## Calibration-based two-frequency projected fringe profilometry: a robust, accurate, and single-shot measurement for objects with large depth discontinuities

Optics Express, Vol. 14, Issue 20, pp. 9178-9187 (2006)

http://dx.doi.org/10.1364/OE.14.009178

Acrobat PDF (583 KB)

### Abstract

An improved method is proposed to perform calibration-based fringe projected profilometry using a two-frequency fringe pattern for the 3D shape measurements of objects with large discontinuous height steps. A fabrication scheme for the two-frequency pattern is described as well. The proposed method offers following major advantages: (1) only one phase measurement needed for operation, (2) easiness for calibration, (3) robust performance, especially for automatic phase unwrapping, and (4) more flexible data acquisition for complex objects. This makes it possible for a single-shot measurement of dynamic objects with discontinuities. Both theoretical descriptions and experimental demonstrations are provided.

© 2006 Optical Society of America

## 1. Introduction

1. G. Indebetouw, “Profile measurement using projection of running fringes,” Appl. Opt. **17**, 2930–2933 (1978). [CrossRef] [PubMed]

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

3. M. Takeda and K. Motoh, “Fourier transform profilometry for the automatic measurement of 3-D object shaped,” Appl. Opt. **22**, 3977–3982 (1983). [CrossRef] [PubMed]

4. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. **35**, 263–284 (2001). [CrossRef]

5. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A **9**, 1740–1748 (1992). [CrossRef]

7. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**, 51–60 (1996). [CrossRef] [PubMed]

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

9. W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, “Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry,” Opt. Eng. **42**, 1730–1740 (2003). [CrossRef]

10. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A **11**, 107- (1994). [CrossRef]

13. J.-J. Chyou, S.-J. Chen, and Y.-K. Chen, “Two-dimensional phase unwrapping with a multichannel least-mean-square algorithm,” Appl. Opt. **43**, 5655–5661 (2004). [CrossRef] [PubMed]

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

15. H. O. Saldner and J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. **36**, 610–615 (1997). [CrossRef]

16. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. **26**, 2810–2816 (1987). [CrossRef] [PubMed]

20. 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**, 1561–1569 (2005). [CrossRef] [PubMed]

21. M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, “Frequency-multiplex Fourier-transform profilomery: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. **36**, 5347–5354 (1997). [CrossRef] [PubMed]

22. J. L. Li, H. J. Su, and X. Y. Su, “Two-frequency grating used in phase-measuring profilometry,” Appl. Opt. **36**, 277–280 (1997). [CrossRef] [PubMed]

21. M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, “Frequency-multiplex Fourier-transform profilomery: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. **36**, 5347–5354 (1997). [CrossRef] [PubMed]

22. J. L. Li, H. J. Su, and X. Y. Su, “Two-frequency grating used in phase-measuring profilometry,” Appl. Opt. **36**, 277–280 (1997). [CrossRef] [PubMed]

## 2. Properties of the two-frequency fringe pattern

*N*intervals. Shown in Fig. 1 is an example, in which

*N*=16, and

*P*=5.

*M*levels as well. Its quantized transmittance

*t*

_{n}for the

*n*th interval can be found as

*M*cells are used as a group to represent a quantized transmittance level

*t*

_{n}. Different transmittance levels are implemented by setting the total number of transparent cells to the represented level. The patterns resulted from such an encoding method for several transmittance levels are outlined as Fig. 2.

## 3. Phase-extraction for two-frequency patterns

*φ*

_{h}and

*φ*

_{l}are phases of the high frequency and the low frequency, respectively. They can be expressed more specifically as

*Δφ*is the distorted phase caused by depth variation.

*j*=√-1, and “∗” represents a complex conjugate. Complex modulation

*h̃*(

*x, y*) and

*l̃*(

*x, y*) are of the following form,

*A*(

*f*

_{x}

*, y*),

*H̃*(

*f*

_{x}

*, y*), and

*L̃*(

*f*

_{x}

*, y*), are complex Fourier amplitudes. The terms

_{h}and Φ

_{l}are within the interval between -

*π*and

*π*, and need to be unwrapped to obtain the absolute phases

*φ*

_{h}and

*φ*

_{l}. In the above phase calculation, the unwanted DC term

*a(x,y)*is eliminated during the filtering process and the multiplicative noise is cancelled out by the division operation in Eq. (10a) and (10b).

## 4. Phase unwrapping

_{h}and the low frequency phase Φ

_{l}. It is reasonable to assume that the frequency of the lower one is low enough that no discontinuity is larger than its equivalent wavelength. Thus, errors of phase unwrapping for the high frequency pattern can be corrected from the absolute phase map of the lower frequency one.

