Experimental Study on the Reconstruction of a Light Field through a Four-Step Phase-Shift Method and Multiple Improvement Iterations of the Least Squares Method for Phase Unwrapping

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors studies the reconstruction of light field through four step phase shift method and multiple iterations of least squares method for phase unwrapping, to solve the problem that the true phase information extracted with traditional phase unwrapping technology is easily affected by noise, shadows, and fractures. The authors summarize and analyze various phase unwrapping algorithms, and ultimately selects the four step phase shift method to reconstruct the phase of the optical field, and then combining it with the least squares method to unwrap the phase through multiple iterations. Matlab simulations are carried out to verify their methods. The theoretical analysis presented in the article is relatively detailed and comprehensive. The theoretical analysis presented in the article, including "Reconstruction of Optical Field Phase" and "Method for Unwrapping Phase," is relatively detailed and comprehensive. In the simulation analysis and experimental verification sections, the author's presentation of the effectiveness of the proposed method seems to be relatively straightforward as well. However, I believe that the author still needs to improve in aspects such as the importance of the problem, the mechanism analysis of the proposed method, and the description of simulation and experimental conditions.

1. In the Introduction section, it is recommended that the author presents specific problems in specific application scenarios, using figures or data tables for illustration. In lines 37-44, what are the quantitative evaluation metrics for the "Reconstruction of Light Field" in existing methods? How is the superiority or inferiority of a method judged? At the same time, it is recommended that the author provide quantitative descriptions for the simulation and experimental figures in the following sections based on the proposed quantitative metrics, which will make it easier for readers to understand the effectiveness of the proposed method.

2. In the "Theoretical Analysis" section, the author provides a detailed theoretical derivation of the traditional method. However, I believe that further theoretical derivation is needed to explain the mechanism by which the author's proposed method can suppress the impact of adverse factors such as noise, shadows, and fractures. The current theoretical derivation does not show how noise is suppressed.

3. In the "Experimental Verification" section, it is recommended that the author add "experimental verification figures with added noise".

4. Additionally, there are some formatting issues that need to be revised.

(1)What is the horizontal axis in Figure 1?

(2)In lines 65 and 66, what do "bright lines" and "dark lines" refer to?

(3)The entire text should be written in English and should not be a mixture of Chinese and English. For example, in Figures 6, 12, 15, and 16.

Comments on the Quality of English Language

The English writing level of the article is satisfactory, with no obvious grammatical errors, spelling mistakes, or other issues.

Author Response

1. In the Introduction section, it is recommended that the author presents specific problems in specific application scenarios, using figures or data tables for illustration. In lines 37-44, what are the quantitative evaluation metrics for the "Reconstruction of Light Field" in existing methods? How is the superiority or inferiority of a method judged? At the same time, it is recommended that the author provide quantitative descriptions for the simulation and experimental figures in the following sections based on the proposed quantitative metrics, which will make it easier for readers to understand the effectiveness of the proposed method.

In the Introduction section, it is recommended that the author presents specific problems in specific application scenarios, using figures or data tables for illustration.

In response to the problem of phase unwrapping in situations with high noise or steep gradients, this paper proposes a unwrapping algorithm based on multiple iterations of least squares optimization. In order to minimize the difference between the partial derivative of the true phase and the wrapped phase difference, this algorithm proposes an optimization algorithm using the Poisson equation derived from differentiation, combining with boundary conditions and iterative least squares solution strategy. This algorithm transforms the solution wrapping problem into an equivalent continuous minimization problem, ultimately obtaining the optimal solution.

The specific algorithm has been supplemented in the "theoretical analysis".

In lines 37-44, what are the quantitative evaluation metrics for the "Reconstruction of Light Field" in existing methods? How is the superiority or inferiority of a method judged?

Lines 37-44 describe the design of the "unwrapping phase" algorithm. Due to the complexity and variability of measurement objects, noise interference, measurement system errors, and other complex factors, efficient and accurate phase unwrapping has become a very difficult problem. At present, there is no algorithm that can completely solve all the problems in phase unwrapping. Therefore, based on the characteristics of interference fringes, selecting and optimizing algorithms for different phase wrapping patterns plays a key role.

