increase the detection effect of the algorithm. These papers have constructed infrared small target detection models, in which the symbols are given specific physical meaning.

In summary, RCPA is a mathematical problem. It can be said that the infrared small target detection method based on sparse and low-rank matrices recovery is an extension of the RPCA problem in practical physical applications. Moreover, there is no infrared small target detection algorithm called “RPCA”. Therefore, the comparison between the method and RPCA cannot be added in the experimental results.

In addition to this we have added reference [55] (See line 714-715) and gave a brief introduction to the development trend of infrared small targets detection (See line 575-577).

Author Response File: Author Response.pdf

Reviewer 3 Report

A novel infrared small target detection method based on non-convex optimization with Lp-norm constraint (NOLC) has been proposed in this paper. The proposed method shows the higher detection accuracy compared to the existing methods and is computationally efficient.

Few comments needs to be addressed:

1. Isn't the assumption that "infrared images are a linear combination of target image, background image, and noise image" is bit restrictive to implement the proposed NOLC method?

2. In Figure. 9, 13 and 15, can you clearly describe what the cyan coloured circle is representing?

Author Response

Thank you very much for your letter and the comments from the referees about our paper submitted to Remote Sensing.

We have learned a lot from the referee’s comments, which is fair, encouraging and constructive. After studying the advices carefully, we have made corresponding changes. Our response shows as below and the words in red are the changes we made in our manuscript.

If you have any questions about the paper, please do not hesitate to contact us.

Point 1: Isn't the assumption that "infrared images are a linear combination of target image, background image, and noise image" is bit restrictive to implement the proposed NOLC method?

Response 1:

I think this assumption has few restrictions on the implementation of the NOLC method. On the contrary, all theoretical derivations of NOLC are based on this assumption.This is because we assume that "infrared images are a linear combination of target image, background image, and noise image", so it can be regarded as an RPCA problem, and then there is a model building and solving process.

Regarding this assumption, "infrared images are a linear combination of target image, background image, and noise image". Strictly speaking, the noise in the image is not necessarily additive, so the real model may be non-linear. Our assumption is relatively reasonable when the noise is approximately additive. And in the infrared small target images, many noises can be approximated as additive, so this assumption is feasible.

The application of this assumption in the field of infrared small target detection is firstly derived from the literature [10]. Later, Gao [29] applied this assumption and proposed the IPI model, which is also the beginning of the sparse and low-rank matrices recovery based methods. Subsequent such methods also applied this assumption. The validity of this assumption is also confirmed by the large number of infrared small target detection methods based on this assumption.

Thank you very much for your suggestion. After careful consideration, we feel that the statement in the article is not particularly strict and has been revised in the text. The description was changed to: "In reference [10], when the noise can be approximated as additive, the infrared small target image can be seen as a linear combination of target image, background image, and noise image." and it was marked red in the text (See line 183-184).

Point 2: In Figure. 9, 13 and 15, can you clearly describe what the cyan coloured circle is representing?

Response 2:

The marking methods for Figures 9, 13, and 15 are given in lines 374-377, 439-441 and 452-454, respectively. The original text is: "In the figure, the target position in the original image is circled in red, and the position of the clutter is circled with cyan in the processing result."

The cyan circle indicates where the clutter appears in the 3D processing results. Since the clutter is relatively small in 3D display, it is not easy to visually see it, so it is circled in a cyan circle.

We have added a description of the cyan circle for images 9, 13, and 15 in lines 374-377, 439-441 and 452-454, respectively. The corrections are marked in red in the text.

In addition to this we have added reference [55] (See line 714-715) and gave a brief introduction to the development trend of infrared small targets detection (See line 575-577).

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Dear authors,

I think you have to recheck the references especially reference [29], because all of your work quite similar to the work done by [29] which published in 2013. In [29] , they used Robust PCA for detecting small Targets. In addition , most of your methodolgy similar to reference[29]. So authors have to compare their results with RPCA(Robust PCA) and they can download the code for this methodolgy from the homepage of auther [29] (http://gr.xjtu.edu.cn/web/dymeng/3?fbclid=IwAR1GUJ1IJQE0T3gqrNsm9LW3cG2-LBh6ZGqDzkIAwEjQLjofOfbStHJZwa0) number 14.

