ECG Signal Denoising and Reconstruction Based on Basis Pursuit

Round 1

Reviewer 1 Report

Major comments.

This paper proposes an ECG denoising method that includes both an ADMM based algorithm that solves the basis pursuit (BP) optimization problem. More precisely, the BP-ADMM algorithm is developed to minimize the effects induced by perturbations that follow a Gaussian distribution. This methodology also uses a zero-phase filter for removing the baseline wander commonly found in ECG signals. However, this manuscript requires major revisions. The procedures are not clear and the mathematical notation is not consistent. Furthermore, in order to reproduce the yielded results, the parameters of both the BP-ADMM algorithm and the zero-phase filter are not included. Finally, the proposed methodology should be compared with respect to the state-of-the-art methods referenced in [20]-[22].

Specific comments.

• It seems that the BP-ADMM solves the denoising problem in the context of the sparse signal representation framework. In other words, the signal denoising approach from compressive sensing measurements is not sufficiently clear. An example of compressive measurements should be included. Furthermore, the details of the STFT dictionary should be also included. See the STFT example included in

Ramirez, Juan Marcos, and Jose Luis Paredes. "Robust transforms based on the weighted median operator." IEEE Signal Processing Letters 22, no. 1 (2014): 120-124.

• Various claims are contradictory. For example, it is added in the introduction that the ECG signals are contaminated with impulsive noise. Later, the denoising proposed method proposes the BPDN approach and zero-phase filter, both optimized to reduce the Gaussian noise.

• The notation and equations in the Review of the ADMM are not properly presented. This problem is also found in Section 4.3. Please refers to

Boyd, Stephen, Neal Parikh, and Eric Chu. Distributed optimization and statistical learning via the alternating direction method of multipliers. Now Publishers Inc, 2011.

• An algorithm that contains the complete proposed procedure should be incorporated.

• Eq. 1 is not an optimization formulation, it is an acquisition model or a signal model

• What is the relationship between the FIR filter justification in Section 4.2 and the IIR filter in Eq. 14.?

• The following reference should be revised:

Chen, Scott Shaobing, David L. Donoho, and Michael A. Saunders. "Atomic decomposition by basis pursuit." SIAM review 43, no. 1 (2001): 129-159.

• A rigorous comparative evaluation in terms of computational complexity should be introduced.

Author Response

Merry Christmas to you! Please see the attachment. Thank you!

Author Response File: Author Response.docx

Reviewer 2 Report

This is a paper that uses Basis Pursuit to denoise ECG signals. The Alternating Direction Method of Moments is implemented to solve the optimization problem.

Overall, the paper lacks a Discussion section that would put the study into context. The flow of the paper is confusing at times and key references are missing for several methods. Three reference methods are compared against the proposed method, but the implementation details of those methods are not included. This makes the comparisons uninformative, especially since the denoising algorithms seem to reduce SNR. This begs the question - if we are losing signal by denoising, why denoise?