*Q*which plays an important role in determining the ambiguity condition is defined as

*Unwrap*{} denotes the phase-unwrapping operator, and [] denotes rounding to the nearest integer. For measurements of continuous surfaces,

*Q*is equal to zero. Therefore, the unwrapped phase

*Unwrap*{Φ

_{h}}at the discontinuous area can by automatically carried out if

*Q*is not zero. The unwanted errors can be compensated with this term

*Unwrap*{Φ

_{h}}-2

*πQ*.

## 5. System calibrations

*X*

_{g}

*Y*

_{g}plane with fringes normal to axis

*X*

_{g}is projected to the tested object. World coordinate system

*XYZ*is a fixed reference system for representing the shape of the tested object. The CCD sensor array, which is located in the coordinate system

*RC*, records the distorted fringes on the object via the imaging lens. The origin of the camera coordinate system,

*UVW*, is the nodal point of the imaging lens, while the origin of the projection system,

*X*

_{g}

*Y*

_{p}

*Z*

_{p}, is the nodal point of the projection lens. Axis

*W*and

*Z*

_{p}coincides with the optical axis of the imaging lens and projection lens, respectively.

23. H. Liu, W. H. Su, K. R., and S. Yin, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. **216**, 65–80 (2003). [CrossRef]

*a*

_{1}

*, a*

_{o}

*, b*

_{1}, and

*b*

_{o}can be determined by a set of depth values and corresponding transverse positions. The conversion between the phase and the depth information can be determined by:

*z*and

*φ*are the depth position and unwrapped phase, and

*, g*

_{i}, and

*h*

_{i}are undetermined coefficients from the optical system. The first part of Eq. (13) can be recognized as the ideal phase-to-depth relation, while the second term originates from distortions. For well-behaved systems, the distorted image points are very close to their ideal counterparts.

*c*

_{i}and

*c*

_{-i}are coefficients of the polynomial. With a sufficient set of phase values and corresponding depth values, the coefficients can be carried out.

## 6. Experiments

### 6.1 3D shape measurements

### 6.2 Error analysis

9. W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, “Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry,” Opt. Eng. **42**, 1730–1740 (2003). [CrossRef]

*n*, increased, Eq. (14) could become more accurate as well. Our experiments had shown that the coefficients could be carried out in the order of sub-micron accuracy while

*n*=5.

9. W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, “Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry,” Opt. Eng. **42**, 1730–1740 (2003). [CrossRef]

23. H. Liu, W. H. Su, K. R., and S. Yin, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. **216**, 65–80 (2003). [CrossRef]

## 7. Conclusion

## Acknowledgments

## References and links

1. | G. Indebetouw, “Profile measurement using projection of running fringes,” Appl. Opt. |

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

3. | M. Takeda and K. Motoh, “Fourier transform profilometry for the automatic measurement of 3-D object shaped,” Appl. Opt. |

4. | X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. |

5. | K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A |

6. | V. Y. Su, G von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. |

7. | Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. |

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

9. | W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, “Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry,” Opt. Eng. |

10. | D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A |

11. | K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. |

12. | A. Collaro, G. Franceschetti, F. Palmieri, and M. S. Ferreiro, “Phase unwrapping by means of genetic algorithms,” J. Opt. Soc. Am. A |

13. | J.-J. Chyou, S.-J. Chen, and Y.-K. Chen, “Two-dimensional phase unwrapping with a multichannel least-mean-square algorithm,” Appl. Opt. |

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

15. | H. O. Saldner and J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. |

16. | K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. |

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

18. | D. R. Burton and M. J. Lalor, “Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping,” Appl. Opt. |

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

20. | 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 |

21. | M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, “Frequency-multiplex Fourier-transform profilomery: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations,” Appl. Opt. |

22. | J. L. Li, H. J. Su, and X. Y. Su, “Two-frequency grating used in phase-measuring profilometry,” Appl. Opt. |

23. | H. Liu, W. H. Su, K. R., and S. Yin, “Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement,” Opt. Commun. |

24. | L. B. Jackson, |

**OCIS Codes**

(110.6880) Imaging systems : Three-dimensional image acquisition

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

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 14, 2006

Revised Manuscript: September 25, 2006

Manuscript Accepted: September 25, 2006

Published: October 2, 2006

**Citation**

Wei-Hung Su and Hongyu Liu, "Calibration-based two-frequency projected fringe profilometry: a robust, accurate, and single-shot measurement for objects with large depth discontinuities," Opt. Express **14**, 9178-9187 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9178

