## Analysis and identification of phase error in phase measuring profilometry

Optics Express, Vol. 18, Issue 11, pp. 11300-11307 (2010)

http://dx.doi.org/10.1364/OE.18.011300

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### Abstract

Both the analysis of phase errors which occur at the abrupt discontinuities in phase measuring profilometry (PMP) and the identification method are presented in this paper. The sampling effect of CCD will cause a dilution of accuracy in PMP, especially at abrupt discontinuities on the object surface. The existing methods cannot efficiently identify the abrupt discontinuities. We analyze the relationship between the phase, the height and the equivalent wavelength. By viewing the phase as the argument of a vector we find out that CCD sampling introduces errors into the measurement and the phase is nonlinear to the equivalent wavelength at the abrupt discontinuities. Therefore temporal phase unwrapping (TPU) is introduced into the measurement to identify the abrupt discontinuities. Computer simulations and practical experiment validate the feasibility of this method.

© 2010 OSA

## 1. Introduction

1. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. **23**(18), 3105–3108 (1984). [CrossRef] [PubMed]

2. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. **28**(16), 3268–3270 (1989). [CrossRef] [PubMed]

7. 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]

9. E. Hu, Y. He, and Y. Chen, “Study on a novel phase-recovering algorithm for partial intensity saturation in digital projection grating phase-shifting profilometry,” Optik - International Journal for Light and Electron Optics **121**(1), 23–28 (2010). [CrossRef]

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

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

12. M. Huntley and H. O. Saldner, “Error-reduction methods for shape measurement by temporal phase-unwrapping,” J. Opt. Soc. Am. A **14**(12), 3188–3196 (1997). [CrossRef]

## 2. Effect of CCD sampling on PMP

*θ*between principal axes of projector and that of the imaging system. The wrapped phase is obtained by using the Eq. (2) in the

*N*-step phase calculation algorithm.

*i*and

*j*are discrete coordinates in the image.

*i*th column and

*j*th row. The value of the pixel

## 3. Invalid pixel identification

*ω*

^{[12]}.

*s*is the number of gratings. LSF improves the accuracy of the measurement.

*ω*is used to calculate the result of phase unwrapping. The standard deviation of LSF indicates the deviation between the data and the fitting curve. The standard deviation is small at the smooth region on the object surface. Equation (1) is inapplicable at the abrupt discontinuities because of the sampling. The standard deviation is dramatically big. If a suitable threshold of standard deviation is set, the abrupt discontinuities can be identified by comparing the standard deviation to the threshold. The pixels are positioned at the abrupt discontinuities if their standard deviations are bigger than the threshold. They are invalid. The phases of these pixels are deleted from the unwrapped phase map and recalculated by interpolating.

## 4. Simulation and experiment

### 4.1 Simulation demonstration

*θ*is 27°. The deformed fringe patterns are shown in Fig. 6 . For a better demonstration of the effect of CCD sampling, both the shadow and noises are ignored. The height distribution obtained by temporal phase unwrapping is shown in Fig. 7 . The phases and the fitting curves at valid pixels (32, 36), (32, 38) and invalid pixel (32, 37) are shown in Fig. 8 . The true height values of pixel (32, 36) and (32, 38) are 0 cm and 0.5 cm separately. Pixel (32, 37) is positioned at the abrupt discontinuities. The reconstructed height at pixel (32, 37) is 0.59097cm. It is greater than its true value 0.5cm and that of pixel (32, 38). At the valid pixel (32, 36) and (32, 38), the measurements and fitting curves are consistent with the standard deviations 8.2516 × 10

^{−15}rad and 0rad separately. There is a big standard deviation 6.3842rad at the invalid pixel (32, 37).The mask which is shown in Fig. 9 is generated from standard deviation distribution using threshold 0.1rad. The white pixels in the mask are invalid pixels to be deleted and interpolated. The final result is shown in Fig. 10 .

### 4.2 Experiment

## 5. Conclusions

*N*-step algorithm are not in inverse relation to the equivalent wavelength at the abrupt discontinuities. Therefore TPU is employed to identify the abrupt discontinuities. We presented the simulation demonstration and experiment. All results show the correctness of our theoretical analysis and validity of TPU in identification of phase errors caused by CCD sampling and the abrupt discontinuities on the object surface.

## Acknowledgement

## References and links

1. | V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. |

2. | J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. |

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

4. | H. Su, J. Li, and X. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. |

5. | A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. |

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

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

8. | E. Hu, Y. He, and Y. Chen, “Study on a novel phase-recovering algorithm for partial intensity saturation in digital projection grating phase-shifting profilometry,” Opt. Int. J. Light Electron. Opt. (2008), doi:. |

9. | E. Hu, Y. He, and Y. Chen, “Study on a novel phase-recovering algorithm for partial intensity saturation in digital projection grating phase-shifting profilometry,” Optik - International Journal for Light and Electron Optics |

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

11. | J. M. Huntley and H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. |

12. | M. Huntley and H. O. Saldner, “Error-reduction methods for shape measurement by temporal phase-unwrapping,” J. Opt. Soc. Am. A |

**OCIS Codes**

(100.5070) Image processing : Phase retrieval

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

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

(350.5030) Other areas of optics : Phase

(100.5088) Image processing : Phase unwrapping

**ToC Category:**

Image Processing

**History**

Original Manuscript: February 12, 2010

Revised Manuscript: April 4, 2010

Manuscript Accepted: April 22, 2010

Published: May 13, 2010

**Citation**

Feng Chen, Xianyu Su, and Liqun Xiang, "Analysis and identification of phase error in phase measuring profilometry," Opt. Express **18**, 11300-11307 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11300

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### References

- V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23(18), 3105–3108 (1984). [CrossRef] [PubMed]
- J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28(16), 3268–3270 (1989). [CrossRef] [PubMed]
- T. R. Judge and P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Lasers Eng. 21(4), 199–239 (1994). [CrossRef]
- H. Su, J. Li, and X. Su, “Phase algorithm without the influence of carrier frequency,” Opt. Eng. 36(6), 1799–1805 (1997). [CrossRef]
- A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. 37(23), 5416–5420 (1998). [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]
- E. Hu, Y. He, and Y. Chen, “Study on a novel phase-recovering algorithm for partial intensity saturation in digital projection grating phase-shifting profilometry,” Opt. Int. J. Light Electron. Opt. (2008), doi:.
- E. Hu, Y. He, and Y. Chen, “Study on a novel phase-recovering algorithm for partial intensity saturation in digital projection grating phase-shifting profilometry,” Optik - International Journal for Light and Electron Optics 121(1), 23–28 (2010). [CrossRef]
- X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42(3), 245–261 (2004). [CrossRef]
- J. M. Huntley and H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993). [CrossRef] [PubMed]
- M. Huntley and H. O. Saldner, “Error-reduction methods for shape measurement by temporal phase-unwrapping,” J. Opt. Soc. Am. A 14(12), 3188–3196 (1997). [CrossRef]

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