Among the three algorithms introduced in this paper, the residual point search method has a large error in unwrapping due to noise; The row and column search method will result in different results due to the difference in searching first row and then column or searching first column and then row. In the case of multiple iterations, the least squares method not only achieves simplicity and is not limited by phase quality, but also maintains good phase continuity. So the improved least squares method with multiple iterations was chosen for the solution of the unwrapping algorithm.

There are three common methods for reconstructing the phase of light field: phase shift method, diffraction calculation method, and Fourier transform method. The advantages and disadvantages among the three are also explained in the third part "Simulation Verification". Finally, the conclusion is drawn that the four step phase-shifting method can accurately obtain the distribution of object light waves during digital holographic reconstruction, effectively eliminating the influence of random errors and CCD noise. So the "four step phase-shifting method" was chosen for light field reconstruction.

At the same time, it is recommended that the author provide quantitative descriptions for the simulation and experimental figures in the following sections based on the proposed quantitative metrics, which will make it easier for readers to understand the effectiveness of the proposed method.

The following provides explanations for each simulation model: relevant parameter designs and algorithm formulas are explained quantitatively. Seven tables have been added to this paper for clarification.

2. In the "Theoretical Analysis" section, the author provides a detailed theoretical derivation of the traditional method. However, I believe that further theoretical derivation is needed to explain the mechanism by which the author's proposed method can suppress the impact of adverse factors such as noise, shadows, and fractures. The current theoretical derivation does not show how noise is suppressed.

The specific theoretical derivation has been supplemented in the second part "Theoretical Analysis".

3. In the "Experimental Verification" section, it is recommended that the author add "experimental verification figures with added noise".

The optical system design of the entire experimental verification section, both in hardware and algorithm design, considers the elimination of phase gradient truncation caused by undersampling through multiple iterations of the least squares method in the presence of noise, while minimizing error propagation outside the residual region, in order to obtain better unwrapping results. Through simulation and experimental comparison, it has been proven that the least squares method has better suppression of error propagation after multiple iterations, which to some extent improves the stability of the algorithm. Comparison was made in simulation figure 13: no noise, noise of 0.3, and the effect after multiple iterations using the least squares method for improvement.

Comparing with traditional iteration, the improved algorithm solves the following problems. During the process of unwrapping the phase, if it passes through an area of phase inconsistency, the phase error will not propagate throughout the entire space. This paper uses data fusion algorithm to improve the least squares method iteration. and even when the noise is relatively high, the true unwrapping phase can still be obtained normally. The unwrapping phase can be compared with the true phase, which is close to the result when the external noise coefficient is 0.06, as shown in Figures 13(e) and 13(f). It can also smooth out defects in the real phase, which means that if there are spikes or steep slopes, corresponding smoothing effects can be applied. It has improved the stability and accuracy of the algorithm to some extent.

4. Additionally, there are some formatting issues that need to be revised.

(1)What is the horizontal axis in Figure 1?

Pixels, marked as 10, 20, 30, etc. Already modified.

(2)In lines 65 and 66, what do "bright lines" and "dark lines" refer to?

There is a problem with the translation, it is a combination of bright and dark stripes. Already modified.

(3)The entire text should be written in English and should not be a mixture of Chinese and English. For example, in Figures 6, 12, 15, and 16.

Already modified.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript by Li et al, the authors investigated the reconstruction of light field through four step phase shift method and multiple iterations of least squares method for phase unwrapping. In the transmission process of light field, phase plays a crucial role, so the research content of this manuscript has important scientific significance. In this manuscript, the authors mainly used two methods, namely four step phase shift method and multiple iterations of least squares method, to conduct an in-depth analysis of the phase reconstruction problem of the light field. They obtained the wrapped phase of the four step phase-shifting interferometric digital hologram, and the least squares algorithm was used for multiple iterations to unwrap the phase of the object light field on the holographic recording surface reconstructed by the four step phase-shifting method, thereby obtaining the phase distribution of the interference information field. The writing of this manuscript is standardized, and the results obtained are of scientific significance. Therefore, I recommend that this manuscript can be accepted. Here are some small modification suggestions.