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### References

- G. Indebetouw, "Profile measurement using projection of running fringes," Appl. Opt. 17, 2930-2933 (1978). [CrossRef] [PubMed]
- V. Srinivasan, H. C. Liu, and M. Halioua, "Automated phase-measuring profilometry of 3-D diffuse objects," Appl. Opt. 23, 3105-3108 (1984). [CrossRef] [PubMed]
- M. Takeda and K. Motoh, "Fourier transform profilometry for the automatic measurement of 3-D object shaped," Appl. Opt. 22, 3977-3982 (1983). [CrossRef] [PubMed]
- X. Su, and W. Chen, "Fourier transform profilometry: a review," Opt. Lasers Eng. 35, 263-284 (2001). [CrossRef]
- K. G. Larkin and B. F. Oreb, "Design and assessment of symmetrical phase-shifting algorithms," J. Opt. Soc. Am. A 9, 1740-1748 (1992). [CrossRef]
- V. Y. Su, G von Bally, and D. Vukicevic, "Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation," Opt. Commun. 98, 141-150 (1993). [CrossRef]
- Y. Surrel, "Design of algorithms for phase measurements by the use of phase stepping," Appl. Opt. 35, 51-60 (1996). [CrossRef] [PubMed]
- F. Chen, G. M. Brown, and M. Song, "Overview of three-dimensional shape measurement using optical methods," Opt. Eng. 39, 10-22 (2000). [CrossRef]
- W. H. Su, H. Liu, K. Reichard, S. Yin, and F. T. S. Yu, "Fabrication of digital sinusoidal gratings and precisely conytolled diffusive flats and their application to highly accurate projected fringe profilometry," Opt. Eng. 42, 1730-1740 (2003). [CrossRef]
- D. C. Ghiglia and L. A. Romero, "Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods," J. Opt. Soc. Am. A 11, 107 (1994). [CrossRef]
- K. A. Stetson, J. Wahid, and P. Gauthier, "Noise-immune phase unwrapping by use of calculated wrap regions," Appl. Opt. 36, 4830-4838 (1997). [CrossRef] [PubMed]
- A. Collaro, G. Franceschetti, F. Palmieri, and M. S. Ferreiro, "Phase unwrapping by means of genetic algorithms," J. Opt. Soc. Am. A 15, 407-418 (1998). [CrossRef]
- J.-J. Chyou, S.-J. Chen, and Y.-K. Chen, "Two-dimensional phase unwrapping with a multichannel least-mean-square algorithm," Appl. Opt. 43, 5655-5661 (2004). [CrossRef] [PubMed]
- J. M. Huntley and H. O. Saldner, "Temporal phase-unwrapping algorithm for automated inteferogram analysis," Appl. Opt. 32, 3047-3052 (1993). [CrossRef] [PubMed]
- H. O. Saldner and J. M. Huntley, "Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector," Opt. Eng. 36, 610-615 (1997). [CrossRef]
- K. Creath, "Step height measurement using two-wavelength phase-shifting interferometry," Appl. Opt. 26, 2810-2816 (1987). [CrossRef] [PubMed]
- H. Zhao, W. Chen, and Y. Tan, "Phase-unwrapping algorithm for the measurement of three-dimensional object shapes," Appl. Opt. 33, 4497-4500 (1994). [CrossRef] [PubMed]
- D. R. Burton and M. J. Lalor, "Multichannel Fourier fringe analysis as an aid to automatic phase unwrapping," Appl. Opt. 33, 2939-2948 (1994) [CrossRef] [PubMed]
- Y. Hao, Y. Zhao, and D. Li, "Multifrequency grating projection profilometry based on the nonlinear excess fraction method," Appl. Opt. 38, 4106-4110 (1999). [CrossRef]
- 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, 1561-1569 (2005). [CrossRef] [PubMed]
- M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilomery: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," Appl. Opt. 36, 5347-5354 (1997). [CrossRef] [PubMed]
- J. L. Li, H. J. Su, and X. Y. Su, "Two-frequency grating used in phase-measuring profilometry," Appl. Opt. 36, 277-280 (1997). [CrossRef] [PubMed]
- H. Liu, W. H. Su, K. R., and S. Yin, "Calibration-based phase-shifting projected fringe profilometry for accurate absolute 3D surface profile measurement," Opt. Commun. 216, 65-80 (2003). [CrossRef]
- L. B. Jackson, Digital Filters and Signal Processing (Toppan, 1996), Chap. 6.

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