Extended Smoothing Methods for Sparse Test Data Based on Zero-Padding

Round 1

Reviewer 1 Report

Substitute all occurrences of the adverb "where" by the relative "in which". Parts of the text should be adjusted to be comprehensible, for instance "After the above two smoothing, it can be found that the method improves the data density of discrete data without changing the value size" can be written as "After the preceding smoothing process, it is possible to verify that the method improves the density of the discrete data without changing the value size."

Author Response

If you find it difficult to read this, please see the attachment.

Dear Reviewer,

Thank you for your comments concerning our manuscript entitled Extended smoothing methods for sparse test data based on zero-padding. We apologize for the poor language of our manuscript. We have carefully checked and improved the English writing in the revised manuscript. We really hope that the flow and language level have been substantially improved. Revised portion are marked in Red in the paper. The responds to your comments are as following:

Comment 1: Substitute all occurrences of the adverb "where" by the relative "in which".

Response: Thanks for your suggestion. All occurrences of the adverb "where" had been substituted by the relative "in which". They appear in Line 121, 126, 132, 205, 210, 213, and 214, respectively.

 

Comment 2: Parts of the text should be adjusted to be comprehensible, for instance "After the above two smoothing, it can be found that the method improves the data density of discrete data without changing the value size" can be written as "After the preceding smoothing process, it is possible to verify that the method improves the density of the discrete data without changing the value size."

Response 1: Thanks for your suggestion. "After the above two smoothing, it can be found that the method improves the data density of discrete data without changing the value size" had been substituted by " After the preceding zero-padding process, it is possible to verify that the method improves the density of the discrete data without changing the value size. " They appear in Line 356.

Response 2: We have fully embellished the article. The specific modifications are as follows:

Line 27: "Smoothing sparse test data can not only solve the problem of too little test data, but al-so speed up the process of data acquisition, processing and practical application to a certain extent." had been substituted by "Smoothing sparse test data can not only solve the problem of too little test data but also, to a certain extent, speed up the processes of data acquisition, processing, and practical application. "

Line 40: "Oraiche et al. provide an algebraic hyperbolic cubic Hermite spline interpolation method, using which data from second-order systems can be fitted more easily " had been substituted by "Data from second-order systems can be more easily fitted using the algebraic hyperbolic cubic Hermite spline interpolation approach provided by Oraiche et al. "

Line 56: "The advantage is that they greatly improve the smoothness and accuracy of the curve, the limitation is that they can only be used under specific conditions, not universal." had been substituted by "The advantage is that they greatly improve the smoothness and accuracy of the curve; the limitation is that they can only be used under specific conditions and are not universal. "

Line 75: "Based on previous research, this paper proposes an extended smoothing method for two-dimensional discrete data, whose main core idea is based on the FFT operation and zero-padding." had been substituted by "The basic core concept of this study, which is based on the DFT operation and zero-padding, is an extended smoothing method for two-dimensional discrete data. "

Line 104: "This procedure complements the original sequence with zeroes and adds no additional information to the angular spectrum of the wave-number domain." had been substituted by " This procedure complements the original sequence with zeroes but does not add any additional information to the angular spectrum in the wavenumber domain."

Line 107: "This regenerated discrete signal is the result of smoothing." had been substituted by " Smoothing produced the discrete signal that was regenerated."

Line 149: "After the angular spectrum value P of the original two-dimensional spatial discrete signal is obtained, the angular spectrum is expanded by zero-padding, thereby obtaining a generated angular spectrum P'." had been substituted by " The generated angular spectrum P' is acquired by zero-padding of the original angular spectrum P. "

Line 199: "The newly generated wave-number domain angular spectrum is transformed from the wave-number domain to the spatial domain by two-dimensional IFFT, and a new two-dimensional discrete signal is obtained. " had been substituted by " A new two-dimensional discrete signal is obtained after the newly generated domain angular spectrum in wavenumber is translated domain to spatial domain via two-dimensional IDFT. "