Thanks

Author Response

Please see the file attached. Thanks.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors has incorporated the relevant concerns in the revised version of a paper.

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

P { margin-bottom: 0.08in; }

This paper proposes a method for small target enhancement with application to infrared images. The method is formulated as finding the best decomposition of the image into their background, target and noise components. The decomposition is computed from the minimization of an objective function that includes an Lp norm constraint. Optimization is approached using ADMM, a very suitable method for Lp based problems.

This paper deals with a very difficult problem in remote sensing: the detection of small targets in infrared images. I found little methodological difference with respect to previous authors work “Infrared small target detection via non-convex rank approximation minimization joint l2,1 norm”. Indeed the authors did not include the results from this previous method in the experimental section.

I believe that the authors should state more clear the actual differences with respect to previous work in order to be able to assess whether this is an incremental or a substantial contribution. In addition, the comparison of both methods should be included in the manuscript with a proper discussion on why the new method outperforms previous work.

Author Response

Response to Reviewer 1 Comments

Thank you very much for your letter and the comments from the referees about our paper submitted to Remote Sensing.

We have learned a lot from the referee’s comments, which is fair, encouraging and constructive. After studying the advices carefully, we have made corresponding changes. Our response shows as below and the words in red are the changes we made in our manuscript.

If you have any questions about the paper, please do not hesitate to contact us.

Point 1: This paper deals with a very difficult problem in remote sensing: the detection of small targets in infrared images. I found little methodological difference with respect to previous authors work “Infrared small target detection via non-convex rank approximation minimization joint l2,1 norm”. Indeed the authors did not include the results from this previous method in the experimental section.


Response 1: Thank you very much for your comments and suggestions. The paper titled " Infrared small target detection via non-convex rank approximation minimization joint l2,1 norm " is indeed a great method, and we have added a comparison with this method in various parts of the experiment.

Point 2: I believe that the authors should state more clear the actual differences with respect to previous work in order to be able to assess whether this is an incremental or a substantial contribution.

Response 2: Thank you very much for your comments and suggestions. Actually, to our knowledge, this idea has never been applied to the field of infrared small target detection.

Point 3: In addition, the comparison of both methods should be included in the manuscript with a proper discussion on why the new method outperforms previous work.

Response 3: Thank you very much for your comments and suggestions. We have added comparison with NRAM which you mentioned in Point 1 in the experimental section. At the same time, the differences between multi-scene detection effect, ROC curve, SCR Gain, BSF, noise interference image detection effect and iteration number are compared in detail. The analysis for the NRAM method is given in Section 3.4, paragraph "To further demonstrate...".

Author Response File: Author Response.pdf

Reviewer 2 Report

The proposed manuscript illustrates the application to infrared images of small target detection techniques. In general, the work is very good with and technically well expressed. I have only a few changes to propose to the authors.

Introduction. It is very short. It would be useful to add the state of the art of the use of these techniques to emphasize the results obtained in the conclusions. In general, in the following paragraphs 1.1 and 1.2 the state of the art is mentioned; however I think it is useful to insert  a small  part it in the first part of the introduction.

Figure 7 differentiating the different images (i.e. a, b, c, d). The result is good however I suggest commenting  the difficult of NOLC to  separate background and target in images with low contrast (example second image figure 7), and how in your case of study,  IPI ( example second image figure  8) allow a better result. In general, I suggest to improve the comment  in the text of image 7 and image 8 given the results obtained.

Author Response

Response to Reviewer 2 Comments

Thank you very much for your letter and the comments from the referees about our paper submitted to Remote Sensing.

We have learned a lot from the referee’s comments, which is fair, encouraging and constructive. After studying the advices carefully, we have made corresponding changes. Our response shows as below and the words in red are the changes we made in our manuscript.

If you have any questions about the paper, please do not hesitate to contact us.

Point 1: Introduction. It is very short. It would be useful to add the state of the art of the use of these techniques to emphasize the results obtained in the conclusions. In general, in the following paragraphs 1.1 and 1.2 the state of the art is mentioned; however I think it is useful to insert  a small  part it in the first part of the introduction.


Response 1: Thank you very much for your comments and suggestions. We have lengthened the description in the first part of the introduction.

Point 2: Figure 7 differentiating the different images (i.e. a, b, c, d).

Response 2: Thank you very much for your comments and suggestions. We have replaced all the symbols used to distinguish different images in (a) (b) (c) (d).

Point 3: The result is good however I suggest commenting the difficult of NOLC to separate background and target in images with low contrast (example second image figure 7), and how in your case of study, IPI (example second image figure 8) allow a better result.

Response 3: Thank you very much for your comments and suggestions. In fact, Figure 7 illustrates that NOLC is able to separate the background and target well (as is the second image). Moreover, the IPI in Figure 8 does not give a better result. In order to better show the detection effect, we circled the position of the background clutter in cyan in the three-dimensional display. Actually, the IPI processing result in Figure 8 has the most background clutter, while the NOLC method gives a cleaner background which means NOLC performs better.

Point 4: In general, I suggest to improve the comment in the text of image 7 and image 8 given the results obtained.

Response 4: Thank you very much for your comments and suggestions. We put the comments in Figure 7 in the paragraph in section 3.2 (a), and the comments in Figures 8 and 9 in the paragraph in section 3.2 (b) and respectively describe and analyse the results in the figure.

Author Response File: Author Response.pdf

Reviewer 3 Report

In this work, the object detection solution is original and valuable. The paper presents theoretical issues and results elaboration in an exhaustive way.
I suggest to consider the following corrections:
- the distribution of results applied by the authors should be specified. Is there a normal distribution?
- The robustness of the NOLC model is described unclearly (for me).
How robust are these algorithms?

- Conclusions may be slightly extended - in relation to the balance of the Article.

In the future, the robustness of algorithms could be verified by distorting the data (e.g. by noise) according to the assumed data distribution model. I also suggest in the preliminary analysis (in future research) to consider the use of the Modulation Transfer Function (MTF). This could be used to analyze the characteristics (limits) of the optical-electronic system.

Author Response

Response to Reviewer 3 Comments

Thank you very much for your letter and the comments from the referees about our paper submitted to Remote Sensing.

We have learned a lot from the referee’s comments, which is fair, encouraging and constructive. After studying the advices carefully, we have made corresponding changes. Our response shows as below and the words in red are the changes we made in our manuscript.

If you have any questions about the paper, please do not hesitate to contact us.

Point 1: the distribution of results applied by the authors should be specified. Is there a normal distribution?

Response 1: Thank you very much for your comments and suggestions. In fact, our work does not involve distribution issues. The main purpose for our processing of images is the low rank and sparse matrix decomposition, where the distribution is not involved in the theory.

Point 2: The robustness of the NOLC model is described unclearly (for me). How robust are these algorithms?

Response 2: Thank you very much for your comments and suggestions. We added a description of the robustness of the algorithm in the text. Simultaneously, we have added a comparison of the processing results of noisy images in the text. Experiments show that the NOLC method can accurately detect the target when the noise is strong, while ensuring a pure background. In contrast, other methods do not have a good effect, which further illustrates the robustness of the NOLC method.

Point 3: Conclusions may be slightly extended - in relation to the balance of the Article.

Response 3: Thank you very much for your comments and suggestions. The conclusions have been extended. The conclusions about the experimental part are added to verify that the robustness of the NOLC method is better than other algorithms.

Point 4: In the future, the robustness of algorithms could be verified by distorting the data (e.g. by noise) according to the assumed data distribution model.

Response 4: Thank you very much for your comments and suggestions. Experiments have been added as mentioned in Point 2.

Point 5: I also suggest in the preliminary analysis (in future research) to consider the use of the Modulation Transfer Function (MTF). This could be used to analyze the characteristics (limits) of the optical-electronic system.

Response 5: Thank you very much for your comments and suggestions. In fact, our focus is on the process of processing the image, rather than the lens or optical sensing device.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors demonstrated in the experiments how the proposed Lp norm improves their previous proposal with L21 norm. However, I would expect some discussion on the reasoning why is this Lp norm outperforming L21 in some cases. Indeed what is specifically the difference between both methods? Are the authors just changing the metric or they are including any other algorithmic difference in the ADMM optimization? The authors go over a thorough treatment of this issue in their response (point 2).

Reviewer 2 Report

All requested corrections were accepted by the authors

Reviewer 3 Report

All my concerns have been taken into account.