- Email: [email protected]
Iterative two-step temporal phase-unwrapping applied to high sensitivity three-dimensional profilometry
Optics and Lasers in Engineering 79 (2016) 22–28
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Iterative two-step temporal phase-unwrapping applied to high sensitivity three-dimensional profilometry Guangliang Du a, Chaorui Zhang a, Canlin Zhou a,n, Shuchun Si a, Hui Li a, Yanjie Li b a b
School of Physics, Shandong University, Jinan 250100, China School of Civil Engineering and Architecture, University of Jinan, Jinan 250022, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 September 2015 Received in revised form 29 October 2015 Accepted 16 November 2015 Available online 17 December 2015
Although temporal phase unwrapping method can be applied to solve some problems to measure an object with steep shapes, isolated parts or fringe undersampling in three-dimensional (3D) shape measurement, it needs to acquire and process a sequence of fringe pattern images which would take much time. Servin et al. proposed a 2-step temporal phase unwrapping algorithm, which only needs the 2 extreme phase-maps to achieve exactly the same results as standard temporal unwrapping method. But this method is only validated by computer simulation, shortage of experimental demonstration, its sensitivity coefficient G is limited, and it cannot be used when the G value is larger. We proposed an iterative two-step temporal phase-unwrapping algorithm which is an extension of Servin's method. First, add a fringe pattern with an intermediate sensitivity, project the fringe patterns of different sensitivity onto the tested object’s surface, and collect deformed fringe patterns with a CCD camera. Then we obtain the unwrapped phase with larger sensitivity coefficient G by cascading the sensitivity coefficients. And we derive the initial phase conditions of the 2-step temporal phase unwrapping algorithm. Finally, the experimental evaluation is conducted to prove the validity of the proposed method. The results are analyzed and compared with Servin's method. The experimental results show that the proposed method can achieve higher sensitivity and more accurate measurement, and it can overcome the main disadvantages encountered by Servin's method. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Temporal phase unwrapping Sensitivity Phase measuring profilometry (PMP)
1. Introduction Phase measuring profilometry is an important method in three-dimensional (3D) shape measurement. It has been extensively investigated and widely used in numerical fields for its simple device and higher accuracy [1–4]. To unwrap the wrapping phase in 3D surface measurement, many spatial and temporal phase retrieval methods have been presented. However, spatial phase retrieval method often leads to errors because of discontinuous morphology, noise and fringe undersampling [5–7]. Temporal phase retrieval method [8,9] can solve this problem. But the method needs multiple frames of fringe images which would take much time. To solve this problem, Xi et.al [10–12] proposed a temporal shift unwrapping technique based on projection of patterns of two selected frequencies. Chen [13] proposed a method making use of the three primary color channels associated with digital projectors. Fu [14] proposed a modified temporal phase unwrapping algorithm with the exponent of 4 and changing n
Corresponding author. Tel.: þ 86 13256153609. E-mail address: [email protected] (C. Zhou).
http://dx.doi.org/10.1016/j.optlaseng.2015.11.006 0143-8166/& 2015 Elsevier Ltd. All rights reserved.
phase-shifting step. Goldstein [15] proposed a smart temporal unwrapping that temporally unwraps the phase data such that small motion between frames is accounted for and phase data are unwrapped consistently between frames. Song [16] proposed a multi-sensitivity temporal phase unwrapping algorithm that does not need to calculate the equivalent wavelengths and the equivalent phases. Liu [17] proposed tri-Frequency Heterodyne Method. Liu [18] proposed a phase retrieval method using a composite fringe with multi-frequency. These methods make great progress in phase unwrapping. Recently, Manuel Servin proposed a 2-step temporal phase unwrapping algorithm [19], which only needs the 2 extreme phase-maps to achieve exactly the same results as standard temporal unwrapping method. However, according to our own experience with the method, Servin's method has the following disadvantages: (1) It is only validated by computer simulation, shortage of experimental demonstration; (2) The sensitivity coefficient G is limited, and it is difficult to process when the G is becoming larger; (3) It does not provide the initial phase conditions. Here, we present an iterative two-step temporal phase unwrapping algorithm which is an extension of Servin's method.
G. Du et al. / Optics and Lasers in Engineering 79 (2016) 22–28
23
Fig. 1. Optical path of phase measuring profilometry.
In our method, the sensitivity coefficient G can take as larger value as possible only if the hardware conditions such as CCD and projector can permit. We test the proposed method on many experiments, and compare our results with Servin’s method. In all cases, our method can give desired results. The paper is organized as follows. Section 2 introduces the principle of the system. Section 3 presents the experimental results and discusses the different results. Section 4 summarizes this paper.
2. Theory 2.1. Two-step temporal phase unwrapping algorithm The temporal phase unwrapping method is made in the temporal domain, a sequence of maps is acquired while the fringe pitch is changed. Then the phase at each pixel is unwrapped along the time axis. This method can be used to realize the 3-D shape measurement of complex object surface with steep shapes or isolated parts and fringe undersampling [8,9]. Servin proposed a 2-step temporal phase unwrapping algorithm based on temporal phase unwrapping method. The details for Servin’s method can be found in [19]. What follows is a brief synopsis of the method. Assuming that the intensity mathematical formula for two fringe patterns with different phase modulation sensitivities is as follows, I1ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½φðx; yÞ; φðx; yÞ A ð π ; π Þ; I2ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½Gφðx; yÞ; ðG 4 4 1Þ; G A R:
ð1Þ
where φðx; yÞ is a 1λ sensitive phase (λ is wavelength) and G φðx; yÞ is G-times more sensitive. We can use the phase demodulation algorithm [2,20] to obtain the 2 demodulated wrapped phasemaps as,
φ1ðx; yÞ ¼ W½φðx; yÞ; φðx; yÞ A ð π ; π Þ φ2w ðx; yÞ ¼ W½Gφðx; yÞ; ðG 4 4 1Þ; G A R:
ð2Þ
where W is the wrapping phase operator. The first demodulation φ1(x,y) is not wrapped because it is less than 1λ. So, we have, φ1(x,y)¼ φðx; yÞ. Based on Ref. [19], we can obtain the unwrapped phase of φ2w (x,y),
φ2ðx; yÞ ¼ Gφ1ðx; yÞ þ W½φ2w ðx; yÞ Gφ1ðx; yÞ where φ2(x,y) is the continuous phase of φ2w (x,y).
ð3Þ
Fig. 2. The tested object.
Eq. (3) is effective, only when the following is satisfied, ½φ2ðx; yÞ Gφ1ðx; yÞ A ð π ; π Þ
ð4Þ
2.2. The iterative two-step temporal phase unwrapping algorithm 2.2.1. The initial phase In Ref. [19], Eq. (1) says, φðx; yÞ is the modulation phase of the projected fringe whose frequency is 1; Eq. (13) says, φ1ðx; yÞ A ð π ; π Þ. These two requirements seem to contradict each other. But we think that they are same in essence, that is, the initial modulation phase must be continuous phase distribution which needs not a phase unwrapping process. Here, we give a method for determining the initial phase according to Ref. [21]. Fig. 1 shows the optical path of phase measuring profilometry, where P is the projection center of the projector, C is the camera imaging center, and D is an arbitrary point on the tested object. Its modulation phase can be calculated by,
ϕBD ¼
dh ϕ ðl hÞl0 0
ð5Þ
where l0 is the maximum width of field-of-view (FOV) observed by CCD camera, ϕ0 is the corresponding phase values within l0 ϕBD does not occur 2 π jump, that means, ϕBD o 2π . So we can determine the starting point (initial phase conditions) according to the phase value of ϕ0 . If there is no phase jump until ϕ0 ¼ kπ , then the initial phase conditions should be ϕ0 ¼ ðk 1Þ π . 2.2.2. The algorithm with larger sensitivity coefficient G The method in Ref. [19] can unwrap the phase correctly when the value of G is not larger. But when G is too large to satisfy Eq. (4), the results are not accurate. By Eq. (4) in Ref. [19], the greater the sensitivity G is, the greater the SNR of the demodulation phase is. Therefore, in order to improve the measurement accuracy, the G value should be as large as possible if the hardware conditions such as CCD and projector permit. So, we propose an algorithm with larger G. The basic idea of the method is that we can obtain the larger sensitivity coefficient G by adding an intermediate sensitivity fringe pattern and cascading the sensitivity coefficients. Assuming that the intensity mathematical formula for three fringe patterns with different phase modulation sensitivities is as follows, I1ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½φðx; yÞ; φðx; yÞ A ð π ; π Þ;
24
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Fig. 3. (a) The captured images when the frequency is 5, (b) the captured images when G ¼ 5, (c) the demodulated phase of Fig. 3(a), (d) the demodulated phase of Fig. 3(b), and (e) processing results by Servin's method.
I2ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½g 1 φðx; yÞ; ðg 1 4 1Þ; g 1 A R; I3ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½Gφðx; yÞ; ðg 2 4 1Þ; g 2 A R:
where G ¼g1*g2. ð6Þ
φ1ðx; yÞ ¼ W½φðx; yÞ; φðx; yÞ A ð π ; π Þ φ2w ðx; yÞ ¼ W½g 1 φðx; yÞ; ðg1 4 1Þ; g1 A R;
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Fig. 4. (a) The captured images when G ¼ 15 and (b) the results processed by Servin's method. Fig. 6. (a) The results processed by Servin's method and (b) the results processed by the proposed method.
Fig. 5. The processing results by the proposed method.
φ3w ðx; yÞ ¼ W½g 1 g2 φðx; yÞ; ðg2 4 1Þ; g2 A R:
ð7Þ
Eq. (7) is the phase data obtained by the phase demodulation algorithm, where φ2w ðx; yÞ and φ3w ðx; yÞ are the wrapped phase, φ1 (x,y) is not wrapped.
Fig. 7. Plastic board with a big hole.
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Fig. 8. (a) The captured images when the frequency is 5, (b) the captured images when G ¼ 5, (c) the demodulated phase of Fig. 8(a), (d) the demodulated phase of Fig. 8(b), and (e) processing results by Servin's method.
Based on Eq. (3), we can obtain
φ2ðx; yÞ ¼ g 1 φ1ðx; yÞ þ W½φ2w ðx; yÞ g1 φ1ðx; yÞ φ3ðx; yÞ ¼ g 2 φ2ðx; yÞ þ W½φ3w ðx; yÞ g2 φ2ðx; yÞ
ð8Þ
The main stages of our algorithm are summarized as follows: (1) obtain the 3 demodulated phase-maps φ1, φ2w and φ3w from different fringe patterns corresponding to the different sensitivity by the phase demodulation algorithm [20]. (2) unwrap φ2w with φ1 by Eq. (3), obtain the continuous phase φ2. (3) unwrap φ3w with φ2 by Eq. (3), obtain the continuous phase data φ3 with a high sensitivity G. (4) obtain the 3D topography of the object from the continuous phase φ3 after the measurement system is calibrated. These are the,main procedures of the algorithm with larger sensitivity coefficient G by cascading with 2 level different sensitivities,(G ¼g1*g2). Similarly, the larger G can be obtained by cascading with 3 or more level different sensitivities. Some following experiments are used to verify the proposed algorithm.
3. Experiments In this section, for evaluating the real performance of our method, we test our method on a series of experiments. Below, we will describe these experiments and practical suggestions for the above procedure. To verify the proposed algorithm, we develop a fringe projection measurement system, which consists of a DLP projector (Optoma EX762) driven by a computer and a CCD camera ( DHSV401FM). Fig. 1 shows the schematic of fringe-projection
profilometry system. The captured image is 592 pixels wide by 496 pixels high. The surface measurement software is programmed by Matlab with I5-4570 CPU @ 3.20 GHz. Firstly, an experiment is provided to demonstrate the feasibility of the proposed algorithm. The tested object is a face model and an isolated cup, as shown in Fig. 2. In this measurement system, d¼ 135 cm, l ¼120 cm, l0 ¼ 50 cm, and the biggest h is about 7 cm. According to Eq. (5), when the frequency of projected fringe is 6, the modulation phase was wrapped, so the initial phase frequency we chose is 5. Fig. 3 shows the captured images and processing results obtained by Servin's method when the sensitivity coefficient G is smaller. Where Fig. 3(a) is the captured images when the frequency of the projected fringe is 5, Fig. 3(b) is the captured images when G ¼5, that is, the frequency of the projected fringe is 25. Fig. 3(c) and (d) shows the demodulated phase distribution from Fig. 3(a) and (b) by standard phase-step algorithm respectively. The phase in Fig. 3(c) is continuous while the phase in Fig. 3(d) is wrapped. Fig. 3(e) is the restored 3D surface shape by Servin's method. It is clear that we can obtain the desired results by Servin's method when the sensitivity coefficient G is smaller. Fig. 4 shows the captured images and processing results by Servin's method when the sensitivity coefficient G is larger. Fig. 4 (a) shows the captured images when G ¼15,that is, the frequency of the projected fringe is 75, Fig. 4(b) is the results processed by Fig. 3(a) and Fig. 4(a) by Servin's method. From Fig. 4(b), we can see that many errors occur in the results by Servin's method when G is larger. Fig. 3(a), (b) and Fig. 4(a) can be analyzed to get the 3D surface shape by our proposed method. In this case, we set the parameters g1 ¼5,g2 ¼3. We can restore 3D surface shape shown in Fig. 5 according to the steps in Section 2.2.2.
G. Du et al. / Optics and Lasers in Engineering 79 (2016) 22–28
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Fig. 9. (a) The captured images when G ¼ 15 and (b) the results processed by Servin's method. Fig. 11. (a) The results processed by Servin's method and (b) the results processed by the proposed method.
Fig. 10. The processing results by the proposed method.
Fig. 6 shows the results of the 300th rows of the two methods when the sensitivity coefficient G is larger. Fig. 6(a) shows the results processed by Servin's method. Fig. 6(b) is the results processed by the proposed method. The results show that the proposed method can restore 3D surface shape well.
Secondly, we do an experiment on a plastic board with a big hole as shown in Fig. 7. To validate that the methods are applied to objects with hole, we use a black background in the hole area. The experiment procedure is similar. In this experiment, d ¼135 cm, l¼ 120 cm, l0 ¼65 cm, and the biggest h is about 9 cm. According to Eq. (5), the initial phase frequency we chose is 5. And we set the parameters g1 ¼5, g2 ¼3. The captured images and processing results obtained by Servin's method are shown in Fig. 8. Where Fig. 8(a) is the captured images when the frequency of the projected fringe is 5, Fig. 8(b) is the captured images when G ¼5, that is, the frequency of the projected fringe is 25, Fig. 8(c) is the demodulated phase of Fig. 8(a). Fig. 8(d) is the demodulated phase of Fig. 8(b), (e) is the processing results by Servin's method when G is smaller. Fig. 9 shows the captured images and processing results of G ¼15. Fig. 9(a) shows the captured images when G ¼15,that is, the frequency of the projected fringe is 75, Fig. 9(b) is the results obtained from Fig. 8(a) and Fig. 9(a) by Servin's method when G is larger. Fig. 10 shows the processing results by the proposed method. Fig. 11 shows the results of the 300th rows of the two methods. Fig. 11(a) shows the results processed by Servin's method, Fig. 11 (b) is the results processed by the proposed method. From this
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G. Du et al. / Optics and Lasers in Engineering 79 (2016) 22–28
experiment, the measured results are more accurate by our proposed method.
4. Conclusion As an alternative, we propose an iterative two-step temporal phase unwrapping algorithm which is an extension of Servin's method. Our method overcomes the main disadvantages that Servin's method encounters. The performance of the proposed methods is demonstrated via application to a series of experiments. It can obtain the unwrapped phase with larger sensitivity coefficient G, which has better visual quality. The measurement results are more accurate.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 11302082 and 11472070). The support is gratefully acknowledged.
References [1] Lei ZK, Wang C, Zhou CL. Multi-frequency inverse-phase fringe projection profilometry for nonlinear phase error compensation. Opt Lasers Eng 2015;66:249–57. [2] Fujun Yang, Min Geng, Meiling Dai, et al. 3D shape measurement of discontinuous objects based on single frequency fringe projection in 4-step phase shifting profilemetry. J Optoelectron Laser 2012;23(8):1535–8. [3] Liu TC, Zhou CL, Si SC, et al. Improved differential 3D shape retrieval. Opt Lasers Eng 2015;73:143–9. [4] Dai SJ, Shao M, Wang ZP, et al. Study on the two-step method to remove the background in 3D measurement of titanium blade. J Optoelectron laser 2014;25(8):1540–7.
[5] de Souza J, Oliveira M, dos Santos P. Branch-cut algorithm for optical phase unwrapping. Opt Lett 2015;40(15):3456–9. [6] Rivera M, Hernandez-Lopez FJ, Gonzalez A. Phase unwrapping by accumulation of residual maps. Opt Lasers Eng 2015;64:51–8. [7] Zhao M, Huang L, Zhang Q, et al. Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies. Appl Opt 2011;50 (33):6214–24. [8] Saldner HO, Huntley JM. Temporal phase unwrapping: application to surface profiling of discontinuous object. Appl Opt 1997;36(13):2770–5. [9] Lulu Li, Xianyu Si, Yunfu Dou, et al. Error analysis and algorithm design of temporal phase unwrapping. J Sichuan Univ (Nat Sci Ed) 2012;49(1):102–8. [10] Cao P, Xi J, Yu Y, et al. 3D shape measurement based on projection of triangular patterns of two selected frequencies. Opt Express 2014;22(23):29234–48. [11] Long J, Xi J, Zhu M, et al. Absolute phase map recovery of two fringe patterns with flexible selection of fringe wavelengths. Appl Opt 2014;53(9):1794–801. [12] Ding Y, Xi J, Yu Y, et al. Absolute phase recovery of three fringe patterns with selected spatial frequencies. Opt Lasers Eng 2015;70:18–25. [13] Chen K, Xi J, Yu Y, et al. Three-dimensional measurement of object surfaces with complex shape and color distribution based on projection of color fringe pattern. Appl Opt 2013;52(30):7360–6. [14] Fu Y, Wei Wang, Huirong Xiao. Three-dimensional profile measurement based on modified temporal phase unwrapping algorithm. Optik 2014;124(6):557–60. [15] Goldstein G, Creath K. Quantitative phase microscopy: automated background leveling techniques and smart temporal phase unwrapping. Appl Opt 2015;54 (16):5175–85. [16] Song L, Chang Y, Xi J, et al. Phase unwrapping method based on multiple fringe patterns without use of equivalent wavelengths. Opt Commun 2015;355:213–24. [17] Liu S, Feng W, Zhang Q, et al. Three-dimensional shape measurement of small object based on tri-frequency heterodyne method. In: Proceedings of International Conference on Optical Instruments and Technology. International Society for Optics and Photonics; 2015. [18] Liu T, Zhou C, Liu Y, et al. Deflectometry for phase retrieval using a composite fringe. Opt Appl 2014;94:451–61. [19] Servin M, Padilla JM, Gonzalez Adonai, et al. Temporal phase-unwrapping of static surfaces with 2-sensitivity fringe-patterns. Opt Express 2015;23 (12):15806–15. [20] Jeught SV d, Soons JA, Dirckx JJ. Real-time microscopic phase-shifting profilometry. Appl Opt 2015;54(15):4953–9. [21] Peng Xiang, Yang Zhaoliang, Niu Hanben. Multi-resolution reconstruction of 3D image with modified temporal unwrapping algorithm. Opt Commun 2003;224(1–3):35–44.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Iterative two-step temporal phase-unwrapping applied to high sensitivity three-dimensional profilometry Guangliang Du a, Chaorui Zhang a, Canlin Zhou a,n, Shuchun Si a, Hui Li a, Yanjie Li b a b
School of Physics, Shandong University, Jinan 250100, China School of Civil Engineering and Architecture, University of Jinan, Jinan 250022, China
art ic l e i nf o
a b s t r a c t
Article history: Received 22 September 2015 Received in revised form 29 October 2015 Accepted 16 November 2015 Available online 17 December 2015
Although temporal phase unwrapping method can be applied to solve some problems to measure an object with steep shapes, isolated parts or fringe undersampling in three-dimensional (3D) shape measurement, it needs to acquire and process a sequence of fringe pattern images which would take much time. Servin et al. proposed a 2-step temporal phase unwrapping algorithm, which only needs the 2 extreme phase-maps to achieve exactly the same results as standard temporal unwrapping method. But this method is only validated by computer simulation, shortage of experimental demonstration, its sensitivity coefficient G is limited, and it cannot be used when the G value is larger. We proposed an iterative two-step temporal phase-unwrapping algorithm which is an extension of Servin's method. First, add a fringe pattern with an intermediate sensitivity, project the fringe patterns of different sensitivity onto the tested object’s surface, and collect deformed fringe patterns with a CCD camera. Then we obtain the unwrapped phase with larger sensitivity coefficient G by cascading the sensitivity coefficients. And we derive the initial phase conditions of the 2-step temporal phase unwrapping algorithm. Finally, the experimental evaluation is conducted to prove the validity of the proposed method. The results are analyzed and compared with Servin's method. The experimental results show that the proposed method can achieve higher sensitivity and more accurate measurement, and it can overcome the main disadvantages encountered by Servin's method. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Temporal phase unwrapping Sensitivity Phase measuring profilometry (PMP)
1. Introduction Phase measuring profilometry is an important method in three-dimensional (3D) shape measurement. It has been extensively investigated and widely used in numerical fields for its simple device and higher accuracy [1–4]. To unwrap the wrapping phase in 3D surface measurement, many spatial and temporal phase retrieval methods have been presented. However, spatial phase retrieval method often leads to errors because of discontinuous morphology, noise and fringe undersampling [5–7]. Temporal phase retrieval method [8,9] can solve this problem. But the method needs multiple frames of fringe images which would take much time. To solve this problem, Xi et.al [10–12] proposed a temporal shift unwrapping technique based on projection of patterns of two selected frequencies. Chen [13] proposed a method making use of the three primary color channels associated with digital projectors. Fu [14] proposed a modified temporal phase unwrapping algorithm with the exponent of 4 and changing n
Corresponding author. Tel.: þ 86 13256153609. E-mail address: [email protected] (C. Zhou).
http://dx.doi.org/10.1016/j.optlaseng.2015.11.006 0143-8166/& 2015 Elsevier Ltd. All rights reserved.
phase-shifting step. Goldstein [15] proposed a smart temporal unwrapping that temporally unwraps the phase data such that small motion between frames is accounted for and phase data are unwrapped consistently between frames. Song [16] proposed a multi-sensitivity temporal phase unwrapping algorithm that does not need to calculate the equivalent wavelengths and the equivalent phases. Liu [17] proposed tri-Frequency Heterodyne Method. Liu [18] proposed a phase retrieval method using a composite fringe with multi-frequency. These methods make great progress in phase unwrapping. Recently, Manuel Servin proposed a 2-step temporal phase unwrapping algorithm [19], which only needs the 2 extreme phase-maps to achieve exactly the same results as standard temporal unwrapping method. However, according to our own experience with the method, Servin's method has the following disadvantages: (1) It is only validated by computer simulation, shortage of experimental demonstration; (2) The sensitivity coefficient G is limited, and it is difficult to process when the G is becoming larger; (3) It does not provide the initial phase conditions. Here, we present an iterative two-step temporal phase unwrapping algorithm which is an extension of Servin's method.
G. Du et al. / Optics and Lasers in Engineering 79 (2016) 22–28
23
Fig. 1. Optical path of phase measuring profilometry.
In our method, the sensitivity coefficient G can take as larger value as possible only if the hardware conditions such as CCD and projector can permit. We test the proposed method on many experiments, and compare our results with Servin’s method. In all cases, our method can give desired results. The paper is organized as follows. Section 2 introduces the principle of the system. Section 3 presents the experimental results and discusses the different results. Section 4 summarizes this paper.
2. Theory 2.1. Two-step temporal phase unwrapping algorithm The temporal phase unwrapping method is made in the temporal domain, a sequence of maps is acquired while the fringe pitch is changed. Then the phase at each pixel is unwrapped along the time axis. This method can be used to realize the 3-D shape measurement of complex object surface with steep shapes or isolated parts and fringe undersampling [8,9]. Servin proposed a 2-step temporal phase unwrapping algorithm based on temporal phase unwrapping method. The details for Servin’s method can be found in [19]. What follows is a brief synopsis of the method. Assuming that the intensity mathematical formula for two fringe patterns with different phase modulation sensitivities is as follows, I1ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½φðx; yÞ; φðx; yÞ A ð π ; π Þ; I2ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½Gφðx; yÞ; ðG 4 4 1Þ; G A R:
ð1Þ
where φðx; yÞ is a 1λ sensitive phase (λ is wavelength) and G φðx; yÞ is G-times more sensitive. We can use the phase demodulation algorithm [2,20] to obtain the 2 demodulated wrapped phasemaps as,
φ1ðx; yÞ ¼ W½φðx; yÞ; φðx; yÞ A ð π ; π Þ φ2w ðx; yÞ ¼ W½Gφðx; yÞ; ðG 4 4 1Þ; G A R:
ð2Þ
where W is the wrapping phase operator. The first demodulation φ1(x,y) is not wrapped because it is less than 1λ. So, we have, φ1(x,y)¼ φðx; yÞ. Based on Ref. [19], we can obtain the unwrapped phase of φ2w (x,y),
φ2ðx; yÞ ¼ Gφ1ðx; yÞ þ W½φ2w ðx; yÞ Gφ1ðx; yÞ where φ2(x,y) is the continuous phase of φ2w (x,y).
ð3Þ
Fig. 2. The tested object.
Eq. (3) is effective, only when the following is satisfied, ½φ2ðx; yÞ Gφ1ðx; yÞ A ð π ; π Þ
ð4Þ
2.2. The iterative two-step temporal phase unwrapping algorithm 2.2.1. The initial phase In Ref. [19], Eq. (1) says, φðx; yÞ is the modulation phase of the projected fringe whose frequency is 1; Eq. (13) says, φ1ðx; yÞ A ð π ; π Þ. These two requirements seem to contradict each other. But we think that they are same in essence, that is, the initial modulation phase must be continuous phase distribution which needs not a phase unwrapping process. Here, we give a method for determining the initial phase according to Ref. [21]. Fig. 1 shows the optical path of phase measuring profilometry, where P is the projection center of the projector, C is the camera imaging center, and D is an arbitrary point on the tested object. Its modulation phase can be calculated by,
ϕBD ¼
dh ϕ ðl hÞl0 0
ð5Þ
where l0 is the maximum width of field-of-view (FOV) observed by CCD camera, ϕ0 is the corresponding phase values within l0 ϕBD does not occur 2 π jump, that means, ϕBD o 2π . So we can determine the starting point (initial phase conditions) according to the phase value of ϕ0 . If there is no phase jump until ϕ0 ¼ kπ , then the initial phase conditions should be ϕ0 ¼ ðk 1Þ π . 2.2.2. The algorithm with larger sensitivity coefficient G The method in Ref. [19] can unwrap the phase correctly when the value of G is not larger. But when G is too large to satisfy Eq. (4), the results are not accurate. By Eq. (4) in Ref. [19], the greater the sensitivity G is, the greater the SNR of the demodulation phase is. Therefore, in order to improve the measurement accuracy, the G value should be as large as possible if the hardware conditions such as CCD and projector permit. So, we propose an algorithm with larger G. The basic idea of the method is that we can obtain the larger sensitivity coefficient G by adding an intermediate sensitivity fringe pattern and cascading the sensitivity coefficients. Assuming that the intensity mathematical formula for three fringe patterns with different phase modulation sensitivities is as follows, I1ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½φðx; yÞ; φðx; yÞ A ð π ; π Þ;
24
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Fig. 3. (a) The captured images when the frequency is 5, (b) the captured images when G ¼ 5, (c) the demodulated phase of Fig. 3(a), (d) the demodulated phase of Fig. 3(b), and (e) processing results by Servin's method.
I2ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½g 1 φðx; yÞ; ðg 1 4 1Þ; g 1 A R; I3ðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos ½Gφðx; yÞ; ðg 2 4 1Þ; g 2 A R:
where G ¼g1*g2. ð6Þ
φ1ðx; yÞ ¼ W½φðx; yÞ; φðx; yÞ A ð π ; π Þ φ2w ðx; yÞ ¼ W½g 1 φðx; yÞ; ðg1 4 1Þ; g1 A R;
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Fig. 4. (a) The captured images when G ¼ 15 and (b) the results processed by Servin's method. Fig. 6. (a) The results processed by Servin's method and (b) the results processed by the proposed method.
Fig. 5. The processing results by the proposed method.
φ3w ðx; yÞ ¼ W½g 1 g2 φðx; yÞ; ðg2 4 1Þ; g2 A R:
ð7Þ
Eq. (7) is the phase data obtained by the phase demodulation algorithm, where φ2w ðx; yÞ and φ3w ðx; yÞ are the wrapped phase, φ1 (x,y) is not wrapped.
Fig. 7. Plastic board with a big hole.
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Fig. 8. (a) The captured images when the frequency is 5, (b) the captured images when G ¼ 5, (c) the demodulated phase of Fig. 8(a), (d) the demodulated phase of Fig. 8(b), and (e) processing results by Servin's method.
Based on Eq. (3), we can obtain
φ2ðx; yÞ ¼ g 1 φ1ðx; yÞ þ W½φ2w ðx; yÞ g1 φ1ðx; yÞ φ3ðx; yÞ ¼ g 2 φ2ðx; yÞ þ W½φ3w ðx; yÞ g2 φ2ðx; yÞ
ð8Þ
The main stages of our algorithm are summarized as follows: (1) obtain the 3 demodulated phase-maps φ1, φ2w and φ3w from different fringe patterns corresponding to the different sensitivity by the phase demodulation algorithm [20]. (2) unwrap φ2w with φ1 by Eq. (3), obtain the continuous phase φ2. (3) unwrap φ3w with φ2 by Eq. (3), obtain the continuous phase data φ3 with a high sensitivity G. (4) obtain the 3D topography of the object from the continuous phase φ3 after the measurement system is calibrated. These are the,main procedures of the algorithm with larger sensitivity coefficient G by cascading with 2 level different sensitivities,(G ¼g1*g2). Similarly, the larger G can be obtained by cascading with 3 or more level different sensitivities. Some following experiments are used to verify the proposed algorithm.
3. Experiments In this section, for evaluating the real performance of our method, we test our method on a series of experiments. Below, we will describe these experiments and practical suggestions for the above procedure. To verify the proposed algorithm, we develop a fringe projection measurement system, which consists of a DLP projector (Optoma EX762) driven by a computer and a CCD camera ( DHSV401FM). Fig. 1 shows the schematic of fringe-projection
profilometry system. The captured image is 592 pixels wide by 496 pixels high. The surface measurement software is programmed by Matlab with I5-4570 CPU @ 3.20 GHz. Firstly, an experiment is provided to demonstrate the feasibility of the proposed algorithm. The tested object is a face model and an isolated cup, as shown in Fig. 2. In this measurement system, d¼ 135 cm, l ¼120 cm, l0 ¼ 50 cm, and the biggest h is about 7 cm. According to Eq. (5), when the frequency of projected fringe is 6, the modulation phase was wrapped, so the initial phase frequency we chose is 5. Fig. 3 shows the captured images and processing results obtained by Servin's method when the sensitivity coefficient G is smaller. Where Fig. 3(a) is the captured images when the frequency of the projected fringe is 5, Fig. 3(b) is the captured images when G ¼5, that is, the frequency of the projected fringe is 25. Fig. 3(c) and (d) shows the demodulated phase distribution from Fig. 3(a) and (b) by standard phase-step algorithm respectively. The phase in Fig. 3(c) is continuous while the phase in Fig. 3(d) is wrapped. Fig. 3(e) is the restored 3D surface shape by Servin's method. It is clear that we can obtain the desired results by Servin's method when the sensitivity coefficient G is smaller. Fig. 4 shows the captured images and processing results by Servin's method when the sensitivity coefficient G is larger. Fig. 4 (a) shows the captured images when G ¼15,that is, the frequency of the projected fringe is 75, Fig. 4(b) is the results processed by Fig. 3(a) and Fig. 4(a) by Servin's method. From Fig. 4(b), we can see that many errors occur in the results by Servin's method when G is larger. Fig. 3(a), (b) and Fig. 4(a) can be analyzed to get the 3D surface shape by our proposed method. In this case, we set the parameters g1 ¼5,g2 ¼3. We can restore 3D surface shape shown in Fig. 5 according to the steps in Section 2.2.2.
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Fig. 9. (a) The captured images when G ¼ 15 and (b) the results processed by Servin's method. Fig. 11. (a) The results processed by Servin's method and (b) the results processed by the proposed method.
Fig. 10. The processing results by the proposed method.
Fig. 6 shows the results of the 300th rows of the two methods when the sensitivity coefficient G is larger. Fig. 6(a) shows the results processed by Servin's method. Fig. 6(b) is the results processed by the proposed method. The results show that the proposed method can restore 3D surface shape well.
Secondly, we do an experiment on a plastic board with a big hole as shown in Fig. 7. To validate that the methods are applied to objects with hole, we use a black background in the hole area. The experiment procedure is similar. In this experiment, d ¼135 cm, l¼ 120 cm, l0 ¼65 cm, and the biggest h is about 9 cm. According to Eq. (5), the initial phase frequency we chose is 5. And we set the parameters g1 ¼5, g2 ¼3. The captured images and processing results obtained by Servin's method are shown in Fig. 8. Where Fig. 8(a) is the captured images when the frequency of the projected fringe is 5, Fig. 8(b) is the captured images when G ¼5, that is, the frequency of the projected fringe is 25, Fig. 8(c) is the demodulated phase of Fig. 8(a). Fig. 8(d) is the demodulated phase of Fig. 8(b), (e) is the processing results by Servin's method when G is smaller. Fig. 9 shows the captured images and processing results of G ¼15. Fig. 9(a) shows the captured images when G ¼15,that is, the frequency of the projected fringe is 75, Fig. 9(b) is the results obtained from Fig. 8(a) and Fig. 9(a) by Servin's method when G is larger. Fig. 10 shows the processing results by the proposed method. Fig. 11 shows the results of the 300th rows of the two methods. Fig. 11(a) shows the results processed by Servin's method, Fig. 11 (b) is the results processed by the proposed method. From this
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experiment, the measured results are more accurate by our proposed method.
4. Conclusion As an alternative, we propose an iterative two-step temporal phase unwrapping algorithm which is an extension of Servin's method. Our method overcomes the main disadvantages that Servin's method encounters. The performance of the proposed methods is demonstrated via application to a series of experiments. It can obtain the unwrapped phase with larger sensitivity coefficient G, which has better visual quality. The measurement results are more accurate.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant nos. 11302082 and 11472070). The support is gratefully acknowledged.
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