 

1. This article focuses on two methods, namely four step phase shift method and multiple iterations of least squares method. The author did not provide any relevant references on the achievements of these two methods in previous applications. It is suggested that the author increase the introduction of the methods used in this manuscript, the progress in this field, the differences between the results of this manuscript and before, and highlight the importance of achieving results in this manuscript.

2. The introduction of this manuscript needs to be revised and improved. In the current version, only four references are cited in the introduction section, which does not reflect the relevant progress in this field and the important scientific significance of this study. Phase plays an important role in optical transmission, and different phases can generate different transmission states. For example, cross phase can generate rotation of the optical field, such as Chaos, Solitons and Fractals, 2024, 178: 114398; and complex variable optical fields can generate diverse mode transformations, such as Communications in Nonlinear Science and Numerical Simulation, 2021, 103: 106005 and Applied Mathematics Letters, 2022, 125: 107755. I suggest the author should enrich the content of the introduction section.

3. On Page 4, in Eqs. (10) and (13), it is recommended to change “imag” to “Image” and “real” to “Real”, and use regular font instead of italics.

4. Some formulas in the text have font sizes that are too large and should be revised, such as lines 150-152, lines 160-161 on page 6.

5. Line 178 on page 7, it is suggested to modify Figure 4-a to Figure 4(a); line 184, Figure 4-c-f to Figures 4(c)-4(f); similar to others.

6. Some graphic coordinates have font sizes that are too small to distinguish. It is recommended to increase font sizes, such as in Figure 4(h), Figure 11(a), Figure 12(b); Some graphics still contain Chinese characters, it is recommended to delete them.

7. The format of references needs to be unified, and currently the format of references is inconsistent.

Author Response

1. This article focuses on two methods, namely four step phase shift method and multiple iterations of least squares method. The author did not provide any relevant references on the achievements of these two methods in previous applications. It is suggested that the author increase the introduction of the methods used in this manuscript, the progress in this field, the differences between the results of this manuscript and before, and highlight the importance of achieving results in this manuscript.

The introduction of the methods used and the progress in this field：

Kazuyoshi Itoh conducts mathematical analysis on one-dimensional phase unwrapping algorithm^[5]; Takeda extended this theoretical foundation to two-dimensional phase unwrapping and proposed a row and column point by point algorithm for unwrapping phase ^[6]; Goldstein proposed the Branch Cut phase unwrapping algorithm^[7]; Bone proposed the quality graph oriented phase unwrapping algorithm^[8]; Ghiglia proposed least squares phase unwrapping algorithm^[9]; Phase unwrapping technology has flourished in recent years. Qian Xiaofan et al. proposed a solution phase unwrapping algorithm based on mask and least squares iteration^[10]. Jin Guowang et al. from the University of Information Engineering proposed a novel phase unwrapping algorithm for instantaneous frequency estimation based on the relationship between complex light field and interference fringe frequency ^[11]. Servin M proposed a "network planning based algorithm" that can effectively suppress error propagation without identifying residual points ^[12]. These algorithms have emerged one after another, driving the further development of phase unwrapping technology.

Although many experts and scholars have proposed many algorithms at home and abroad, there is currently no algorithm that can completely solve all the problems faced in phase unwrapping. Therefore, it is necessary to conduct in-depth research on phase unwrapping algorithms and find the optimal and suitable phase unwrapping algorithm for various complex situations in digital holographic microscopy measurements.

[5] Kazuyoshi Itoh, Analysis of the phase unwrapping algorithm[J]. Appl. Opt, 1982,21(14):2470.

[6] Mitsuo Takeda, Hideki Ina, and Seiji Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry[J], Journal of the Optical Society of America, 1982,72(1):156-160.

[7] Goldstein R.M., Zebker H.A., Werner C.L, Satellite radar interferometry: Two-dimensional phase unwrapping[J], Radio Science, 1988,23(4):713-720.

[8] Bone D.J, Fourier fringe analysis: the two-dimensional phase unwrapping problem[J], Applied Optics, 1991,30(25):3627-3632.

[9] Ghiglia D.C., Romero L.A, Minimum L^P-norm two-dimensional phase unwrapping[J], Journal of the Optical Society of America A, 1996,13(10):1999-2013.

