Real-time profilometry by bicolor grating video projection.

A novel real-time 2+1 three-dimensional(3D) measuring method based on bicolor grating video projection is proposed. Firstly, only two frames of bicolor gratings, in which the red channels are two sinusoidal fringes with a shifting phase of π/2 and the blue channels are the same background light equivalent to the DC component of the two sinusoidal fringes are encoded and arranged alternatively to synthesize into a repetitive bicolor grating video, While this video is projected onto the measured object, the real-time bicolor deformed pattern video can be recorded by using a color CMOS camera, and the bicolor deformed pattern sequence at different moments can be extracted by computer processing, so that the 2+1 algorithm can be used to accomplish real-time 3D measurement of moving object. Before measuring, we used the same method to design two sinusoidal fringes with a difference of π in their red channels, respectively, to calibrate the sensitivity ratio between the red and blue channels of the CMOS camera, which can effectively eliminate the chromaticity imbalance between R and B channels and reduce the color crosstalk. Experimental results and analysis confirm the feasibility and effectiveness of the proposed method. Because the proposed method needs a repetitive bicolor grating video synthesized with only two-frame bicolor gratings to be projected, the 3D measurement acquisition speed and real-time accuracy will be improved compared with the traditional 2+1 3D measuring method.

1 Introduction

With the rapid development of science and technology in photoelectric information technology, image data processing technology, computer technology and other relevant fields, optical three-dimensional(3D) surface measurement technology has also been developed rapidly [1-3]. Nowadays, optical 3D surface measurement technology is applied in many fields such as mechanical engineering, industrial monitoring, computer vision, biomedicine, physical modeling, and so on [4-6]. The commonly used 3D surface measuring methods based on structured light illumination [7, 8] include Fourier transform profilometry (FTP) [9, 10] and phase measurement profilometry (PMP) [11, 12]. FTP requires only one frame of phase shift diagram, it is capable of retaining the fringe frequency in the frequency domain, removing high-frequency noise and carrier waves, recovering the fringe pattern from the frequency domain to the spatial domain via the inverse Fourier transform, reproducing the fringe field distribution, and lastly calculating the phase value of the fringe field [13, 14]. PMP usually uses three or more frames of phase shift diagrams to retrieve the phase information of the measured object. As opposed to FTP, PMP exhibits a lower speed but a higher measuring accuracy, a wider range of heights of measuring object, less effect by different reflectivity values on object surface, and less effect exerted by drastic variations or fractures on object surface [15, 16]. The modern 3D measurement is committed to investigating a higher measuring speed, higher measuring accuracy, as well as a wider measurement application range; thus, PMP is recognized as a more effective choice [17-21].

When testing objects, vibrations are likely to break the method because of the fuzziness and random phase errors caused by sequential exposures. To reduce the measurement error caused by its influence, Peter L. Wizinowich proposed a 2+1 phase shift algorithm [22, 23]. This method requires two phase-shifted π/2 fringe patterns and two phase-shifted π fringe patterns, respectively. The first two are taken as the first and second frames of the common three-step PMP and the last two take the average as the third frame image. The proposed algorithm has the limited technical conditions and a narrow application scope [24]. The high-speed movement of objects will bring errors similar to vibration, to solve the error problem of moving objects, ZHANG S proposed an improved 2+1 phase shift algorithm [25]. The third frame directly projects the uniform plane illumination generated by the computer to replace the average effect of the third frame achieved by the two frames in the method Peter L.Wizinowich proposed. The traditional three-step PMP contains the phase information of the object in three pictures, respectively, however, in ZHANG S’ improved 2+1 phase shift algorithm, the phase information of the object is only contained in two pictures. Since the sensitivity of the uniform plane graph to motion blur is significantly lower than that of the fringe graph, this method, compared with the conventional 2+1 phase shift algorithm, employs one frame less and exhibits a higher acquisition speed. At the same time, the digital projector projection directly eliminates the influence of nonlinear error of phase shifter, so ZHANG S’ improved 2+1 phase shift algorithm can effectively reduce the measurement error caused by object motion.

In this paper, a novel bicolor 2+1 3D real-time measuring method based on grating video projection was proposed. First, only two frames of bicolor patterns are used, as against the three frames which are essential in the traditional 2+1 3D real-time measuring method. These two frames are encoded and arranged alternately to synthesize a repetitive bicolor grating video. Then the video is projected onto the object and the bicolor deformed patterns captured by the color camera that covered two frames of phase-shifting sinusoidal deformed patterns plus one background pattern could be extracted through the computer to reconstruct the 3D shape of the object measured. Thus, the 3D measurement acquisition speed and real-time measuring accuracy will be improved compared with the traditional 2+1 3D measuring method.

2 Principle

2.1 ZHANG S’ 2 +1 phase shift algorithm

When a sinusoidal grating pattern is projected onto the surface of a 3D diffuse reflective object, the deformed pattern acquired from the imaging system may be represented as:

In the Eq (1), (x,y) represents the pixel point coordinates of the captured image, a expresses the component of DC term of the fringe, b expresses the fringe contrast; R(x,y) expresses the reflectivity of different positions on the surface of the object; φ(x,y) expresses the phase modulated by the height of the measured object; δi represents the shifted phase of the i-frame fringe pattern.

ZHANG S’ 2 +1 phase shift algorithm projects two frames of sinusoidal grating with phase shift π/2 and one frame of uniform background light with gray scale equal to the DC term of the sinusoidal grating. The two frames of deformed patterns and one frame of background image modulated by the surface shape of the object obtained from the imaging system can be expressed as:

This method considers that the light intensity distribution of the background map collected by the camera is equal with the light intensity distribution of the DC term of the deformed fringe pattern, i.e., a = a’. From this, the difference between Eqs (2)–(4) respectively can obtain two light intensity distributions with only AC term. Finally, the expression of φ(x,y) can be obtained through calculation:

2.2 Theory of the proposed method

As we all know, the responsiveness of color CMOS camera to red, green, and blue light is different, and because the wavelength difference between red and blue light is relatively large, only red and blue color channels are used in this paper to reduce the phenomenon of color crosstalk. In practical application, the projected chromaticity and projected light intensity of the red and blue channels of the color projector [26] are different, and the sensitivity of the red and blue channels to light is also different when a color CMOS camera is collected. If two frames of red and blue uniform background lights with the identical value as the DC component of the sinusoidal fringe are separately projected, this can address the chromaticity imbalance problem attributed to the different sensitivities of the CMOS camera to red and blue. Subsequently, the red color of the identical color image in the practical experiment is ignored. To possibly eliminate the chromaticity imbalance attributed the different sensitivities of the CMOS camera to the red and blue lights, two bicolor gratings are designed in advance:

The red channel () in Eq (6) expresses the sinusoidal fringe, the blue channel () expresses the background light corresponding to the sinusoidal fringe; The red channel () in Eq (7) expresses a fringe pattern complementary to the sinusoidal fringe of Eq (6),the blue channel () expresses still the corresponding background light; The green channels () expresses all zero set to avoid interference to the red and blue channels as well cut the interference between the red and blue channels to some extent.

These two bicolor gratings are arranged alternatively to synthesize into a repetitive bicolor grating video. While this video is projected onto the reference plane, the fringe patterns of the reference plane captured by the color CMOS camera can be expressed as:

Rr(x,y) and Rb(x,y) denote the reflectivity of the red channel and the blue channel, respectively, and φ0(x,y) expresses the phase modulated by the reference plane. RGB tricolor extraction is carried out on the fringe image to obtain 2 monochromatic fringe patterns and 2 monochromatic background images:

As indicated from the in-depth analysis, Rr(x,y) and Rb(x,y) exhibit the similar characteristics. The ratio of reflectivity of the red channel and the blue channel is a constant. This is attributed to the chromaticity imbalance attributed to the different sensitivities of the CMOS camera to the red and blue lights. We introduce a chromaticity imbalance coefficient K(x,y) caused by different sensitivities of CMOS camera to red and blue light, and we can obtain:

During real-time measurement, two other bicolor gratings with π/2 phase difference are arranged alternatively to synthesize into a new repetitive bicolor grating video. While this new video is projected onto measured object, the bicolor deformed patterns obtained from the imaging system can be expressed as follows:

Similarly, 2 monochromatic deformed patterns and 2 monochromatic background images can be extracted:

Considering the chromaticity imbalance coefficient K(x,y) caused by different sensitivities of CMOS camera to red and blue light, it can be solved simultaneously as follows:

The phase calculated by the Eq (23) is wrapped within the principal value range of the inverse trigonometric function (-π, π] [27, 28]. The wrapped phase can be restored to a continuous phase through the phase unwrapping algorithm [29]. Finally, the 3D surface shape of the object can be reconstructed through the phase height mapping calculation of each point on the surface of the object.

3 Experimental results and analysis

To verify the measurement effect of the proposed method on real objects, an experimental system shown in Fig 1 is built. The digital projector used was the PLED-W200 model produced by ViewSonic. The camera used was DFM 72AUCO2, a color industrial camera produced by The Imaging Source. The size of the images captured by the camera is 640pixel×480pixel.

Fig 1

Experimental system.

3.1 Comparison experiment

Before measuring the 3D shape of the measured object, we make the two-frame bicolor gratings Ibi1(x,y) and Ibi2(x,y) into a repetitive video and project it onto the reference plane through PLED-W200. Since we only need to use the red and blue channels in this paper, we set the green channel empty. The CMOS camera synchronously records the bicolor fringe pattern video in real time, so the bicolor fringe pattern sequences can be extracted through computer processing by selecting the corresponding two adjacent frames of the bicolor fringe patterns Ic1(x,y), Ic2(x,y), and the chromaticity imbalance coefficient can be calibrated in the pre-experimental procedure.

The two-frame bicolor gratings Ibi1(x,y) and Ibi2(x,y) are made into a repetitive video and projected onto a stationary face model object through PLED-W200. The bicolor deformed pattern video is collected synchronously in real time by the CMOS camera, and the experimental data of two adjacent frames of deformed patterns extracted by the computer are shown in Fig 2.

Fig 2

Experimental data.

(a)Measured object, (b) grating pattern, (c)1st deform pattern, (d)2nd deform pattern, (e)1st R deform pattern, (f)2nd R deform pattern, (g) 1st B background image, (h) 2nd B background image.

Fig 2(A) is the face model tested in the experiment. Fig 2(B) are bicolor grating patterns with a phase difference of . Fig 2(C) and 2(D) are the corresponding two frames of bicolor deformed patterns. Fig 2(E) and 2(F) are monochromatic deformed patterns extracted from R channel respectively, and Fig 2(G) and 2(H) are background images extracted from B channel, respectively.

As we all know, the result of traditional eight-step PMP has relatively high measuring accuracy [30], so the reconstruction result of traditional eight-step PMP is regarded as the quasi-truth in our experiments. Fig 3 is the experimental results. Fig 3(A) is a face model reconstructed by ZHANG S’ 2+1 phase shift algorithm, and Fig 3(B) is a face model reconstructed by the bicolor 2+1 3D proposed method, both of which can form a better 3D face profile.

Fig 3

Experimental results and height error distribution.

(a)Object reconstructed by ZHANG S′s algorithm, (b)Object reconstructed by proposed algorithm, (c)Sectional view of the 3D profile of the object, (d)Magnified view of the area A(e)Magnified view of area B, (f) Height error distribution of ZHANG S′s algorithm, (g) Height error distribution of proposed algorithm.

Fig 3(C)–3(E) are cross-sectional views of three methods including the traditional eight-step PMP, ZHANG S’ 2+1 phase shift algorithm, and the proposed method. Dotted lines correspond to eight-step PMP, dash-dotted lines correspond to ZHANG S’ 2+1 phase shift algorithm, and solid lines correspond to proposed method. Fig 3(C) is an overall cross-sectional view of the 319th column of the reconstructed face, Fig 3(D) is an enlarged view of the area A of Fig 3(C) and 3(E) is an enlarged view of area B of Fig 3(C). Analysis shows that the results of the two methods are close to the eight-step PMP, and the proposed method is closer to the traditional eight-step PMP. As opposed to the results obtained with the eight-step PMP results, the algorithm height error of ZHANG S as shown in Fig 3(F), and the algorithm height error of the proposed method is shown in Fig 3(G). It can be clearly seen that the height error of the proposed method is smaller. Calculating the height root mean square error of the two methods, the result of the algorithm of ZHANG S can be obtained as 0.1913, the proposed method can be obtained as 0.0629.

3.2 Real-time experiment

The experiment in this paper only needs to project two frames of bicolor gratings, which can effectively increase the speed of 3D measurement and improve efficiency. When the object moves online with the linear motion guide, if the projected two frames of bicolor grating are made into a periodic repetitive video, the PLED-W200 digital projector is used to project the video onto the surface of the online moving object. The bicolor deformed pattern video is synchronously recorded in real time by the color CMOS camera, and the bicolor deformed pattern sequences at different times are extracted by computer processing, to accomplish real-time online 3D measurement of the moving object.

Here, a double-layer rectangular model moving online is measured in real-time. By starting the stepper motor to control the heart model moving online, DLP stably projects the bicolor grating video onto the double-layer rectangular model. The bicolor deformed patterns at different times are simultaneously collected by the CMOS camera, and each group of deformed patterns are captured at 30 frames per second (30fps), are shown in S1 Visualization. The part of the experimental results are shown in Fig 4.

Fig 4

Experimental results of real-time 3D measurement for moving double-layer rectangular model.

(a)-(c) 1st-3rd deform patterns, (d)-(f) 1st-3rd monochromatic deform pattens, (g)-(i)objects reconstructed of time 1 to 3.

Fig 4(A)–4(C) respectively show the bicolor deformed patterns extracted by the real-time online moving double-layer rectangular model under the conditions of state 1, 2, and 3 respectively. Fig 4(D)–4(F) respectively show the monochrome deformed patterns extracted by the real-time online moving double-layer rectangular model under the corresponding state conditions. Fig 4(G)–4(I) respectively show the 3D shapes reconstructed by the real-time online moving double-layer rectangular model under the corresponding time conditions, more reconstructed results are shown in S2 Visualization. The 3D topography of the real-time moving double-layer rectangular model in different time states can be well reconstructed. Thus, it proves that the proposed method is suitable for real-time 3D measurement.

In order to further verify the measuring accuracy of the proposed method, we here take the traditional eight-step PMP result of measuring the static double-layer rectangular model as the quasi-truth and conduct a real-time online measurement comparison experiment with the ZHANG S’ method. Before the experiment, the measured object is accurately positioned and controlled at a specific position by a stepping motor. The traditional eight-step PMP is used to perform a static measurement at the specific position to obtain the quasi-truth of the reconstructed surface of the measured object, and then the steps are precisely controlled. When the electric motor moves the measured object to a specific position, real-time online measurements of the two methods are carried out to ensure the comparability of the reconstruction results of different methods. The comparison result is shown in Fig 5.

Fig 5

The 235th row comparison results of the reconstructed object.

(a) Sectional view of the 3D profile of the object, (b) Magnified view of the A area, (c) Magnified view of the B area(d) Magnified view of the C area, (e) Height error distribution of ZHANG S′s algorithm, (f) Height error distribution of proposed algorithm.

Fig 5 is a cross-sectional view of a reconstructed double-layer rectangular model 3D surface, in which the dotted lines correspond to eight-step PMP, the dash-dotted lines correspond to ZHANG S’ 2+1 phase shift algorithm, and the solid lines correspond to proposed method. Fig 5(A) is an overall cross-sectional view of the 250th row of the reconstructed double-layer rectangular model 3D surface, Fig 5(B) is an enlarged view of the heart model A of Fig 5(A) and 5(C) is an enlarged view of the part B of Fig 5(A) and 5(D) is an enlarged view of the part C of Fig 5(A). Analysis shows that, the results of the proposed method are close to the eight-step PMP, ZHANG S’ method is not effective in restoring the middle connection of the double-layer rectangle. From Fig 5(B) and 5(D), due to the time difference between real-time motion, real-time projection and capture of the object, the object is slightly deformed during imaging, and the real-time bicolor 2+1 image is closer to the static real result, which shows that the bicolor 2+1 projection of two frames has better real-time performance than the ZHANG S’ phase shift algorithm of three-frame projection. As opposed to the results obtained with the eight-step PMP results, the real-time algorithm height error of ZHANG S as shown in Fig 5(E), and the real-time algorithm height error of the proposed method is shown in Fig 5(F). It can be clearly seen that the height error of the proposed method is smaller. Calculating the height root mean square error of the two methods, the result of the algorithm of ZHANG S can be obtained as 0.1055, the proposed method can be obtained as 0.0517.

According to the above comparison method, we select the speed as 24mm/s, 30mm/s, and 36mm/s respectively by controlling the speed of the stepper motor to complete real-time experiment. Then, we calculate the corresponding root mean square errors and made a table as seen in Table 1.”

Table 1

The height root mean square error at different speeds.

Speed(mm/s)method	24mm/s	30mm/s	36mm/s
ZHANG S	0.0817	0.1055	0.1443
Proposed method	0.0410	0.0517	0.0706

It can be observed that the measuring effect becomes worse as the speed increases. The reason for this result is that the higher the speed of the object was, the greater the amplitude of the object’s vibration would be, the more factors that the captured image would interfere with, the more unstable resulting distorted image will be and the worse the measurement effect would be.

In order to further prove the feasibility of this method, a more complex sector model was measured in real time according to the same method. The bicolor deformed patterns at different times are simultaneously collected by the CMOS camera, and each group of deformed patterns are captured at 30 frames per second (30fps), are shown in S3 Visualization. The part of the experimental results is shown in Fig 6.

Fig 6

Experimental results of real-time 3D measurement for moving sector model.

(a)-(c) 1st-3rd deform patterns, (d)-(f) 1st-3rd monochromatic deform pattens, (g)-(i)objects reconstructed of time 1 to 3.

Fig 6(A)–6(C) respectively show the bicolor deformed patterns extracted by the real-time online moving sector model under the conditions of state 1, 2, and 3 respectively. Fig 6(D)–6(F) respectively show the monochrome deformed patterns extracted by the real-time online moving sector model under the corresponding state conditions. Fig 6(G)–6(I) respectively show the 3D shapes reconstructed by the real-time online moving sector model under the corresponding time conditions, all reconstructed results are shown in S4 Visualization. It can be observed that the 3D reconstruction of the sector model has also been well restored, and its rotating tense can be restored in real time, which further proves the real-time feasibility of the proposed method.

4 Conclusion

In this paper, a novel 2+1 3D real-time measuring method based on bicolor grating video projection is proposed. Compared with the traditional 2+1 3D real-time measurement, which requires three frames of patterns, the proposed method only needs to encode two frames of bicolor grating and arranged alternatively to synthesize into a repetitive bicolor grating video, thus shortening the 3D measurement time and effectively improving the real-time performance of 2+1 measurement. By introducing the chromaticity imbalance coefficient between RGB channels, after the deformation fringe is calibrated, the error caused by chromaticity imbalance is effectively reduced on the one hand, and the color crosstalk problem is also alleviated on the other hand. The validity and feasibility of this method are verified by experiments, and it has a good application prospect in real-time 3D measurement. Of course, this method has certain limitations for measuring three-dimensional objects with colored surfaces.

Moving double-layer rectangular model experiment.

The 12 frames of deformed patterns collected during the movement of the double-layer rectangular model are extracted and made into a gif.

(GIF)

Click here for additional data file.

Reconstruction results of moving double-layer rectangular model.

The 12 frames of the 3D shapes reconstructed by the real-time online moving double-layer rectangular model under the corresponding moving states and made into a gif.

(GIF)

Click here for additional data file.

Moving sector model experiment.

The 10 frames of deformed patterns collected during the movement of the sector model are extracted and made into a gif.

(GIF)

Click here for additional data file.

Reconstruction results of moving sector model.

The 10 frames of the 3D shapes reconstructed by the real-time online moving sector model under the corresponding moving states and made into a gif.

(GIF)

Click here for additional data file.

12 Apr 2021

PONE-D-21-03909

Real-time profilometry by bicolor grating video projection

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Reviewer #2: No

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Reviewer #1: This paper proposed a novel real-time 2+1 three-dimensional measuring method based on a two-color fringe projection. In general, the structure of this paper is clear, and there are some new ideas especially on the fusion and phase recovery of two-color fringes. But the article still has many problems before published.

1. The English writing needs to be revised, because there are too many wording mistakes which influence the readability of the article. For example, in line 40, ‘faster speed’ should be placed by ‘higher speed’; in line 41, ‘wider measurement application range’, the meaning is not clear; in line 48, ‘application scope is very narrow’; many wrong uses of ‘Figs’ in line 170…; in line 174 wrong use of ‘quasi-true’; in line 222, wrong use of ‘cow’; invalid format in line 170 ‘π/2’, etc.

2. There are many descriptions hard to understand. For example, line 33-36; line 38-40; line 97-99; line 123-125…

3. Many references are miscoded. For example, line 31, 36, 42…

4. Suggest that the red number should be placed in the Fig.1.

5. About the experiment, there was no quantitative analysis in the comparative experiment. It will be better to give the residual figure between each of the method and measurement accuracy values.

6. I still have a question why the eight-step PMP was used for a reference not the five-step or the other method?

7. High speed measurements are expected to give speed values and comparisons.

At last, this paper should be revised on the English writing and format. I suggest the paper should be polished. The experiments should be richer. The accuracy values should be given as well as the speed values.

Hope the suggestions helps you.

Reviewer #2: The experimental section does not specify if the data is cherry-picked or averaged or repeatable over several frames. Unfortunately, the manuscript contains lots of English errors, and also, references are displayed as "Error! Reference source not found."

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Reviewer #1: Yes: Fumin Zhang

Reviewer #2: No

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11 May 2021

Dear Editor and Reviewers:

Thank you very much for your excellent editing work of our manuscript entitled “Real-time profilometry by bicolor grating video projection ([PONE-D-21-03909]-[EMID:a2a144e40538be21])”. And we also appreciate the reviewers for their valuable comments about our manuscript. Those comments are all valuable and very helpful for revising and improving our manuscript. We have studied the comments carefully and have made the correction which we hope to be able to meet the approval. The Response to Reviewer and Editor Comments is attached below. Hope our revised manuscript meets the high publication standard set by PLOS ONE.

Sincerely,

Yujiao Zhang, Yiping Cao, Haitao Wu, Haihua An, Xiuzhang Huang Sichuan University

Reviewer 1:

Comments to the Author

This paper proposed a novel real-time 2+1 three-dimensional measuring method based on a two-color fringe projection. In general, the structure of this paper is clear, and there are some new ideas especially on the fusion and phase recovery of two-color fringes. But the article still has many problems before published.

1. The English writing needs to be revised, because there are too many wording mistakes which influence the readability of the article. For example, in line 40, ‘faster speed’ should be placed by ‘higher speed’; in line 41, ‘wider measurement application range’, the meaning is not clear; in line 48, ‘application scope is very narrow’; many wrong uses of ‘Figs’ in line 170…; in line 174 wrong use of ‘quasi-true’; in line 222, wrong use of ‘cow’; invalid format in line 170 ‘π/2’, etc.

Response:

Thank you for raising this question, and I am sorry that my English grammatical errors and improper expressions have caused you difficulties in reading. I have corrected the grammar of the thesis from beginning to end. After receiving the modification comments, I did the real-time experiment again. And this time, I selected the 250th column of the reconstructed heart, so I changed the corresponding part of the experiment expression accordingly.

Revision:

1) In page 2 line 38, “slower speed” has been revised as “a lower speed”.

2) In page 2 line 38, “higher measuring accuracy, wider measurable range” has been revised as “a higher measuring accuracy, a wider range of heights of measuring object”.

3) In page 2 line 41, “faster measurement speed, higher measuring accuracy, and wider measurement application range” has been revised as “a higher measuring speed, higher measuring accuracy, a wider measuring application range”.

4) In page 9 line 172, “Figs 2(c) and 2(d)” has been revised as “Fig 2(c) and Fig 2(d)”.

5) In page 9 line 173, “Figs 2(e) and 2(f)” has been revised as “Fig 2(e) and Fig 2(f)”.

6) In page 9 line 174, “Figs 2(g) and 2(h)” has been revised as “Fig 2(g) and Fig 2(h)”.

7) In page 10 line 181, “Figs 3(c)-(e)” has been revised as “Fig 3(c)–Fig3 (e)”.

8) In page 12 line 208, “Figs 4 (a)-(c)” has been revised as “Fig 4(a)-Fig 4(c)”.

9) In page 12 line 209, “Figs 4(d)-(f)” has been revised as “Fig 4(d)-Fig 4 (f)”.

10) In page 12 line 211, “Figs 4(g)-(i)” has been revised as “Fig 4(g)-Fig 4(i)”.

11) In page 14 line 231, “Figs 5(b) and 5(d)” has been revised as “Fig 5(b) and Fig 5(d)”.

12) In page 9 line 176, page 12 line 216, page 13 line 220 “quasi-true value” has been revised as “quasi-truth”.

13) In page 13 between line 223 and 224, “323th cow” has been revised as “250th column”.

14) In page 14 line 228, “323th cow” has been revised as “250th column”.

15) In page 7 line 131, page 9 line 172 “π/2” has been revised as “ ”.

16) In page 3 line 61, “is” has been revised as “was”.

17) In page 3 line 61, “Firstly” has been revised as “First”.

18) In page 3 line 62, “are” has been revised as “were”.

19) In page 3 line 63, “is” has been revised as “was”.

20) In page 3 line 64, “containing” has been revised as “that covered”.

21) In page 3 line 65, “can” has been revised as “could”.

22) In page 3 line 66, “measured object” has been revised as “object measured”.

23) In page 4 line 73, page 4 line 74, page 4 line 75, page 5 line 104, page 5 line 105, page 5 line 106, page 5 line 107, page 6 line 115, “is” has been revised as “expresses”.

24) In page 6 line 114, “are” has been revised as “denote”.

25) In page 4 line 86, “expressions (2) and (3) and (4)” has been revised as “Eq (2), Eq (3) and Eq (4)”.

26) In page 6 line 114, we add two words “the”.

2. There are many descriptions hard to understand. For example, line 33-36; line 38-40; line 97-99; line 123-125…

Response:

I am sorry that I did not express clearly enough in these places, which caused your reading difficulties. I re-expressed the content of these places.

Revision:

1) In page 2 line 32-36, The content “FTP requires only one frame of phase shift diagram, retains the fringe frequency in the frequency domain, removes high frequency noise and carrier waves, recovers the fringe pattern from the frequency domain to the spatial domain through inverse Fourier transform, reproduces the fringe field distribution, and finally calculates the phase value of the fringe field [13-14].” has been revised as “FTP requires only one frame of phase shift diagram, It is capable of retaining the fringe frequency in the frequency domain, removing high-frequency noise and carrier waves, recovering the fringe pattern from the frequency domain to the spatial domain via the inverse Fourier transform, reproducing the fringe field distribution, and lastly calculating the phase value of the fringe field [13-14]”.

2) In page 2 line 37-42, The content “Compared with FTP, PMP has slower speed but higher measuring accuracy, wider measurable range, less influence by different reflectivity on the surface of the object, and less influence by drastic changes or fractures on the surface of the object[15-16]. Modern 3D measurement is devoted to the research of faster measurement speed, higher measuring accuracy, and wider measurement application range, so PMP is a better choice[17-21] .” has been revised as “As opposed to FTP, PMP exhibits a lower speed but a higher measuring accuracy, a wider measurable range, less effect by different reflectivity values on object surface, and less effect exerted by drastic variations or fractures on object surface[15-16]. The modern 3D measurement is committed to investigating a higher measuring speed, higher measuring accuracy, as well as a wider measurement application range; thus, PMP is recognized as a more effective choice [17-21].”.

3) In page 3 line 48-49, I corrected the sentence “The technical conditions of the proposed method are limited and the application scope is very narrow[24]”to “The proposed algorithm has the limited technical conditions and a narrow application scope [24]”.

4) In page 3 line 55-57, this sentence is a bit difficult to read, so I corrected “Because the sensitivity of the uniform plane graph to motion blur is much lower than that of the fringe graph, moreover, compared with the traditional 2+1 phase shift algorithm, this method uses less one frame of image and has faster acquisition speed.” to “Since the sensitivity of the uniform plane graph to motion blur is significantly lower than that of the fringe graph, this method, compared with the conventional 2+1 phase shift algorithm, employs less one frame of image and exhibits a higher acquisition speed.”

5) In page 5 line 96-99, The content “If two frames of red and blue uniform background light with the same value as the DC component of the sinusoidal fringe are separately projected to solve the chromaticity imbalance problem caused by the different sensitivity of the CMOS camera to red and blue, then the red color of the same color image in the actual experiment is ignored..” has been revised as “If two frames of red and blue uniform background lights with the identical value as the DC component of the sinusoidal fringe are separately projected to address the chromaticity imbalance problem attributed to the different sensitivities of the CMOS camera to red and blue. Subsequently, the red color of the identical color image in the practical experiment is ignored. To possibly eliminate the chromaticity imbalance attributed the different sensitivities of the CMOS camera to the red and blue lights, two bicolor gratings are designed in advance:”.

6) In page 6 line 124-127, The content “Further analysis shows that and have similar characteristics. It can be seen that the ratio of reflectivity of red channel and blue channel is in constant characteristic. This is caused by the chromaticity imbalance caused by the different sensitivities of CMOS camera to red and blue light.” has been revised as “As indicated from the in-depth analysis, and exhibit the similar characteristics. which demonstrates that the ratio of reflectivity of the red channel and the blue channel exhibits the constant characteristic. This is attributed to the chromaticity imbalance attributed to the different sensitivities of the CMOS camera to the red and blue lights.”.

3. Many references are miscoded. For example, line 31, 36, 42…

Response:

Thank you so much for your question, and I am sorry that my carelessness caused this format problem. I have corrected the reference number hyperlink problem.

Revision:

There are three numbering errors in the preview state.

1) In page 2 line 31, Number “[7-8]” has updated the number hyperlink；

2) In page 2 line 36, Number “[13-14]” has updated the number hyperlink；

3) In page 2 line 42, Number “[17-21]” has updated the number hyperlink；

4. Suggest that the red number should be placed in the Fig.1.

Response:

Thank you very much for your valuable comments. After my consideration, I also think that your suggestions for modification are better.

Revision:

So I replace the Fig.1

between lines 155 and 156 in page 8 with the following figure

5. About the experiment, there was no quantitative analysis in the comparative experiment. It will be better to give the residual figure between each of the method and measurement accuracy values.

Response:

Thank you so much for your suggestion. I am sorry that I did not do the quantitative analysis of the experiment. Now I am supplementing the quantitative analysis of the experiment. Compared with the eight-step PMP, ZHANG S and the proposed method are compared to obtain the height error distribution and calculate the height root mean square error. We did a set of static experiments again We respectively added the height error distribution of the ZHANG S algorithm (Fig3 (f)) and the proposed method relative to the eight-step PMP (Fig3 (g)) to Fig.3 in the static face model measurement. Then, we separately calculated the height root mean square error relative to the eight-step PMP, the result of the algorithm of ZHANG S can be obtained as 0.1913, the proposed method can be obtained as 0.0629.

In the same way, we also did a set of real-time experiments again, and the cross-sectional view is replaced with the 250th column. We respectively added the height error distribution of the ZHANG S algorithm (Fig.5(e)) and the proposed method relative to the eight-step PMP(Fig.5(f)) to Fig.5 in the real-time heart model measurement.

Revision:

In page 10, between line 177 and line 178, the measurement results of the face model at rest.

The content of the analysis is added between line 187 and 192 in page 11, the content presents “As opposed to the results obtained with the eight-step PMP results, the algorithm height error of ZHANG S as shown in Fig 3(f), and the algorithm height error of the proposed method is shown in Fig 3(g). It can be clearly seen that the height error of the proposed method is smaller. Calculating the height root mean square error of the two methods, the result of the algorithm of ZHANG S can be obtained as 0.1913, the proposed method can be obtained as 0.0629.”.

In page 13, between line 223 and line 224, the measurement results of the heart model at real time.

The content of the analysis is added between line 235 and 239 in page 14, the content presents “As opposed to the results obtained with the eight-step PMP results, the real-time algorithm height error of ZHANG S as shown in Fig 5(e), and the real-time algorithm height error of the proposed method is shown in Fig 5(f). It can be clearly seen that the height error of the proposed method is smaller. Calculating the height root mean square error of the two methods, the result of the algorithm of ZHANG S can be obtained as 0.1546, the proposed method can be obtained as 0.0513.”.

6. I still have a question why the eight-step PMP was used for a reference not the five-step or the other method?

Response:

Thank you so much for asking this question, and I am sorry for not being able to describe in detail the advantages of 8-step PMP as a quasi-truth method in my article. The reason for using this method is that I read a thesis called “Gamma model and its analysis for phase measuring profilometry”. From the thesis I know that Phase measuring profilometry is a method of structured light illumination whose three-dimensional reconstructions are susceptible to error from nonunitary gamma in the associated optical devices. The effects of this distortion diminish with an increasing number of employed phase-shifted patterns, so the more PMP steps, the better the measurement effect. I added this thesis as number 30 to the reference, located at line 175, and added “30. Liu, Kai, et al. "Gamma model and its analysis for phase measuring profilometry." JOSA A 27.3 (2010): 553-562.” in page 19 line 333-334.

7. High speed measurements are expected to give speed values and comparisons.

Response:

Thank you very much for your valuable suggestion. By controlling the speed of the stepper motor to control the real-time movement speed of the cardioid model, the corresponding root-mean-square is calculated according to the same method as the previous real-time comparison. The speed range of the stepper motor is 0~40mm/s. We choose 24mm/s, 30mm/s, and 36mm/s respectively for experimen

Revision:

The height root mean square is made into a table and the results are as follows and added the table and analysis content to page 15 between line 246 and 252:

Table.1 The height root mean square error at different speeds

24mm/s 30mm/s 36mm/s

ZHANG S 0.1435 0.1546 0.1866

Propose 0.0442 0.0513 0.7462

In page 14 line 240-242, we added the sentence of “According to the above comparison method, we select the speed as 24ms/s, 30ms/s, and 36ms/s respectively by controlling the speed of the stepper motor to complete real-time experiment. Then, to calculate the corresponding root mean square value and make a table1 expressed below:”

In page 15 line 244-246, we added the sentence of “It can be observed that the measuring effect becomes worse as the speed increases. The reason for this result is that the higher the speed of the object was, the greater the amplitude of the object’s vibration would be, the more factors that the captured image would interfere with, the more unstable resulting distorted image will be and the worse the measurement effect would be.”

Reviewer #2: The experimental section does not specify if the data is cherry-picked or averaged or repeatable over several frames. Unfortunately, the manuscript contains lots of English errors, and also, references are displayed as "Error! Reference source not found."

1. The experimental section does not specify if the data is cherry-picked or averaged or repeatable over several frames.

Response:

Thank you for raising this question, and sorry for not being able to express it clearly. This article uses video raster projection, and what the camera captures is also a video image. I selected three sets of adjacent two frames of images for three-dimensional measurement. For example, the video images selected in this paper have a total time of 9s, which corresponds to more than 160 frames of images. I selected frames 29-30, 89-90, and 139-140 to complete the three-dimensional measurement.

2. Unfortunately, the manuscript contains lots of English errors.

Response:

Thank you for raising this question, and I am sorry that my English grammatical errors and improper expressions have caused you difficulties in reading. I have corrected the grammar of the thesis from beginning to end.

Revision:

1) In page 2 line 38, “slower speed” has been revised as “a lower speed”.

2) In page 2 line 38, “higher measuring accuracy, wider measurable range” has been revised as “a higher measuring accuracy, a wider range of heights of measuring object”.

3) In page 2 line 41, “faster measurement speed, higher measuring accuracy, and wider measurement application range” has been revised as “a higher measuring speed, higher measuring accuracy, a wider measuring application range”.

4) In page 9 line 172, “Figs 2(c) and 2(d)” has been revised as “Fig 2(c) and Fig 2(d)”.

5) In page 9 line 173, “Figs 2(e) and 2(f)” has been revised as “Fig 2(e) and Fig 2(f)”.

6) In page 9 line 174, “Figs 2(g) and 2(h)” has been revised as “Fig 2(g) and Fig 2(h)”.

7) In page 10 line 181, “Figs 3(c)-(e)” has been revised as “Fig 3(c)–Fig3 (e)”.

8) In page 12 line 208, “Figs 4 (a)-(c)” has been revised as “Fig 4(a)-Fig 4(c)”.

9) In page 12 line 209, “Figs 4(d)-(f)” has been revised as “Fig 4(d)-Fig 4 (f)”.

10) In page 12 line 211, “Figs 4(g)-(i)” has been revised as “Fig 4(g)-Fig 4(i)”.

11) In page 14 line 231, “Figs 5(b) and 5(d)” has been revised as “Fig 5(b) and Fig 5(d)”.

12) In page 9 line 176, page 12 line 216, page 13 line 220 “quasi-true value” has been revised as “quasi-truth”.

13) In page 13 between line 223 and 224, “323th cow” has been revised as “250th column”.

14) In page 14 line 228, “323th cow” has been revised as “250th column”.

15) In page 7 line 131, page 9 line 172 “π/2” has been revised as “ ”.

16) In page 3 line 61, “is” has been revised as “was”.

17) In page 3 line 61, “Firstly” has been revised as “First”.

18) In page 3 line 62, “are” has been revised as “were”.

19) In page 3 line 63, “is” has been revised as “was”.

20) In page 3 line 64, “containing” has been revised as “that covered”.

21) In page 3 line 65, “can” has been revised as “could”.

22) In page 3 line 66, “measured object” has been revised as “object measured”.

23) In page 4 line 73, page 4 line 74, page 4 line 75, page 5 line 104, page 5 line 105, page 5 line 106, page 5 line 107, page 6 line 115, “is” has been revised as “expresses”.

24) In page 6 line 114, “are” has been revised as “denote”.

25) In page 4 line 86, “expressions (2) and (3) and (4)” has been revised as “Eq (2), Eq (3) and Eq (4)”.

26) In page 6 line 114, we add two words “the”.

27) In page 2 line 32-36, The content “FTP requires only one frame of phase shift diagram, retains the fringe frequency in the frequency domain, removes high frequency noise and carrier waves, recovers the fringe pattern from the frequency domain to the spatial domain through inverse Fourier transform, reproduces the fringe field distribution, and finally calculates the phase value of the fringe field [13-14].” has been revised as “FTP requires only one frame of phase shift diagram, It is capable of retaining the fringe frequency in the frequency domain, removing high-frequency noise and carrier waves, recovering the fringe pattern from the frequency domain to the spatial domain via the inverse Fourier transform, reproducing the fringe field distribution, and lastly calculating the phase value of the fringe field [13-14]”.

28) In page 2 line 37-42, The content “Compared with FTP, PMP has slower speed but higher measuring accuracy, wider measurable range, less influence by different reflectivity on the surface of the object, and less influence by drastic changes or fractures on the surface of the object[15-16]. Modern 3D measurement is devoted to the research of faster measurement speed, higher measuring accuracy, and wider measurement application range, so PMP is a better choice[17-21] .” has been revised as “As opposed to FTP, PMP exhibits a lower speed but a higher measuring accuracy, a wider measurable range, less effect by different reflectivity values on object surface, and less effect exerted by drastic variations or fractures on object surface[15-16]. The modern 3D measurement is committed to investigating a higher measuring speed, higher measuring accuracy, as well as a wider measurement application range; thus, PMP is recognized as a more effective choice [17-21].”.

29) In page 3 line 48-49, I corrected the sentence “The technical conditions of the proposed method are limited and the application scope is very narrow[24]”to “The proposed algorithm has the limited technical conditions and a narrow application scope [24]”.

30) In page 3 line 55-57, this sentence is a bit difficult to read, so I corrected “Because the sensitivity of the uniform plane graph to motion blur is much lower than that of the fringe graph, moreover, compared with the traditional 2+1 phase shift algorithm, this method uses less one frame of image and has faster acquisition speed.” to “Since the sensitivity of the uniform plane graph to motion blur is significantly lower than that of the fringe graph, this method, compared with the conventional 2+1 phase shift algorithm, employs less one frame of image and exhibits a higher acquisition speed.”

31) In page 5 line 96-99, The content “If two frames of red and blue uniform background light with the same value as the DC component of the sinusoidal fringe are separately projected to solve the chromaticity imbalance problem caused by the different sensitivity of the CMOS camera to red and blue, then the red color of the same color image in the actual experiment is ignored..” has been revised as “If two frames of red and blue uniform background lights with the identical value as the DC component of the sinusoidal fringe are separately projected to address the chromaticity imbalance problem attributed to the different sensitivities of the CMOS camera to red and blue. Subsequently, the red color of the identical color image in the practical experiment is ignored. To possibly eliminate the chromaticity imbalance attributed the different sensitivities of the CMOS camera to the red and blue lights, two bicolor gratings are designed in advance:”.

32) In page 6 line 124-127, The content “Further analysis shows that and have similar characteristics. It can be seen that the ratio of reflectivity of red channel and blue channel is in constant characteristic. This is caused by the chromaticity imbalance caused by the different sensitivities of CMOS camera to red and blue light.” has been revised as “As indicated from the in-depth analysis, and exhibit the similar characteristics. which demonstrates that the ratio of reflectivity of the red channel and the blue channel exhibits the constant characteristic. This is attributed to the chromaticity imbalance attributed to the different sensitivities of the CMOS camera to the red and blue lights.”.

3. references are displayed as "Error! Reference source not found.

Response:

Thank you so much for your question, and I am sorry that my carelessness caused this format problem. I have corrected the reference number hyperlink problem.

Revision:

There are three numbering errors in the preview state.

1) In page 2 line 31, Number “[7-8]” has updated the number hyperlink；

2) In page 2 line 36, Number “[13-14]” has updated the number hyperlink；

3) In page 2 line 42, Number “[17-21]” has updated the number hyperlink；

Submitted filename: Response to Reviewers (10).docx

Click here for additional data file.

28 May 2021

PONE-D-21-03909R1

Real-time profilometry by bicolor grating video projection

PLOS ONE

Dear Dr. Cao,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

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Academic Editor

PLOS ONE

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: No

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: No

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #2: No

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

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Reviewer #2: No

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #2: About the data - the authors have indicated in their response to my review that only three pairs of frames were used for the entire paper, out of 9 seconds of video with 160 frames. Some statistical error analysis of the other frames would be desirable, and the raw data should be made available as PLOS ONE's requirement is "PLOS journals require authors to make all data necessary to replicate their study's findings..."

After revision, the manuscript is much more readable. But typos, grammatical mistakes and or omissions remain.

Some suggestions -

1. Line 20 - feasibility and effectiveness of the proposed (method)?

2. Line 32 - add a space after [9-10]

3. Line 33 - It is capitalized after a comma. Either change the comma to a period or don't capitalize.

4. Line 36 - add a period before beginning the sentence with PMP

5. Line 45, Peter L. Wizinowich is in normal case, while in Line 50 and later throughout the document, ZHANG is in ALL CAPS. Normal case would be desirable.

6. Line 51 - "directly projects the uniform plan generated by the computer" - perhaps the authors mean uniform plane illumination generated by the computer? This phrase is also repeated throughout the manuscript.

7. Line 56-57 - employs less one frame of image - perhaps can be phrased as employs one frame less?

8. Line 61-62 - First, only two frames of bicolor patterns substituting for the three frames of essential patterns in the traditional 2+1 3D real-time measuring method... - maybe the authors mean "First, only two frames of bicolor patterns are used, as against the three frames which are essential in the traditional 2+1 3D real-time measuring method. These two frames were encoded and arranged alternately to synthesize a repetitive bicolor grating video.

9. Line 63 - could be better written as "Then the video was projected onto the object and the bicolor..."

10. Line 96 to 98 - The statement starts with an If, but there is no then... perhaps the sentence should have ... separately projected, this can address ... in line 97?

11. Line 123-125 - No capitalization after the period in line 123, and the sentence doesn't make sense. Maybe the authors mean ...Rr(x,y) and Rb(x,y) exhibit similar characteristics. The ratio of reflectivity of the red channel and the blue channel is a constant. This is attributed ....

12. Line 152 - The digital projector used was the PLED-W200 model produced by ViewSonic?

13. Line 153 - The camera used was ...

14. Line 158 - maybe the words "to code in this section" can be omitted?

15. Line 159 - Since we only need to use.....

16. Line 160 - Maybe "At the same time" can be omitted?

17. Line 163 - Maybe have a period after Ic2(x,y) or maybe join the next phrase with an "and" - "... and the chromaticity imbalance coefficient can be..."

18. Line 164 - maybe this line can be omitted?

19. Line 165 - Maybe omit "coded in this section" ?

20. Line 167 - by the CMOS camera

21. Line 175 - replace comma with a period?

22. Line 240-241 - Then, we calculates the corresponding root mean square errors and and made a table as seen in Table 1.

23. In Table 1 - Proposed method instead of Propose

24. Line 247 - It can be observed that the measuring error becomes

25. Line 250 - measurement error

**********

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Reviewer #2: No

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9 Jul 2021

Dear Editor and Reviewers:

Thank you very much for your excellent editing work of our manuscript entitled “Real-time profilometry by bicolor grating video projection ([PONE-D-21-03909R1]-[EMID:72c44d7d061b8290])”. And we also appreciate the reviewers for their valuable comments about our manuscript. Those comments are all valuable and very helpful for revising and improving our manuscript. We have studied the comments carefully and have made the correction which we hope to be able to meet the approval. The Response to Reviewer and Editor Comments is attached below. Hope our revised manuscript meets the high publication standard set by PLOS ONE.

Sincerely,

Yujiao Zhang, Yiping Cao, Haitao Wu, Haihua An, Xiuzhang Huang Sichuan University

Reviewer #2:

1. About the data - the authors have indicated in their response to my review that only three pairs of frames were used for the entire paper, out of 9 seconds of video with 160 frames. Some statistical error analysis of the other frames would be desirable, and the raw data should be made available as PLOS ONE's requirement is "PLOS journals require authors to make all data necessary to replicate their study's findings..."

Response:

I am sorry that I only provided three frames of data. In order to be able to provide real-time motion data more accurately, I kept the heart-shaped data and chose a new double-layer rectangular model for experimentation. The multi-frame deformed fringe information and real-time imaging information captured in the experiment are displayed in S1Visualization and S2 Visualizatio, respectively. To further prove the feasibility of real-time exercise, I used this method to measure more complex rotating sectors such as sectors. The multi-frame deformed fringe information and real-time imaging information captured in the experiment are displayed in S3 Visualization and S4 Visualization.

Revision:

1) In page 11 line 201, “heart” has been revised as “double-layer rectangular”.

2) In page 11 line 203, “heart” has been revised as “double-layer rectangular”.

3) In Fig4, “heart” has been revised as “double-layer rectangular”.

4) In page 12 line 209, line 211, line 212, line 213, line 217 “heart” has been revised as “double-layer rectangular”.

5) In page 13 line 226, “heart” has been revised as “double-layer rectangular”.

6) In page 14 line 229, “heart” has been revised as “double-layer rectangular”.

7) In page 12, between line 206 and line 207, I replaced the heart-shaped model in Fig.4 with a double-layer rectangular model

8) In page 13, between line 224 and line 225, I replaced the heart-shaped model in Fig.5 with a double-layer rectangular model.

9) In page 14 line 231-line233, “the results of the two real-time methods are close to the eight-step PMP, however, from Fig 5(b) and Fig 5(d),” has been revised as “the results of the proposed method are close to the eight-step PMP, ZHANG's method is not effective in restoring the middle connection of the double-layer rectangle. From Fig 5(b) and Fig 5(d)”.

10) In page 14 line 241, “0.1456” has been revised as “0.1055”.

11) In page 14 line 241, “0.0153” has been revised as “0.0157”.

12) In page 14, between line 244 and line 246, I replaced the new data in Table1 with a double-layer rectangular model.

Table.1 The height root mean square error at different speeds

24mm/s 30mm/s 36mm/s

ZHANG S 0.0817 0.1055 0.1443

Proposed method 0.0410 0.0517 0.0706

13) In page 11, line 205, I add more deformed patterns with a double-layer rectangular model, and made a S1Visualization of them.

14) In page 12, line 213, I add more reconstructed results with a double-layer rectangular model, and made a S2 Visualization of them.

15) In page 15, line 253, I add more deformed patterns with a more complex sector model, and made a S3 Visualization of them.

16) In page 16, line 261, I add more reconstructed results with a more complex sector model, and made a S4 Visualization of them.

17) In page 15-16, between line 250 and line 263, I add the new reconstruction results with a more complex sector model.

In order to further prove the feasibility of this method, a more complex sector model was measured in real time according to the same method. The bicolor deformed patterns at different times are simultaneously collected by the CMOS camera, and each group of deformed patterns are captured at 30 frames per second (30fps), are shown in Visualization 3. The part of the experimental results is shown in Fig 6.

Fig 6(a)-Fig 6(c) respectively show the bicolor deformed patterns extracted by the real-time online moving sector model under the conditions of state 1, 2, and 3 respectively. Fig 6(d)-Fig 6 (f) respectively show the monochrome deformed patterns extracted by the real-time online moving sector model under the corresponding state conditions. Fig 6(g)-Fig 6(i) respectively show the 3D shapes reconstructed by the real-time online moving sector model under the corresponding time conditions, all reconstructed results are shown in Visualization 4. It can be observed that the 3D reconstruction of the sector model has also been well restored, and its rotating tense can be restored in real time, which further proves the real-time feasibility of the proposed method.

2. Unfortunately, the manuscript contains lots of English errors.

Response:

Thank you for raising this question, and I am sorry that my English grammatical errors and improper expressions have caused you difficulties in reading. I am very grateful for the grammatical errors you pointed out to me, and I corrected the corresponding content according to your suggestions.

Revision:

2) In page 1 line 20, “the proposed” has been revised as “the proposed method”.

18) In page 2 line 32, “profilometry (FTP) [9-10]and” has been revised as “profilometry (FTP) [9-10] and”.

19) In page 2 line 33, “, It” has been revised as “, it”.

20) In page 2 line 36, “[13-14] PMP” has been revised as “[13-14]. PMP”.

21) In page 3 line 51, “uniform plan” has been revised as “uniform plane illumination”.

22) In page 4 line 87, “plane” has been revised as “camera”.

23) In page 3 line 57, “less one frame of image” has been revised as “one frame less.

24) In page 3 line 62-64, “First, only two frames of bicolor patterns substituting for the three frames of essential patterns in the traditional 2+1 3D real-time measuring method were encoded and arranged alternatively to synthesize into a repetitive bicolor grating video” has been revised as “First, only two frames of bicolor patterns are used, as against the three frames which are essential in the traditional 2+1 3D real-time measuring method. These two frames were encoded and arranged alternately to synthesize a repetitive bicolor grating video”.

25) In page 3 line 64-65, “Then, the video was projected onto the object, the bicolor deformed patterns” has been revised as “Then the video was projected onto the object and the bicolor deformed patterns”.

26) In page 5 line 99, “separately projected to address” has been revised as “separately projected, this can address”.

27) In page 6 line 126, “which demonstrates that the ratio of reflectivity of the red channel and the blue channel exhibits the constant characteristic” has been revised as “The ratio of reflectivity of the red channel and the blue channel is a constant”.

28) In page 7 line 153, “VeiwSonic” has been revised as “ViewSonic”.

29) In page 7 line 154, “ImagingSource Company.” has been revised as “The Imaging Source”.

30) In page 7 line 153, “adopts” has been revised as “used was”.

31) In page 7 line 154, “adopts” has been revised as “used was”.

32) In page 8 line 159, to delete “to code in this section”

33) In page 8 line 160, “it only needs” has been revised as “we only need”.

34) In page 8 line 161, “At the same time, the CMOS” has been revised as “The CMOS”.

35) In page 8 line 163, to add “and” between , and the.

36) In page 8 line 165, to delete “coded in this section”.

37) In page 8 line 165, to delete “Next, the chromaticity imbalance coefficient is used in the bicolor 2+1 3D measurement experiment.”.

38) In page 8 line 167, “through” has been revised as “by the”.

39) In page 9 line 175, “,” has been revised as “.”.

40) In page 14 line 243-244 “Then, to calculate the corresponding root mean square value and make a table1 expressed below:” has been revised as “Then, we calculate the corresponding root mean square errors and made a table as seen in Table 1.”.

41) In Table 1, “Propose” has been revised as “Proposed method”.

42) In page 14 line 246, “effect” has been revised as “error”.

Submitted filename: Response to Reviewers.docx

Click here for additional data file.

24 Aug 2021

PONE-D-21-03909R2

Real-time profilometry by bicolor grating video projection

PLOS ONE

Dear Dr. Cao,

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Comments to the Author

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Reviewer #3: All comments have been addressed

Reviewer #4: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #3: Yes

Reviewer #4: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #3: Yes

Reviewer #4: N/A

**********

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Reviewer #3: Yes

Reviewer #4: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #3: Yes

Reviewer #4: Yes

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Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #3: (No Response)

Reviewer #4: This manuscript proposes a bicolor 2+1 3D real-time measuring method based on bicolor grating video projection. The method is interesting and will be of potential application in manufacturing and robotics. Compared with the 2+1 method proposed by ZHANG S, the proposed method reduces one frame of projected fringes, and only requires to use the red and blue two-channel coded fringes with the smallest overlap range of spectrum response, which reduces the crosstalk of different color channels and improves the 3D measuring speed. As they have addressed the previous major concerns, I think it is suitable to be published in PLOS ONE. Some minor issues should be corrected as follows.

1, The label ‘A’ in Fig 3(c) and the label ‘B’ in Fig 5(a) should be moved to their correct positions.

2, For some sentences, the authors need to pay attention to the usage of tense, such as ‘were’ in line 64 should be ‘are’ and ‘was’ in line 65 should be ‘is’.

3. ‘ ZHANG S' s algorithm’ in both Fig.3 and Fig.5 should be ‘ ZHANG S' algorithm’.

**********

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Reviewer #3: No

Reviewer #4: No

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2 Sep 2021

Dear Editor and Reviewers:

Thank you very much for your excellent editing work of our manuscript entitled “Real-time profilometry by bicolor grating video projection ([PONE-D-21-03909R2] - [EMID:aa41d53618b9d2b1])”. And we also appreciate the reviewers for their valuable comments about our manuscript. Those comments are all valuable and very helpful for revising and improving our manuscript. We have studied the comments carefully and have made the correction which we hope to be able to meet the approval. The Response to Reviewer and Editor Comments is attached below. Hope our revised manuscript meets the high publication standard set by PLOS ONE.

Sincerely,

Yujiao Zhang, Yiping Cao, Haitao Wu, Haihua An, Xiuzhang Huang Sichuan University

Reviewer 4:

1. The label ‘A’ in Fig 3(c) and the label ‘B’ in Fig 5(a) should be moved to their correct positions.

Response:

I'm sorry for making a low-level formatting error. I have changed the format of the two pictures as required.

2. For some sentences, the authors need to pay attention to the usage of tense, such as ‘were’ in line 64 should be ‘are’ and ‘was’ in line 65 should be ‘is’.

Response:

Thank you for checking carefully, I have modified as required.

1) In line 63, “were” has been revised as “are”.

2) In line 64, “was” has been revised as “is”.

3. ‘ ZHANG S' s algorithm’ in both Fig.3 and Fig.5 should be ‘ ZHANG S' algorithm’.

Response:

Thank you for your careful inspection. I have checked the keyword in full and modified it.

1) In line 218, “ZHANG S method.” has been revised as “ZHANG S' method”.

2) In line 232, “ZHANG's method” has been revised as “ZHANG S' method”

Submitted filename: Response to Reviewers(minor revision).docx

Click here for additional data file.

22 Oct 2021

Real-time profilometry by bicolor grating video projection

PONE-D-21-03909R3

Dear Dr. Cao,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

Ireneusz Grulkowski, PhD

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #3: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #3: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #3: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #3: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #3: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #3: (No Response)

**********

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Reviewer #3: No

15 Nov 2021

PONE-D-21-03909R3

Real-time profilometry by bicolor grating video projection

Dear Dr. Cao:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

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Thank you for submitting your work to PLOS ONE and supporting open access.

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on behalf of

Dr. Ireneusz Grulkowski

Academic Editor

PLOS ONE