## A novel encoded-phase technique for phase measuring profilometry |

Optics Express, Vol. 19, Issue 15, pp. 14137-14144 (2011)

http://dx.doi.org/10.1364/OE.19.014137

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### Abstract

Three-dimensional (3-D) shape measurement using a novel encoded-phase grating is proposed. The projected sinusoidal fringe patterns are designed with wrapped and encoded phase instead of monotonic and unwrapped phase. Phase values of the projected fringes on the surface are evaluated by phase-shift technique. The absolute phase is then restored with reference to the encoded information, which is extracted from the differential of the wrapped phase. To solve the phase errors at some phase-jump areas, Hilbert transform is employed. By embedding the encoded information in the wrapped phase, there is no extra pattern that needs to be projected. The experimental results identify its feasibility and show the possibility to measure the spatially isolated objects. It will be promising to analyze dynamic objects.

© 2011 OSA

## 1. Introduction

1. F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. **39**(1), 10–22 (2000). [CrossRef]

3. Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express **13**(8), 3110–3116 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3110. [CrossRef] [PubMed]

4. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. **42**(3), 245–261 (2004). [CrossRef]

5. T. R. Judge and P. J. Bryanston-Cross, “Review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. **21**(4), 199–239 (1994). [CrossRef]

6. H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. **33**(20), 4497–4500 (1994). [CrossRef] [PubMed]

10. E. B. Li, X. Peng, J. Xi, J. F. Chicharo, J. Q. Yao, and D. W. Zhang, “Multi-frequency and multiple phase-shift sinusoidal fringe projection for 3D profilometry,” Opt. Express **13**(5), 1561–1569 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-5-1561. [CrossRef] [PubMed]

11. Y. Li, C. F. Zhao, Y. X. Qian, H. Wang, and H. Zh. Jin, “High-speed and dense three-dimensional surface acquisition using defocused binary patterns for spatially isolated objects,” Opt. Express **18**(21), 21628–21635 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-21-21628. [CrossRef] [PubMed]

12. S. Zhang, “Flexible 3D shape measurement using projector defocusing: extended measurement range,” Opt. Lett. **35**(7), 934–936 (2010). [CrossRef] [PubMed]

13. W.-H. Su, “Color-encoded fringe projection for 3D shape measurements,” Opt. Express **15**(20), 13167–13181 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-13167. [CrossRef] [PubMed]

## 2. Principles

### 2.1 The encoded-phase fringe pattern

*n*= 0,1,...,N-1,

*I*means the

_{n}*n*’th image, and

*N*is the total number of phase shifts.

*φ*is the phase limited in [-π, π] called wrapped phase, which can be unwrapped as the continuous phase

_{w}*φ = 2πfx*. Here

*x*is the pixel index of the digital projector, and

*f*is the frequency of the projected sinusoidal fringe.

*2π*phase jumps. The phase differential is positive in type one and negative in type two as shown in Fig. 1 . In general the two types could not appear at the same time. If a code value 1 or 0 is set for each period with positive and negative phase differential separately, a code sequence will be formed by collecting the code value of each period, e.g. it is “1111…” in Fig. 1(a), and “0000…” in Fig. 1(b).

^{6}= 64 different sub-code sequences. Therefore the longest length of code sequence can be 64 to ensure each period is identified by a unique sub-code sequence. For example, any sub-code sequence with a length of six codes appears only once within the whole sequence ˝111111 0000001000011000101000111001001011001101001111010101110110˝. Figure 2(a) shows a part of this code sequence.

*k*is the length of the sub-code sequence,

*M*is the number of the periods in the sinusoidal fringe patterns.

### 2.2 Decoding procedure

*m*(

*m*= 0,1,...,N-1,where N is the length of the code sequence), is obtained using the subsequence matching method. Figure 3 shows how to obtain m. There are 64 codes in the whole sequence as shown in section 2.1. The global position m of the subsequence ‘111110’ equals 1 which is obtained by string matching. So, the fringe orders

*m*of all periods in this subsequence should be 1, 2, 3, 4, 5, and 6, respectively.

### 2.3 Correction for phase errors

14. L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. **34**(15), 2363–2365 (2009). [CrossRef] [PubMed]

*N*-frame phase-shift technique, the wrapped phase can be retrieved by two components, e.g. the

*cos*fringe

*Ic*and

*sin*fringe

*Is*, which are calculated by

*cos*fringe is less influenced. Therefore the error in Fig. 4(e) is mainly introduced by the accumulative

*sin*fringe, in which the direction of phase-shift reverses at the area with codeword 10 or 01. Here we set the accumulative

*cos*fringe as the reference. We choose some part of a give line in the

*cos*fringe, which includes both the A and B fringes in Fig. 5(e). There is no-zero constant value in the

*cos*fringe, therefore Hilbert transform can produce a good

*sin*fringe. A new phase can be recalculated by arc tangent function. We choose the part P

_{1}P

_{2}to replace the corresponding original phase, because some errors will be involved by Hilbert transform at the ends of the chosen part with respect to Fourier transform’s characteristics. The Hilbert transform could not work well, when the captured A or B fringe failed to reach the point P

_{1}or P

_{2}. The reference data is got by classical three-frame phase-shift technique. The difference before correction is shown in Fig. 5(f). Figure 5(g) shows the difference after correction. It shows that the error is dramatically reduced.

## 3. Experimental results

4. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. **42**(3), 245–261 (2004). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

1. | F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. |

2. | J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. |

3. | Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express |

4. | X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. |

5. | T. R. Judge and P. J. Bryanston-Cross, “Review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. |

6. | H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. |

7. | W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. |

8. | H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. |

9. | Y. Hao, Y. Zhao, and D. Li, “Multifrequency grating projection profilometry based on the nonlinear excess fraction method,” Appl. Opt. |

10. | E. B. Li, X. Peng, J. Xi, J. F. Chicharo, J. Q. Yao, and D. W. Zhang, “Multi-frequency and multiple phase-shift sinusoidal fringe projection for 3D profilometry,” Opt. Express |

11. | Y. Li, C. F. Zhao, Y. X. Qian, H. Wang, and H. Zh. Jin, “High-speed and dense three-dimensional surface acquisition using defocused binary patterns for spatially isolated objects,” Opt. Express |

12. | S. Zhang, “Flexible 3D shape measurement using projector defocusing: extended measurement range,” Opt. Lett. |

13. | W.-H. Su, “Color-encoded fringe projection for 3D shape measurements,” Opt. Express |

14. | L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. |

**OCIS Codes**

(110.6880) Imaging systems : Three-dimensional image acquisition

(120.4630) Instrumentation, measurement, and metrology : Optical inspection

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 26, 2011

Revised Manuscript: June 21, 2011

Manuscript Accepted: June 23, 2011

Published: July 8, 2011

**Citation**

Yuankun Liu, Xianyu Su, and Qican Zhang, "A novel encoded-phase technique for phase measuring profilometry," Opt. Express **19**, 14137-14144 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14137

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### References

- F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000). [CrossRef]
- J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of the art in structured light patterns for surface profilometry,” Pattern Recognit. 43(8), 2666–2680 (2010). [CrossRef]
- Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13(8), 3110–3116 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-3110 . [CrossRef] [PubMed]
- X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004). [CrossRef]
- T. R. Judge and P. J. Bryanston-Cross, “Review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994). [CrossRef]
- H. Zhao, W. Chen, and Y. Tan, “Phase-unwrapping algorithm for the measurement of three-dimensional object shapes,” Appl. Opt. 33(20), 4497–4500 (1994). [CrossRef] [PubMed]
- W. Nadeborn, P. Andra, and W. Osten, “A robust procedure for absolute phase measurement,” Opt. Lasers Eng. 24(2–3), 245–260 (1996). [CrossRef]
- H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997). [CrossRef] [PubMed]
- Y. Hao, Y. Zhao, and D. Li, “Multifrequency grating projection profilometry based on the nonlinear excess fraction method,” Appl. Opt. 38(19), 4106–4110 (1999). [CrossRef] [PubMed]
- E. B. Li, X. Peng, J. Xi, J. F. Chicharo, J. Q. Yao, and D. W. Zhang, “Multi-frequency and multiple phase-shift sinusoidal fringe projection for 3D profilometry,” Opt. Express 13(5), 1561–1569 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-13-5-1561 . [CrossRef] [PubMed]
- Y. Li, C. F. Zhao, Y. X. Qian, H. Wang, and H. Zh. Jin, “High-speed and dense three-dimensional surface acquisition using defocused binary patterns for spatially isolated objects,” Opt. Express 18(21), 21628–21635 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-21-21628 . [CrossRef] [PubMed]
- S. Zhang, “Flexible 3D shape measurement using projector defocusing: extended measurement range,” Opt. Lett. 35(7), 934–936 (2010). [CrossRef] [PubMed]
- W.-H. Su, “Color-encoded fringe projection for 3D shape measurements,” Opt. Express 15(20), 13167–13181 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-13167 . [CrossRef] [PubMed]
- L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34(15), 2363–2365 (2009). [CrossRef] [PubMed]

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