Line 310: "The original angular spectrum is zero-padded out to expand the sequence, and the angular spectrum value at a specific position is modified." had been substituted by "To expand the sequence, the original angular spectrum is zero-padded, and the value of the angular spectrum at a particular location is modified. "

Line 322: "The IFFT is performed on the angular spectrum after zero-padding to obtain two-dimensional discrete data after smoothing once, as shown in Fig. 9." had been substituted by " IDFT is applied to the angular spectrum after once zero-padding to produce two-dimensional discrete data after smoothing shown in Fig. 9. "

Line 355: "The above numerical results verify the correctness of the two-dimensional discrete data smoothing theory proposed in this paper. After the above two smoothing, it can be found that the method improves the data density of discrete data without changing the value size. In addition, the smoothing theory of two-dimensional discrete data has the function of multiple computation, which can adjust the data density to the appropriate degree of computation." had been substituted by " The two-dimensional discrete data smoothing theory put forth in this paper is verified, as shown by the aforementioned numerical results. After the preceding zero-padding process, it is possible to verify that the method improves the density of the discrete da-ta without changing the value size. In addition, the multiple computation feature of the smoothing theory for two-dimensional discrete data allows for the adjustment of data density to the proper level of computation."

Line 357: "Aiming at the problem of limited test data in engineering practice, this paper pro-poses a two-dimensional discrete data smoothing method based on FFT and zero-padding." had been substituted by "This work proposes a two-dimensional discrete data smoothing method based on DFT and zero-padding in order to address the issue of limited test data in engineering practice. "

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors of the article propose a method to smooth sparse 2D test data. While the method can be a suitable solution in many cases, the paper has some issues that need to be addressed.

A) The authors mention several other methods that can be used to smooth test data, and the drawbacks of other methods, however they fail to describe the conditions and drawbacks of their own method.

B) The datasets the method is aimed at are sampled datasets, for this reason, they should be satisfying the sampling theorem. If not, the aliasing effect may cause severe artifacts, this fact should be mentioned in the article.

Note: in the article the method of generating the test sample may cause several artifacts in further processing (since the sampling theorem may be violated), however the zero padding of the high frequency components of the spectrum may be really beneficial.

Note2: since the sampling theorem should be satisfied the Whittaker–Shannon interpolation formula could also be used to smooth the surface. However, the presented method may be a computationally effective way to get the same results.

C) Since the FFT assumes the periodic extension of the input data, the method may cause unwanted artefacts at the edge of the area. This artefact can be seen in Figure 4. at the left corner of the smoothed data, where the values abruptly increase.

D) Chapter 5.2: I do not understand why a need for a second smoothing is there. I assume that the zero padding could be bigger even in one iteration.

 

Other remarks:

E) While the x and y values are clearly non-negative in equation (1), the equation (2) take their absolute values. I assume that the formula given in equation (2) is not centered to be in accordance with equation (1).

F) In equations (3) to (5) the indexing of n_x and n_y coordinates go from 0 to N_x-1 and Ny-1 respectively. In Figure 2. they start from index 1 which while not really confusing, but not the best solution. This also affect the Figures (3), (5), (8) and (10) respectively.

G) It would be much better to call the transformations DFT (and inverse DFT), since there is no reason to limit N_x and N_y to be the power of two. However, in this case, the authors should give a solution that work with odd numbered columns or rows for zero padding.

H) The Figure 2. the shape looks oddly asymmetric: the line between the A1 and B1 should be in the middle of the figure (in case of even numbered columns).

I) Line 183: There is no reference to the filter proposed by Veronesi and Maynard.

Line 186: Is there any more information on the empirical formula?

J) Line 273: “it can be seen that the data density increases and the data becomes richer after a smoothing of discrete data”

I would not say that it becomes rich, since no further information is added to the data set. It simply becomes simpler (interpolated using sinusoid signals).

 

General note: while the method does not introduce surprising new elements, the idea may not be trivial for many researchers, the details provided can make it a valuable article.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf