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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24139–24150
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Phase coding method for absolute phase retrieval with a large number of codewords

Dongliang Zheng and Feipeng Da  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24139-24150 (2012)
http://dx.doi.org/10.1364/OE.20.024139


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Abstract

A recently proposed phase coding method for absolute phase retrieval performs well because its codeword is embedded into phase domain rather than intensity. Then, the codeword can determine the fringe order for the phase unwrapping. However, for absolute phase retrieval with a large number of codewords, the traditional phase coding method becomes not so reliable. In this paper, we present a novel phase coding method to tackle this problem. Six additional fringe images can generate more than 64( 2 6 ) unique codewords for correct absolute phase retrieval. The novel phase coding method can be used for absolute phase retrieval with high frequency. Experiment results demonstrate the proposed method is effective.

© 2012 OSA

1. Introduction

Fringe projection technique is one of most widely used techniques for generating three-dimensional (3-D) shape measurement in many practical applications, because of its qualities of non-contact, low-cost, high-resolution, etc [1

1. S. Zhang, “High-resolution 3-D profilometry with binary phase-shifting methods,” Appl. Opt. 50(12), 1753–1757 (2011). [CrossRef] [PubMed]

5

5. E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express 17(10), 7818–7830 (2009). [CrossRef] [PubMed]

]. There are several techniques proposed to extract the wrapped phase for fringe projection technique, such as Fourier transform, wavelet transform and phase shifting algorithm [6

6. C. Quan, W. Chen, and C. J. Tay, “phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng. 48(2), 235–243 (2010). [CrossRef]

]. The transform techniques require complicated computation and cannot retrieve the wrapped phase accurately for measurement objects involving complex shape. Phase shifting algorithm suitable for measuring objects with complex shape has been extensively used owning to its speed and accuracy [7

7. T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010). [CrossRef] [PubMed]

]. Since the wrapped phase contains 2π discontinuous jumps, a phase unwrapping method should be employed to retrieve the continuous absolute phase. However, when the measured objects have complex surface shape or some separated parts, how to implement correctly and rapidly absolute phase retrieval is a notable challenge for the 3-D measurement. There are many absolute phase retrieval methods proposed to tackle this problem [8

8. Z. Wang, D. A. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010). [CrossRef]

11

11. Y. Ding, J. Xi, Y. Yu, W. Cheng, S. Wang, and J. F. Chicharo, “Frequency selection in absolute phase maps recovery with two frequency projection fringes,” Opt. Express 20(12), 13238–13251 (2012). [CrossRef] [PubMed]

]. Temporal phase unwrapping methods can deal with this problem, but need to project multiple frames of fringe images with different frequencies [12

12. J. M. Huntley and H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993). [CrossRef] [PubMed]

,13

13. J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for threee-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008). [CrossRef]

]. Binary encoding grating methods can detect the phase order for phase unwrapping by encoding a unique codeword, and then decoding this codeword [14

14. G. Sansoni, M. Carocci, and R. Rodella, “Three-dimensional vision based on a combination of Gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt. 38(31), 6565–6573 (2005). [CrossRef] [PubMed]

16

16. Y. Liu, X. Su, and Q. Zhang, “A novel encoded-phase technique for phase measuring profilometry,” Opt. Express 19(15), 14137–14144 (2011). [CrossRef] [PubMed]

]. Gray-code method is the most commonly used binary encoding method [17

17. Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012). [CrossRef]

,18

18. D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng. 50(8), 1130–1139 (2012). [CrossRef]

]. A recent study proposed a novel phase coding method to solve this problem, which is inherently better than the Gray-code method, because it uses phase instead of intensity to determine the codeword. As we know, phase is less sensitive to object surface contrast, ambient light, and camera noise [19

19. Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012). [CrossRef] [PubMed]

].

For the phase coding method, the codeword is embedded into the coding phase ϕS(x,y) ranging from π to +π of phase-shifted fringe images (e.g., three images for a three step phase shifting algorithm). To generate N codewords, the coding phase ϕS(x,y) is quantized into N levels with a stair height of 2πN. By this means, three additional fringe images can represent more than 8(23) unique codewords. Figure 1(a)
Fig. 1 (a) One cross section of the coding phase and wrapped phase, (b) The unwrapped absolute phase.
illustrates the phase coding method when N=8. The stair phase-change is aligned with the 2π discontinuities in the wrapped phase ϕ(x,y), and each phase stair can denote one codeword. The result codeword can determine the fringe order to retrieve the absolute phase as shown in Fig. 1(b).

According to the phase coding method with three additional fringe images, the number of generated unique codeword is more than 8(23). In general, for a phase shifting algorithm, the longer the wavelength used, the larger the phase noise is induced. It has been demonstrated the phase error in the low frequency fringes is larger than the high frequency fringes for fringe projection technique [20

20. S. Zhang, “Phase unwrapping error reduction framework for a multiple-wavelength phase-shifting algorithm,” Opt. Eng. 48(10), 105601 (2009). [CrossRef]

]. It should be noted that wavelength here indicates the spatial width of the fringe images, i.e., number of pixels per fringe image. Therefore, for the fringe images with the same width, more codewords need much smaller wavelength. If the wrapped phase with 8(23) fringe periods is accurate enough, more codewords are unnecessary. If 8(23) unique codewords are not enough for measurement requirement, a large number of codewords are needed. In practical, how to retrieve the absolute phase with high frequency effectively remains a challenge [21

21. Y. Ding, J. Xi, Y. Yu, and J. Chicharo, “Recovering the absolute phase maps of two fringe patterns with selected frequencies,” Opt. Lett. 36(13), 2518–2520 (2011). [CrossRef] [PubMed]

]. For example, the maximum unique number it can generate for a Gray-code method is 2M for M fringe images [17

17. Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012). [CrossRef]

]. The mulitiple-wavelength phase unwrapping method needs multiple frames of fringe images [20

20. S. Zhang, “Phase unwrapping error reduction framework for a multiple-wavelength phase-shifting algorithm,” Opt. Eng. 48(10), 105601 (2009). [CrossRef]

]. For dual-wavelength or two-frequency phase unwrapping method, less fringe images are used, but the choice of fringe frequency is restricted to a certain range [21

21. Y. Ding, J. Xi, Y. Yu, and J. Chicharo, “Recovering the absolute phase maps of two fringe patterns with selected frequencies,” Opt. Lett. 36(13), 2518–2520 (2011). [CrossRef] [PubMed]

].

As we know, both of ϕS(x,y) and ϕ(x,y) are embedded by three step phase shifting algorithm. Owning to its easy availability, low cost and high flexibility, a digital video projector (DVP) has been commonly used in fringe projection technique. The phase-shift error can be totally avoided because the fringe images are generated by computer [22

22. B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009). [CrossRef] [PubMed]

]. However, the commercial DVP and the CCD camera used in fringe projection technique are generally fabricated to be nonlinear devices. This nonlinear intensity response of a fringe projection measurement system inevitably leads to the deviation of the captured fringe images from ideal distribution and introduces an additional phase error [23

23. S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46(1), 36–43 (2007). [CrossRef] [PubMed]

]. In general, for commonly used DVP and the CCD camera, when ϕS(x,y) are quantized into small levels, such as N=8, this additional phase error will not influence the result codeword. However, when ϕS(x,y) is quantized into large levels, this additional phase error will lead to wrong codeword, which cannot be used for correct phase unwrapping. The following theoretical analysis and experiment demonstrate this problem.

We propose a novel phase coding method to solve this problem. Through coding two phase information, two sets of embedding codeword are determined, which can generate a large number of codewords by simple calculation. Six additional fringe images can generate more than 64(26) unique codewords for absolute phase retrieval. Compared with the traditional phase unwrapping methods, the proposed method is flexible and effective. The main work of this paper is given as follows. Section 2 gives the principle of the novel phase coding method. Section 3 demonstrates the proposed method through experiments. Section 4 summarizes this paper.

2. Principle of the novel phase coding method

2.1 The problem of a large number of codewords phase retrieval

Because of the nonlinear intensity response generated by the nonlinear devices of the DVP and the CCD camera [23

23. S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46(1), 36–43 (2007). [CrossRef] [PubMed]

], the real response IiC(x,y)of the projector to the input intensity Ii for the ith ideal fringe image is expressed as the function f(Ii) shown in Eq. (1).
IiC(x,y)=f(Ii),
(1)
where f(Ii) is the nonlinear response function of Ii. The calculated coding phase using three step phase shifting algorithm ϕS'(x,y) can be considered as the sum of the actual phase ϕS(x,y) and the phase error ΔϕS(x,y) caused by nonlinear response in the fringe projection measurement system.

ϕS'(x,y)=ϕS(x,y)+ΔϕS(x,y).
(2)

The actual codeword C and the result codeword C' can be calculated by Eq. (3) and Eq. (4), respectively.
C=[N(ϕS+π)/2π]
(3)
C'=[N(ϕS'+π)/2π]=[N(ϕS+π)/2π+NΔϕS/2π],
(4)
where N is the number of the unique codeword, [x] determines the closest integer. In general, if the number of the quantized levels N is small, the value of NΔϕS/2π is small which will not influence the result codeword. Therefore, we can get C=C'. When N is a large number, the value of NΔϕS/2π is large. Even small phase error will cause CC'. The wrong codeword cannot be used for correct absolute phase retrieval. The following experiment demonstrates that the wrong places in the codeword become more during the process of increasing of the quantized levels. For the measurement requiring high frequency fringe images, a large number of codewords are needed to guide the phase unwrapping.

2.2 The novel phase coding method for a large number of codewords

We propose a novel method to solve this problem through coding two phase information. Two sets of embedding codeword can be calculated. Then, a large number of codewords can be determined using six additional fringe images. Experiment results verify that the proposed approach is robust for absolute phase retrieval. The details are given as follows. Figure 2
Fig. 2 (a) One cross section of the coding phase and wrapped phase, (b) The result Codeword 1 and Codeword 2.
shows the framework of the proposed method. Figure 2(a) shows the wrapped phase ϕ(x,y), and two coding phase ϕ1S(x,y) andϕ2S(x,y). The result codeword for the coding phase is illustrated in Fig. 2(b). For the first coding phaseϕ1S(x,y), its fringe period is 32 pixels, which is the same as the wrapped phase. There are 32 fringe periods for ϕ1S(x,y) which is divided into four equal parts. For each part, it is quantized into 8 levels with N=8. In the first 8 periods, the coding phase decreases from +π to π with a stair height of 2π8. For the second 8 periods, the coding phase increases from π to +π with a stair height of 2π8. The third 8 periods, the coding phase distribution is the same as the first 8 periods, and the fourth 8 periods is the same as the second. As we know, the coding phase should change slowly in adjacent phase periods with a stair height of 2πN. If the phase of adjacent periods changes sharply, the intensity of the adjacent periods will change sharply. In the boundary area of the adjacent periods, the intensity of the adjacent periods will strongly influence each other. As a result, the phase achieved from the CCD camera captured fringe images will be not the same as we designed, and it is not suitable for absolute phase retrieval. Therefore, in the boundary area of adjacent 8 periods, the phase changes slowly. For example, the phase information in the 8th period is the same as the 9th period. That is the reason the phase decreases from +π to π in the first 8 periods, and then increases from π to +π in the second 8 periods.

In conclusion, the coding phase is identical in the nonadjacent part of 8 periods for ϕ1S(x,y). There are 4 phase periods for the second phase ϕ2S(x,y), and its phase period is 256 pixels which are eight times of the 32 pixels for ϕ1S(x,y). ϕ2S(x,y) is quantized into 4 levels with N=4 which decreases from +π to π with a stair height of 2π4. All of ϕ(x,y), ϕ1S(x,y) and ϕ2S(x,y)are embedded by three step phase shifting algorithm. Therefore, three phase shifting fringe images are needed for each coding phase. Figures 3(a)
Fig. 3 (a)-(c) Phase shifting fringe images for wrapped phase, (d)-(f) Phase shifting fringe images for the first coding phase, (g)-(i) Phase shifting fringe images for the second coding phase.
-3(c) are used for generating ϕ(x,y), Figs. 3(d)-3(f) are for ϕ1S(x,y), and Figs. 3(g)-3(i) are for ϕ2S(x,y).

Through simple calculations, we get the result codeword shown in Fig. 2(b). C_1 denotes the codeword for ϕ1S(x,y), and the codeword in each 8 periods is unique. The codeword C_2 for ϕ2S(x,y) can determine the order of each 8 periods for ϕ1S(x,y). Therefore, based on C_1 and C_2, the new unique codeword C can be calculated by Eq. (5) as follows,
C(i,j)={(n+1C_2(i,j))×m+1C_1(i,j),C_2(i,j)even(nC_2(i,j))×m+C_1(i,j),C_2(i,j)odd
(5)
where arbitrary integer m and n are the quantized levels of ϕ1S(x,y) and ϕ2S(x,y), respectively.

Shown in Fig. 4
Fig. 4 The result unique codeword.
, the resulted new codeword C ranges from 1 to 32. Each unique codeword can determine one fringe order for the phase unwrapping. The coding phase ϕ1S(x,y) is divided into n parts. In each part, ϕ1S(x,y) is quantized into m levels in the range of [+π,π) or (π,+π]. In this condition, m=8 and n=4. For example, when C_2(i,j)=4 and C_1(i,j)=8, the result codeword C(i,j)=1 which determines the 1th phase order. When C_2(i,j)=3 and C_1(i,j)=7, we can get C(i,j)=10 which determines the 10th phase order.

In this way, 32 unique codewords can be determined for absolute phase retrieval. The number of the codewords equals to m×n, and large unique codewords can be easily get by increasing m and n. For example, m=8 and n=8 which can generate 64 unique codewords. However, when m=16 and n=16, the novel method can generate 256 unique codewords, which is more than 64 codewords of the Gray-code method generated. Here we summarize the procedures of the proposed method.

Step 1: Set two integers m and n, embed the coding phase ϕ1S(x,y) by a stair phase function:
ϕ1S(x,y)={π[x/P]×2πm,[x/(m×P)]evenπ+([x/P]+1)×2πm,[x/(m×P)]odd,
(6)
where P is the fringe pitch, and [x] determines the closest integer. Embed the coding phase ϕ2S(x,y) by a stair phase function:
ϕ2S(x,y)=π[x/(m×P)]×2πn.
(7)
Step 2: Put the stair phase into three step phase shifting fringe patterns:
Ik(x,y)=I'(x,y)+I"(x,y)cos(ϕS+δk),
(8)
whereI'(x,y) is the average intensity, I"(x,y) the intensity modulation, δk the phase shift, δk=0,2π/3,4π/3.

Step 3: Obtain the wrapped phase ϕ(x,y), the coding phase ϕ(x,y) and ϕ1S(x,y) by three phase shifting algorithm, calculate the codeword of C_1 and C_2 from the coding phase:
C_1=[m(ϕ1S+π)/2π],
(9)
C_2=[n(ϕ2S+π)/2π].
(10)
Step 4: Calculate the new unique codeword C from Eq. (5), which can be used to calculate the absolute phase Φ(x,y) based on Eq. (11).

Φ(x,y)=ϕ(x,y)C×2π.
(11)

The proposed novel phase coding method can generate a large number of codewords for absolute phase retrieval through coding two sets of phase, which can be used for high frequency absolute phase retrieval. For traditional gray-code method, the captured gray-code fringe images always are not so sharp cut-off in the black and white conversion boundaries, which may introduce wrong decoding phase orders and lead to absolute phase errors. The proposed phase coding method performs better than the gray-code method because its codeword is embedded into phase domain rather than intensity. However, in the boundary of stair phase-change, the fringe images are also not so sharp cut-off. It is mainly because of the impact of defocus and sharp change intensity in the boundary.

A self-correction phase correction phase unwrapping method is used to correct the unique codeword C [18

18. D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng. 50(8), 1130–1139 (2012). [CrossRef]

]. All the positions of 2π phase jump pixels in the wrapped phase map are marked first. Then, based on the calculated codeword C, all the positions of phase order jump pixels are fixed. In the same row, each position of 2π phase jump can be used to correct the position of its nearest phase order jump. The corrected new codeword can yield reliable absolute phase for 3-D shape measurement. A following experiment is used to verify this.

3. Experiment

A fringe projection measurement system is developed, which consists of a black-and-white CCD camera (UNIQ UP1800) and a DLP projector (Optoma EP737) with the resolution of 1024×768. The distance between the camera and the projector is about 20 cm and the tested object is placed in front of the system about 1.3 m.

Firstly, an experiment is provided to demonstrate the problem of a large number of codewords phase retrieval for traditional phase coding method. We quantize the coding phase ϕS(x,y) into different levels withN=16, N=24 and N=32. For each number of quantized levels, three fringe images are designed for its coding phase [19

19. Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012). [CrossRef] [PubMed]

]. Accordingly, three sinusoidal fringe images with a phase shift of 2π/3 are used. The wrapped phase period includes 32 pixels. One arbitrary row of the phase distribution is illustrated in Fig. 5(a)
Fig. 5 (a) One cross section of the coding phase and wrapped phase, (b) The result codeword, (c) The unwrapped absolute phase.
. Then, we calculate the codeword for the stair phase as the Fig. 5(b) shown.

When the coding phase is quantized into 16 levels, the codeword can determine the correct fringe order. The unwrapped absolute phase is shown in Fig. 5(c). However, when the number of quantized levels increases to N=24, the additional phase error leads to some wrong codewords where the codeword jumps are not aligned with the 2π discontinuities. When the number N increases to 32, more incorrect positions exist. Therefore, these positions lead to wrong absolute phase shown in Fig. 5(c). The wrong places in the codeword become more during the process of increasing of the number of the quantized levels. The experiment result is coincided with the theory analysis in the Section 2. We can conclude that when the number of codewords is large, the phase coding method becomes not reliable. A novel phase coding method to tackle this problem for high frequency fringe projection measurement is necessary.

Secondly, we compare the existed phase error with different lengths of fringe image period. The nonlinear intensity response introduced phase error can be alleviated or eliminated by using a large number of phase shifting algorithm [23

23. S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46(1), 36–43 (2007). [CrossRef] [PubMed]

]. Therefore, the coding phase embedded by eight step phase shifting algorithm can be seemed correct. The coding phase embedded by three step phase shifting algorithm contains the additional phase error. Three sets of coding phase are designed as the first coding phase illustrated before, which is quantized into 8 levels. The width of the fringe image is 1024 pixels. The fringe period length T for each set of the coding phase is 32 pixels,64 pixels and 128 pixels, respectively. These fringe images are projected onto a flat board, then, captured by the CCD camera.

Based on the principle of phase shifting algorithm, two wrapped phase can be calculated for each set of coding phase. Subtracting the coding phase with additional phase error from the correct coding phase achieved by eight phase step phase shifting algorithm, we get the distribution for the phase error. One arbitrary row of the phase error is shown in Fig. 6
Fig. 6 The phase error for phase coding method with different lengths of fringe period.
. There is no obvious difference of the phase error when the fringe images are with different fringe period lengths. The mean phase error for 300 rows for the phase error is −0.0016 rad, 0.0025 rad and −0.0019 rad, respectively. Therefore, the coding phase with different lengths will not have obvious influence on the phase coding method.

Thirdly, in the same measurement, a series of 1024×768 fringe images including three sinusoidal intensity distribution fringe images, and six phase coding fringe images as shown in Fig. 3 are projected onto a flat board to verify the effective of the proposed phase coding method. Then, these fringe images are captured. Based on the principle of the proposed method, the achieved absolute reference phase maps are shown in Fig. 7
Fig. 7 (a) One view of the absolute reference phase before correction, (b) Another view of The absolute reference phase before correction, (c) One view of the absolute reference phase after correction, (d) Another view of the absolute reference phase after correction.
. Figures 7(a)-7(b) show the unwrapped absolute reference phase without using the self-correction method [18

18. D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng. 50(8), 1130–1139 (2012). [CrossRef]

]. There are obvious sharp phase jumps in the unwrapped reference caused by the fringe not so sharp cut-off phenomenon. The unique codeword in the boundary of stair phase-change needs to be corrected. The brief correction process is illustrated before. The corrected unwrapped absolute phase is smooth and without obvious phase errors shown in Figs. 7(c)-7(d). Therefore, based on the above proposed method, the number of the codewords for phase coding method will be not a problem. When m=8 and n=4 which can generate 32 unique codewords for reliable measurement. However, 32 unique codewords for traditional phase coding method is difficult to realize.

It should be noted that the number of quantized levels for one coding phase is limited. Much better device or more fringe patterns helps to increase the number of levels. For our measurement system, the phase coding method is reliable when the number increases to 16. So six additional fringe images can determine up to 256(28) unique codewords, which is more than traditional gray-code method determined 64(26) unique codewords for six additional fringe images. For general measurement system, more than 8 quantized levels can be achieved. It can be concluded that the novel phase coding method can generate more than 64(26) unique codewords for reliable absolute phase retrieval. Then, these fringe images are projected onto a plastic board to verify the proposed phase coding method used for measuring objects with complex surface. A plastic board is used with discontinuous holes and non-uniform surface reflectivity shown in Fig. 8(a)
Fig. 8 (a) The tested object, (b) One of the sinusoidal distribution fringe images, (c) One of the fringe images for the first coding phase, (d) One of the fringe images for the second coding phase, (e) The wrapped phase of the tested object, (f) The absolute phase of the object.
. Figure 8(b) is one of the captured three-step phase shifting fringe images. Figure 8(c) is one of the captured fringe images for the first coding phase. Figure 8(d) illustrates one of the captured fringe images for the second coding phase. The wrapped phase achieved by three step phase shifting algorithm is shown in Fig. 8(e). We retrieve the absolute phase by the proposed method, which illustrated in Fig. 8(f) is smooth and without obvious errors.

At last, traditional Gray-code method is used to compare with the proposed phase coding method. Subtracted the unwrapped phase from the reference phase, the phase information used for 3-D reconstruction can be achieved. Two views of the 3-D phase information are shown in Fig. 9
Fig. 9 (a) One view of 3-D reconstruction phase for Gray-code method, (b) Another view of 3-D reconstruction for Gray-code method, (c) One view of 3-D reconstruction phase for the proposed method, (d) Another view of 3-D reconstruction phase for the proposed method.
. The phase information is achieved by the Gray-code method shown in Figs. 9(a)-9(b). There is obvious phase error in the absolute phase map. These factors of surface contrast, the ambient light, and camera noise may bring errors in the binarization process for Gray-code method, which leads to wrong codeword for the absolute phase retrieval. The novel phase coding method achieved absolute phase is shown in Figs. 9(c)-9(d). There is no phase error caused by the phase unwrapping. The proposed phase coding method is less sensitive to the surface contrast, ambient light, and camera noise [19

19. Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012). [CrossRef] [PubMed]

]. The novel phase coding method is reliable for absolute phase retrieval with high frequency.

4. Conclusion

As a conclusion, we present a novel phase coding method for absolute phase retrieval with a large number of codewords. In general, six additional fringe images can represent more than 64(26) unique codewords. The novel phase coding method is effective, which can be used for measurement requiring high frequency fringe images. Experiment results demonstrate the performance of the proposed method. In addition, this method can be easily combined with other commonly used phase shifting algorithms, for example, four step phase shifting algorithm or five step phase shifting algorithm. Therefore, the novel phase coding method has extensive application in 3-D phase measurement.

Acknowledgments

This research is supported by the National Natural Science Foundation of P. R. China (51175081) and (61107001), Natural Science Foundation of Jiangsu Province (BK2010058), Graduate Scientific Innovation Research Foundation of Jiangsu Province and Scientific Research Foundation of Graduate School of Southeast University.

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J. M. Huntley and H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993). [CrossRef] [PubMed]

13.

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for threee-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008). [CrossRef]

14.

G. Sansoni, M. Carocci, and R. Rodella, “Three-dimensional vision based on a combination of Gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt. 38(31), 6565–6573 (2005). [CrossRef] [PubMed]

15.

J. H. Pan, P. S. Huang, and F. P. Chiang, “Color-coded binary fringe projection technique for 3-D shape measurement,” Opt. Eng. 44(2), 023606 (1999). [CrossRef]

16.

Y. Liu, X. Su, and Q. Zhang, “A novel encoded-phase technique for phase measuring profilometry,” Opt. Express 19(15), 14137–14144 (2011). [CrossRef] [PubMed]

17.

Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng. 50(4), 574–579 (2012). [CrossRef]

18.

D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng. 50(8), 1130–1139 (2012). [CrossRef]

19.

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012). [CrossRef] [PubMed]

20.

S. Zhang, “Phase unwrapping error reduction framework for a multiple-wavelength phase-shifting algorithm,” Opt. Eng. 48(10), 105601 (2009). [CrossRef]

21.

Y. Ding, J. Xi, Y. Yu, and J. Chicharo, “Recovering the absolute phase maps of two fringe patterns with selected frequencies,” Opt. Lett. 36(13), 2518–2520 (2011). [CrossRef] [PubMed]

22.

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009). [CrossRef] [PubMed]

23.