[10] Xiaofan Qian, Yong′an Zhang, Xinyu Li, Hui Ma, Phase unwrapping method based on mask and least squares iteration[J], Acta Optica Sinica, 2010,30(2):440-444。

[11] Guowang Jin, Guohua Xu, Maoxun Yu, Xiong Tan, Yuchi Zhang, InSAR phase unwrapping based on instantaneous frequency estimation[J], Journal of Surveying and Mapping Science and Technology, 2009,1:33-35,40.

[12] Servin M., Padilla M., Garnica G., Gonzalez A, Profilometry of three-dimensional discontinuous solids by combining two-steps temporal phase unwrapping, co-phased profilometry and phase-shifting interferometry[J], Optics and Lasers in Engineering, 2016,87:75-82.

The paper has been modified.

The differences between the results of this manuscript and before:

In response to the problem of phase unwrapping in situations with high noise or steep gradients, this paper proposes a unwrapping algorithm based on multiple iterations of least squares optimization. In order to minimize the difference between the partial derivative of the true phase and the wrapped phase difference, this algorithm proposes an optimization algorithm using the Poisson equation derived from differentiation, combining with boundary conditions and iterative least squares solution strategy. This algorithm transforms the solution wrapping problem into an equivalent continuous minimization problem, ultimately obtaining the optimal solution.

 The paper has been modified.

The highlight the importance of achieving results:

This algorithm can effectively obtain the unwrapping phase. The idea of multiple iterations is: (1) To a certain extent, reducing errors and improving the accuracy of unwrapping phase under noise; (2) Improving the quality of phase unwrapping while minimizing the difference between the partial derivative of the true phase and the phase difference of the wrapped phase, reducing computation time. (3) The boundary conditions are all integers during the iteration process, further improving the robustness of the wrapped algorithm.

 The paper has been modified.

2. The introduction of this manuscript needs to be revised and improved. In the current version, only four references are cited in the introduction section, which does not reflect the relevant progress in this field and the important scientific significance of this study. Phase plays an important role in optical transmission, and different phases can generate different transmission states. For example, cross phase can generate rotation of the optical field, such as Chaos, Solitons and Fractals, 2024, 178: 114398; and complex variable optical fields can generate diverse mode transformations, such as Communications in Nonlinear Science and Numerical Simulation, 2021, 103: 106005 and Applied Mathematics Letters, 2022, 125: 107755. I suggest the author should enrich the content of the introduction section.

The reason why digital holography technology has become a symbol of accurate measurement of three-dimensional information in the field of interferometry is that phase information is the most important part of three-dimensional information. Phase plays an important role in optical transmission, and different phases will produce different transmission states.^[1-3] Measuring phase information can accurately largely explain the basic three-dimensional condition of the object. The necessary technical step is the phase unwrapping algorithm in this important technical field.

 

[1] Zhuoyue Sun, Duo Deng, Zhaoguang Pang, Zhenjun Yang, Nonlinear transmission dynamics of mutual transformation between array modes and hollow modes in elliptical sine-Gaussian cross-phase beams[J], Chaos, Solitons & Fractals, 2024,178:114398.

[2] Shuang Shen, Zhenjun Yang, Xingliang Li, Shumin Zhang，Periodic propagation of complex-valued hyperbolic-cosine-Gaussian solitons and breathers with complicated light field structure in strongly nonlocal nonlinear media[J], Communications in Nonlinear Science & Numerical Simulation, 2021,103:106005.

[3] Shuang Shen, Zhenjun Yang, Zhaoguang Pang, Yanrong Ge, The complex-valued astigmatic cosine-Gaussian soliton solution of the nonlocal nonlinear Schrödinger equation and its transmission characteristics[J]，Applied Mathematics Letters, 2022,125:107755.

3. On Page 4, in Eqs. (10) and (13), it is recommended to change “imag” to “Image” and “real” to “Real”, and use regular font instead of italics.

Im represents the imaginary part, Re represents the imaginary part, using regular fonts.

Already modified.