Comments to the Authors

1. Introduction: Earlier references should be included for Compressive Sensing, such as [1] https://onlinelibrary.wiley.com/doi/full/10.1002/mrm.21391, [2] Candès E, Romberg J, Tao T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 2006;52:489–509. [3] Donoho D. Compressed sensing. IEEE Trans Inf Theory 2006;52:1289–1306.
2. Introduction: Reconstruction algorithm is not the core content of CS theory. The two core principles are that the signal should be sparse in some transform domain and the artefacts due to sub-sampling should be incoherent in a transform domain. If these are not satisfied, no reconstruction algorithm can recover the original signal, whereas multiple algorithms can recover the signal to different degrees of success as long as these conditions are satisfied.
3. Page 2 line 68. Need references for ADMM.
4. Page 2 line 76. The word “Solution” pops up in the middle of the sentence.
5. Page 2 line 68. This paragraph is disorganized, please consider rephrasing.
6. Throughout the paper, please use Ref. [X] when a reference begins a sentence.
7. Page 2 line 95. The abbreviation “BPDN” is used before it is defined.
8. Section 2.1 The first paragraph is very unorganized and confusing. Please rephrase the sentences here. For example, one sentence states “However, this premise …”. This begs the question: “which premise?”
9. Section 2.1 “Base pursuit”. Basis pursuit?
10. Page 3 line 125. “The multiplier method”. Please do not drop words from well known method names such as ADMM.
11. (7): please be consistent in use of italicized and Roman typeface. This issue repeats throughout the paper.
12. Page 4 lines 155-156. Two sentences seem to be merged together.
13. (8) has both w and omega and lacks the term to be integrated.
14. Line 159. Please give a reference for the use of L1 norm for sparsity.
15. (9). Please consider using another Greek letter instead of lambda to reduce confusion with Section 2.2.
16. Line 165. Row vectors of M?
17. Line 208. Orthogonality + equal energy -> scaled orthonormality?
18. Line 251. y=Hx?
19. (16). Please define F(x) and the minimization cost function line separately.
20. Section 4.3 is extremely confusing. Complex equations show up without any clear explanations or references.
21. Line 280. Instead of saying “let it be 0” as if it was an arbitrary choice, the authors should explain that this derivative would be equal to 0 at a local minimum or a local maximum.
22. Different fonts are used for lambda and rho and other parameters throughout the paper. Please unify the style.
23. Line 311. Sentence incomplete.
24. Sections 2-4 lack an overall coherence. The authors switch between different concepts back and forth and the narrative is quite confusing at times. Please reconsider the flow of the paper. While I realize that such a vague comment is not very helpful and often unwelcomed by the authors, this is a very large issue that I cannot correct as a reviewer. Nevertheless, I had to jump back and forth between sections. Please have someone unrelated to the field read the paper.
25. Line 314. “Root square error” -> root mean squared error? Otherwise it is just the magnitude operation.
26. (3) there is no root in root squared error. Also, please see the previous comment.
27. Line 324. SNR and MSE are both L2 norm based and are inversely related and therefore, do not provide much additional information with respect to each other. Please consider using another metric to provide additional information, such as mean magnitude error (L1) or maximum magnitude error (L-infinity).
28. 3. 15 dB noise, or 15 dB peak SNR?
29. Line 338. No verb.
30. Table 1. I am assuming 5 dB and 15 dB noise refer to the peak SNR of the signal after noise is added. It is interesting to note that the reference methods reduce SNR. The proposed method is quite effective at low SNR, but at high SNR, the SNR seems to be reduced? Does this mean the denoising of the proposed method leads to loss of information?
31. The reference methods need references. How were they implemented? The bad performance raises questions about how those methods are implemented and how their parameters were chosen. Which wavelet family were used, for example?
32. TV used as an abbreviation before it was defined.
33. 5. It is quite impossible to tell apart the curves as the lines are very thin and have the same thickness. Where is the blue line, for example?
34. Figures and results tables come before corresponding text, making it confusing. The authors need to improve the flow of the paper.
35. The paper needs more comparisons. It is shown on three signals from a database, whereas it should be demonstrated on preferably >100 to demonstrate reliability. If this is not possible, please explain why.
36. The paper lacks a Discussion section. Why is the proposed method better?
37. The paper lacks a Discussion section that would put it into context in the wider literature. Please see below comments for some questions.
38. (The paper lacks a Discussion section.) What are similar methods. Is this the only paper that uses Compressive Sensing for ECG?
39. (The paper lacks a Discussion section.) Are there other methods that use basis pursuit for ECG?
40. (The paper lacks a Discussion section.) Are there other methods that use ADMM for ECG signal processing?

Author Response

Dear professor,

  Merry Christmas to you! Please see the attachment. Thank you!

Ruixia Liu.et al.

Author Response File: Author Response.docx

Reviewer 3 Report

The paper proposes an algorithm to mitigate the noise effect on ECG signal. In addition, a low pas filter and an alternating direction method of multipliers optimization algorithm are used to eliminate noise and for signal reconstruction.

The paper is clearly presented and well organized. The proposed methods are well described. It has good informational value in showing the state-of-the-art achieved.

There are some minor observations:

On page 7, line 263, it appears "1" in line with equation 16.

On page 8, line 180 - "let it be 0"...who has to be 0?

On page 11, on results section, the TV noise reduction algorithm should be described in a few words. 

On page 13, lines 361...i think a "that" is extra.  

Author Response

Merry Christmas to you! Please see the attachment. Thank you!

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The main revisions were not properly addressed. The layout of both the abstract and the introduction should be improved to provide significant information on the proposed approach. The procedures are not clear and the mathematical description is poorly presented. Furthermore, in order to reproduce the yielded results, the parameters of both the BP-ADMM algorithm and the zero-phase filter are not included. Finally, the proposed methodology should be compared with respect to state-of-the-art methods.

Minor comments

1. The abstract is poorly written and the contributions in the abstract are not sufficiently clear. The abstract has a lot of grammar errors. Authors should revise the abstract and provide a well written big picture of the proposed work.
2. The introduction is hard to read and it has contradictory claims. The third paragraph seems a repetition of the second one. The basis pursuit optimization problem is a widely known topic in sparse signal representation. The work reported in [15] does not introduce the BP problem, this paper formulates the BP problem with non-smooth constraints which is different from the BP problem. Authors should be precise about what is the problem to solve. It is confusing the mentioning of the impulsive noise in the introduction if the noise that the method attempt to minimize is Gaussian and the baseline error can be considered as a nonlinear bias problem. The review of the ADMM algorithm is not clear. In the previous work, authors include various ECG denoising methods [25, 26, 27] which are not used in Section of results to compare the performance of the proposed approach. I have not clear with the fact that the proposed method uses an FIR filter and the mathematical description of the filter describes an IIR filter.
3. Section 2 has various typos. The mathematical description and the notation of the ADMM algorithm review (Section 2.2) are not clearly presented. Furthermore, the involved variables and parameters are not described. Section 2.2 should be rewritten.
4. In equation (8), the variables W(w) and Hd(e(jω)) are not described.
5. In Section 4.1, the authors do not provide sufficient information about the measurement matrix. The definition in (12) suggests an orthonormal square matrix. Fig. 2 shows that the BP algorithm introduces noise in the reconstructed signal. This typically occurs when the dictionary is not properly selected. In other words, the target signal is not sparse on a selected basis. In section 4.2, the authors describe the characteristics of the FIR filter, and equation (14) defines an IIR filter. This is contradictory and hard to follow. Improve the presentation of the ADMM algorithm. The involved variables are not defined, there are a lot of notation inconsistencies.
6. Section 6 is better written compared to the previous version. However, the proposed approach is not compared with respect to the state-of-the art methods [25, 26, 27]. Furthermore, the proposed method is described as a robust denosing procedure that minimizes impulsive noise and the experiments evaluate the performance of the proposed procedure against Gaussian noise.

Author Response

Thank you. Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

I am sorry to say this as I try to keep a positive attitude in all my reviews, but this revision is highly underwhelming. The changes have been only in the text with minimal effort. So minimal that added sentences have not been merged with the previous or the next sentences. There are repetitions throughout the paper. Many sentences are very confusing. I indicated some as I could not help it, but a through stylistic or grammatical review is not my duty. Also, many of my recommendations were overlooked anyway, which is unmotivating for me as a reviewer who is trying to help you improve the paper from its currently unpublishable format.

I had asked the authors to add more results, which they have not.

I had asked the authors to modify section 4.3 to make it possible to read, which they have not.

I had asked the authors to add a Discussion section to put the study into context, and offered many aspects that needed to be discussed, which they have not. They have provided references for the Discussion section, but in the response to me rather than in the paper. Thank you, but I did not want these for myself.

This paper is not well written. It is incoherent and hard to follow.

Most of the paper discusses several topics that are well known, there are too few results, and there is not a Discussion section that puts the study into context.

The reference methods are questionable as they start with 15 dB peak SNR and yield an output of 6 dB peak SNR. Why do we even use those methods then? I have asked this question previously and the authors have just said their method is better. 

1. Previous comment not addressed. Introduction. I had previously written the following comment: “Introduction: Reconstruction algorithm is not the core content of CS theory. The two core principles are that the signal should be sparse in some transform domain and the artefacts due to sub-sampling should be incoherent in a transform domain. If these are not satisfied, no reconstruction algorithm can recover the original signal, whereas multiple algorithms can recover the signal to different degrees of success as long as these conditions are satisfied.” The authors have just copied what I had written as a comment into the text. I did not advise the authors to just copy what I wrote – it was a comment and therefore, it looks out of place. Please rephrase and discuss accordingly, to introduce the idea to the reader rather than copying what I wrote. Furthermore, the phrases that this comment referred to were not changed – it is now in the next paragraph, still claiming reconstruction to be the core of CS.
   
   
2. Previous comment not addressed. I had indicated that the authors used “base pursuit”, which is not interchangeable with “basis pursuit”. Please go through the paper and find every instance. The same phrase appears in the new version.
   
   
3. Previous comment not addressed. I had commented “please be consistent in use of italicized and Roman typeface. This issue repeats throughout the paper.” Still, variables are sometimes italicized, sometimes not. Please be consistent.
   
   
4. Previous comment. Eqs 12-13. Eq. 12 defines orthogonality. Eq. 13 converts orthogonality to orthonormality. Why not say orthonormal and incorporate Eq. 13 into Eq. 12 to begin with?
   
   
5. Previous comment. Eq. 15 is Y=H.X in frequency domain. The paragraph ends with x=Hy? You have referred to the appendix of a paper, but that paper defines y as the input and x as the output. Do they not? In yours however, it is the opposite.
   
   
6. Previous comment. “Section 4.3 is extremely confusing. Complex equations show up without any clear explanations or references.” The section is pretty much the same. There is still no explanation for the series of equations.
   
   
7. Previous comment: “Line 280. Instead of saying “let it be 0” as if it was an arbitrary choice, the authors should explain that this derivative would be equal to 0 at a local minimum or a local maximum.” The authors have kindly explained their choice to me. I had asked the authors to explain the reasoning behind choosing Eq [] to be 0 to the reader as I am quite aware of Lagrange Method of Multipliers.
   
   
8. Previous comment: “Sections 2-4 lack an overall coherence. The authors switch between different concepts back and forth and the narrative is quite confusing at times. Please reconsider the flow of the paper. While I realize that such a vague comment is not very helpful and often unwelcomed by the authors, this is a very large issue that I cannot correct as a reviewer. Nevertheless, I had to jump back and forth between sections. Please have someone unrelated to the field read the paper.” The sections are still incoherent. Please have someone outside your group read your paper.
   
   
9. Previous comment: “Line 324. SNR and MSE are both L2 norm based and are inversely related and therefore, do not provide much additional information with respect to each other. Please consider using another metric to provide additional information, such as mean magnitude error (L1) or maximum magnitude error (L-infinity).” What I suggested was, these two parameters are highly related. The second one does not add much to the first.
   
   
10. Previous comment: “The paper needs more comparisons. It is shown on three signals from a database, whereas it should be demonstrated on preferably >100 to demonstrate reliability. If this is not possible, please explain why.”
    
    The authors have responded as: “I take a random sample. If using a lot of data, you can't compare their waveforms in this intuitive way. In most data denoising papers, this method is used. If we use more than 100 data, we can use the other performance such as accuracy to measure the noise reduction effect.”
    
    Please perform more analyses and summarize them. A random result does not prove that your method is reliably good. How much does the method’s performance vary across different subjects?
    
    
11. Previous comments suggested to have a Discussion section to put the study into context, which have not been addressed.
    
    
12. The abstract is not well written. The first three sentences are disconnected from each other. ADMM is explained in detail but it should not be – it was suggested years ago. The abstract should focus on what is done in this paper, what the main results are etc, rather than explaining how ADMM works.
13. Line 55 – signal sparse reconstruction? Sparse signal reconstruction?
14. I empathize with the authors that they are writing a paper that is not their native language. But the paper is poorly written and needs linguistic editing. Numerous sentences are just incomprehensible. I am not willing to list all here as it is out of the scope of my responsibility. A couple are below:
15. Line 59 – “they have certain characteristics”. Like what? Very vague sentence.
16. Line 60 – “significant impulse characteristics”. What is significant impulse characteristics versus impulse characteristics?
17. Line 55. Basis Pursuit is a method from 1994. This sentence misrepresents BP by giving a reference from 2019 and saying it is a new method. Following sentences list earlier references as applications of BP.
18. Line 60. “Classical BP …” The sentence starts with BPs inadequacy in addressing impulse noise, ends up with baseline wander.
19. Line 74. “Particularly, each subproblem may be made to have only one design variable. Especially each subproblem has only one design variable.” This is repetition.
20. Line 71. This whole paragraph is very confusing and poorly written. Please re-write.
21. Line 95. These are ADMMs standard properties, not contribution of this paper.
22. Line 111. Please give references, otherwise it seems like you are proposing BP.
23. Line 124. Typo.
24. Line 125. Confusing sentence. Please rephrase.
25. Line 138. Confusing sentence. Please rephrase.
26. Noise is sometimes denoted by s, sometimes by n. Please be consistent.
27. Line 156. Confusing sentence. Please rephrase.
28. Line 160. Please discuss why you chose the p-norm, and what value of p.
29. Line 165. Confusing sentence. Please rephrase.
30. Line 172. “It is …”. Confusing sentence. Please rephrase.
31. Line 175. “When the signal and its differential signal”. Confusing. Please rephrase.
32. Line 188. Please split sentence into two.
33. Line 207. M Column vector?
34. Line 202. s used to denote noise. Now it became something else, which is not defined.
35. So many variables are not defined throughout the paper.
36. Line 214. All of a sudden, we get some results. Many steps are omitted, which makes it impossible for one to replicate the study. What do these results pertain to? Rather than defining orthogonality / orthonormality, which are basic terms, steps that have been omitted should have been given.
37. Line 248. In this paragraph, the new text is just added between two sentences from the previous version. A lot is repeated. Please revise the paragraph to make it coherent.
38. Line 286. Please revise.
39. Line 277. Is this line an equation or a line of text?
40. Line 297. The academic circle does not propose a method to select these, as they depend on the experiment, just like the authors suggest on line 300.
41. Figure 3. This is not 15 dB noise. It is noise that would yield 15 dB peak SNR.

Author Response

Please see the attachment, thanks!

Author Response File: Author Response.docx

Reviewer 3 Report

After the changes made by the author, the paper can be published in its current form.

Author Response

Thank you.

Round 3

Reviewer 1 Report

The authors have introduced major changes that have improved the presentation of the manuscript. Furthermore, the authors have addressed major comments of the previous review. An additional spelling check is required.

Author Response

Thank you. Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

The paper has come a long way. Thank you for the revision.

I had asked the authors to provide more results. They have responded by saying they have more results, but they did not provide those results. I find this confusing. As I had indicated in the previous round, I do not ask for more figures or more columns in the table. Since you already have more results, why not provide them? Please provide a mean pSNR and a standard deviation of pSNR across a larger dataset rather than only 3 signals for the compared methods.

Author Response

Thank you. Please see the attachment.

Author Response File: Author Response.docx