S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46(1), 36–43 (2007). [CrossRef] [PubMed]

OCIS Codes
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.2830) Instrumentation, measurement, and metrology : Height measurements
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure
(100.5088) Image processing : Phase unwrapping

ToC Category:
Image Processing

History
Original Manuscript: July 30, 2012
Revised Manuscript: September 5, 2012
Manuscript Accepted: September 26, 2012
Published: October 8, 2012

Citation
Dongliang Zheng and Feipeng Da, "Phase coding method for absolute phase retrieval with a large number of codewords," Opt. Express 20, 24139-24150 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24139


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References

  1. S. Zhang, “High-resolution 3-D profilometry with binary phase-shifting methods,” Appl. Opt.50(12), 1753–1757 (2011). [CrossRef] [PubMed]
  2. J. Zhong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt.43(26), 4993–4998 (2004). [CrossRef] [PubMed]
  3. S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt.48(36), 6893–6906 (2009). [CrossRef] [PubMed]
  4. S. Zhang, D. Van Der Weide, and J. Oliver, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express18(9), 9684–9689 (2010). [CrossRef] [PubMed]
  5. E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express17(10), 7818–7830 (2009). [CrossRef] [PubMed]
  6. C. Quan, W. Chen, and C. J. Tay, “phase-retrieval techniques in fringe-projection profilometry,” Opt. Lasers Eng.48(2), 235–243 (2010). [CrossRef]
  7. T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett.35(12), 1992–1994 (2010). [CrossRef] [PubMed]
  8. Z. Wang, D. A. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng.48(2), 218–225 (2010). [CrossRef]
  9. Y. Wang, S. Zhang, and J. H. Oliver, “3-D shape measurement technique for multiple rapidly moving objects,” Opt. Express19(9), 8539–8545 (2011). [CrossRef] [PubMed]
  10. W. H. Su, “Projected fringe profilometry using the area-encoded algorithm for spatially isolated and dynamic objects,” Opt. Express16(4), 2590–2596 (2008). [CrossRef] [PubMed]
  11. Y. Ding, J. Xi, Y. Yu, W. Cheng, S. Wang, and J. F. Chicharo, “Frequency selection in absolute phase maps recovery with two frequency projection fringes,” Opt. Express20(12), 13238–13251 (2012). [CrossRef] [PubMed]
  12. J. M. Huntley and H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt.32(17), 3047–3052 (1993). [CrossRef] [PubMed]
  13. J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for threee-dimensional profilometry,” Opt. Lasers Eng.46(4), 336–342 (2008). [CrossRef]
  14. G. Sansoni, M. Carocci, and R. Rodella, “Three-dimensional vision based on a combination of Gray-code and phase-shift light projection: analysis and compensation of the systematic errors,” Appl. Opt.38(31), 6565–6573 (2005). [CrossRef] [PubMed]
  15. J. H. Pan, P. S. Huang, and F. P. Chiang, “Color-coded binary fringe projection technique for 3-D shape measurement,” Opt. Eng.44(2), 023606 (1999). [CrossRef]
  16. Y. Liu, X. Su, and Q. Zhang, “A novel encoded-phase technique for phase measuring profilometry,” Opt. Express19(15), 14137–14144 (2011). [CrossRef] [PubMed]
  17. Q. Zhang, X. Su, L. Xiang, and X. Sun, “3-D shape measurement based on complementary Gray-code light,” Opt. Lasers Eng.50(4), 574–579 (2012). [CrossRef]
  18. D. Zheng and F. Da, “Self-correction phase unwrapping method based on Gray-code light,” Opt. Lasers Eng.50(8), 1130–1139 (2012). [CrossRef]
  19. Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett.37(11), 2067–2069 (2012). [CrossRef] [PubMed]
  20. S. Zhang, “Phase unwrapping error reduction framework for a multiple-wavelength phase-shifting algorithm,” Opt. Eng.48(10), 105601 (2009). [CrossRef]
  21. Y. Ding, J. Xi, Y. Yu, and J. Chicharo, “Recovering the absolute phase maps of two fringe patterns with selected frequencies,” Opt. Lett.36(13), 2518–2520 (2011). [CrossRef] [PubMed]
  22. B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett.34(4), 416–418 (2009). [CrossRef] [PubMed]
  23. S. Zhang and S. T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt.46(1), 36–43 (2007). [CrossRef] [PubMed]

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