4. Some formulas in the text have font sizes that are too large and should be revised, such as lines 150-152, lines 160-161 on page 6.

Already modified.

5. Line 178 on page 7, it is suggested to modify Figure 4-a to Figure 4(a); line 184, Figure 4-c-f to Figures 4(c)-4(f); similar to others.

All similar issues have been modified.

6. Some graphic coordinates have font sizes that are too small to distinguish. It is recommended to increase font sizes, such as in Figure 4(h), Figure 11(a), Figure 12(b); Some graphics still contain Chinese characters, it is recommended to delete them.

All similar issues have been modified.

7. The format of references needs to be unified, and currently the format of references is inconsistent.

All similar issues have been modified.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

In the revised manuscript, the author has made detailed revisions to the "Research Background," "Theoretical Analysis," "Experimental Verification," and other aspects according to my suggestions. The quality of the paper has been greatly improved.

However, I believe there is still room for improvement in the analysis of the noise suppression mechanism. In lines 199-266 of the revised manuscript, it is not clear how the noise is suppressed. I suggest that the author include noise terms in the derivation process to make the theory more complete.

Comments on the Quality of English Language

The quality of the English language is very good. There are no suggestions for revision.

Author Response

Comments and Suggestions for Authors

In the revised manuscript, the author has made detailed revisions to the "Research Background," "Theoretical Analysis," "Experimental Verification," and other aspects according to my suggestions. The quality of the paper has been greatly improved.

However, I believe there is still room for improvement in the analysis of the noise suppression mechanism. In lines 199-266 of the revised manuscript, it is not clear how the noise is suppressed. I suggest that the author include noise terms in the derivation process to make the theory more complete.

 

• The least squares method based on discrete cosine transform (DCT) is used to obtain the unwrapping phase, with the aim of solving the Poisson equation.
• Calculating ;
• Performing discrete cosine transform on to obtain its value  in the DCT domain;
• Finding the exact solution of in the DCT domain;
• Performing a two-dimensional inverse discrete cosine transform on the exact solution can obtain the true phase of the unwrapping phase.

This is the traditional least squares method for unwrapping phase.

• Perform multiple iterations for improvement;

The algorithm derivation process is shown in (27) - (25). Noise is randomly added through the Peaks function. This algorithm can obtain good results for unwrapping phase. The idea of multiple iterations is to address the problem of wrapping phase in the presence of noise. This paper proposes a wrapping phase algorithm based on the least squares method for multiple iterations optimization. In order to minimize the difference between the partial derivative of the true phase and the wrapped phase difference, this algorithm uses the derivative of the Poisson equation, combined with boundary conditions and iterative least squares solution strategy, to propose an optimization algorithm. This algorithm transforms the unwrapping phase problem into an equivalent continuous minimization problem, ultimately obtaining the optimal solution.

3、 Adding noise:

Randomly superimpose a noise on the peak function with a pixel size of 256 * 256, which is represented by a random function with values ranging from -0.1π to 0.1π. The noise coefficient is assumed to be 0.3, thus obtaining a two-dimensional true phase containing noise, as shown in Figure 13 (c). Using traditional algorithms, the phase wrapped in solution 13 (d) is obtained; When there is noise, the improved algorithm model "Multiple Iterations of Least Squares Method" is used to obtain the unwrapped phase of 13 (f). It is found that there is a significant change in 13 (f) compared to 13 (d), and the true phase obtained is better after unwrapping phase

 

It should be noted that through the random superposition of noise on the peaks function, when the noise is not severe, using the least squares algorithm, the noise will be distributed to all points, and the actual noise shared by each point is not much. Therefore, the error between the calculated unwrapping phase and the true phase at each point is not significant. Through multiple iterations, an accurate solution to the unwrapping phase can be obtained. When the noise is severe or very severe, the least squares algorithm also distributes the noise to each point, but the actual noise shared by each point is not the same. The error is mainly concentrated near the noise source and peak, so it is discontinuous in these areas. However, through multiple iterations improved by the least squares method, the error can be eliminated and the unwrapping phase result can be close to the true phase value. The overall result is to reduce errors to a certain extent and improve the accuracy of unpacking under noise.

Author Response File: Author Response.docx