## A study on carrier phase distortion in phase measuring deflectometry with non-telecentric imaging |

Optics Express, Vol. 20, Issue 22, pp. 24505-24515 (2012)

http://dx.doi.org/10.1364/OE.20.024505

Acrobat PDF (1228 KB)

### Abstract

In phase measuring deflectometry (PMD), the fringe pattern deformed according to slope deviation of a specular surface is digitized employing a phase-shift technique. Without height-angle ambiguity, carrier-removal process is adopted to evaluate the variation of surface slope from phase distribution when a quasi-plane is measured. However, the difficulty lies in the fact that the nonlinearity is generally contained in the carrier frequency due to the restrictions of system geometries. This paper investigates nonlinear carrier components introduced by the generalized imaging process in PMD. Furthermore, the analytical expression of carrier components in PMD is presented for the first time. The presented analytical form of carrier components can be extended to analyze and describe various effects of system parameters on carrier distortion. Assuming a pinhole perspective model, carrier phase distribution of arbitrary geometric arrangement is modeled as a function of spatial variables by exploring ray tracing method. As shown by simulation and experimental results, the carrier distortion is greatly affected by non-telecentric camera operation. Experimental results on the basis of reference subtraction technique further demonstrate that restrictions on reflection system geometry can be eliminated when the carrier phase is removed elaborately.

© 2012 OSA

## 1. Introduction

3. D. Pérard and J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” Proc. SPIE **3204**, 74–80 (1997). [CrossRef]

2. M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE **5457**, 366–376 (2004). [CrossRef]

4. M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Optical Metrology in Production Engineering, Proc. SPIE **5457**, 366–376 (2004). [CrossRef]

7. Y. Tang, X. Y. Su, Y. K. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express **16**(19), 15090–15096 (2008). [CrossRef] [PubMed]

*et al*. [4

4. M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Optical Metrology in Production Engineering, Proc. SPIE **5457**, 366–376 (2004). [CrossRef]

*et al*. [5] introduced a reference wire to constrain the direction of some reflected rays. Guo

*et al*. [6

6. H. W. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. **48**(2), 166–171 (2010). [CrossRef]

*et al*. [7

7. Y. Tang, X. Y. Su, Y. K. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express **16**(19), 15090–15096 (2008). [CrossRef] [PubMed]

2. M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE **5457**, 366–376 (2004). [CrossRef]

3. D. Pérard and J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” Proc. SPIE **3204**, 74–80 (1997). [CrossRef]

8. C. Quan, C. J. Tay, and L. J. Chen, “A study on carrier-removal techniques in fringe projection profilometry,” Opt. Laser Technol. **39**(6), 1155–1161 (2007). [CrossRef]

*x*and

*y*direction for a generalized PMD. Section 3 presents computer simulations of carrier phase distortion in

*x*and

*y*direction. Reference subtraction technique is investigated by experimental work for carrier-removal in the PMD. Conclusions are drawn in section 4.

## 2. Analytical description of carrier phase distribution

*xyz*and

*XYZ*are defined as the world coordinate and local coordinate of CCD plane, respectively. Considering a generalized imaging system, the CCD is placed behind a lens having an optical center (

*X*,

_{f}*Z*). The optical axis of the CCD camera crosses the imaging center (

_{f}*X*,

_{0}*Z*) perpendicularly. The LCD plane is vertical to

_{0}*xz*-plane, making an angle

*θ*with

*xy*-plane.

*P*(

*x*,

_{0}*z*) is the original point on LCD, where the phase is set to zero. The light reflected by any point

_{0}*C*on the specular surface, and received by CCD passing through the optical center, can be traced back to its source location on

*B*or

_{x}*B*to calculate the phase distribution.

_{y}## A. Reference phase distribution of carrier fringe in *x* direction

*P*can be expressed by a line representation,Without loss of a generality, assuming a pinhole model for camera operation, a light beam is reflected by a point

_{x}*C*(

*x*,

_{i}*z*) and intersects on the CCD passing through (

_{i}*X*,

_{f}*Z*), making an angle

_{f}*β*with

_{x}*x*-axis. For planar-like surface, coordinate of any point

*C*is set to (

*x*, 0). The reflected light can be described using the slope intercept form of a line representation,The source location

_{i}*B*is traced back by Eq. (2) and (3) according to the reflection law. The

_{x}*x*coordinate of

*B*is described as follow,

_{x}*T*and the corresponding carrier frequency is

_{0}*f*

_{0}. According to the geometry and Eq. (2), the carrier phase can be represented as,Furthermore, from Fig. 2(a) we can write,Substituting tan

*β*from Eq. (6) into Eq. (5) we get the carrier phase distribution on CCD of

_{x}*x*direction carrier fringe pattern,

## B. Reference phase distribution of carrier fringe in *y* direction

*y*direction is calculated according to the geometry shown in Fig. 2(b) and (c). From similar triangles we get, From the geometry in Fig. 2(b) we can write, where

*B*on the virtual image of LCD screen, which can be described according to the system parameters as follow,

_{y}*B*,

_{y}*β*and

_{x}*d*+

_{i}*d*from Eq. (6) and Eq. (11) into Eq. (14), the final carrier phase on CCD plane of

_{i}′*y*direction fringe pattern is determined by Eq. (15),

*x*and

*y*direction carrier components, the

*x*direction carrier phase is described as a function of

*x*direction spatial variable, whereas the

*y*direction phase distortion is modulated by two spatial variables.

## 3. Computer simulation and experimental work

*x*and

*y*direction has been simulated by ray tracing method using the model shown in Fig. 3. According to Fig. 2, the plane object was placed 450.0mm away from the LCD screen and the CCD camera. The angle between CCD optical axis and normal direction of the LCD screen was 90°. Both CCD plane and the LCD screen were placed perpendicular to

*xz*-plane, and

*Y*-axis of CCD coordinate was parallel with

*y*-axis in the world coordinate. The fringe pitch on the LCD screen was 10mm and the focal length of camera lens was 28mm. In order to demonstrate the distortion of carrier phase distribution, the linear components have been subtracted by the plane-fitting technique.

8. C. Quan, C. J. Tay, and L. J. Chen, “A study on carrier-removal techniques in fringe projection profilometry,” Opt. Laser Technol. **39**(6), 1155–1161 (2007). [CrossRef]

8. C. Quan, C. J. Tay, and L. J. Chen, “A study on carrier-removal techniques in fringe projection profilometry,” Opt. Laser Technol. **39**(6), 1155–1161 (2007). [CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

1. | S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are,” Opt. Lasers Eng. |

2. | M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE |

3. | D. Pérard and J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” Proc. SPIE |

4. | M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Optical Metrology in Production Engineering, Proc. SPIE |

5. | R. Muhr, G. Schutte, and M. Vincze, “A triangulation method for 3D-measurement of specular surfaces,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. |

6. | H. W. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. |

7. | Y. Tang, X. Y. Su, Y. K. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express |

8. | C. Quan, C. J. Tay, and L. J. Chen, “A study on carrier-removal techniques in fringe projection profilometry,” Opt. Laser Technol. |

9. | H. W. Guo, M. Y. Chen, and P. Zheng, “Least-squares fitting of carrier phase distribution by using a rational function in profilometry fringe projection,” Opt. Lett. |

10. | B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A, Pure Appl. Opt. |

11. | L. J. Chen and C. J. Tay, “Carrier phase component removal: a generalized least-squares approach,” J. Opt. Soc. Am. A |

12. | L. J. Chen and C. G. Quan, “Fringe projection profilometry with nonparallel illumination: a least-squares approach,” Opt. Lett. |

13. | W. S. Li, T. Bothe, C. von Kopylow, and W. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE |

**OCIS Codes**

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

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

(120.5700) Instrumentation, measurement, and metrology : Reflection

(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: July 2, 2012

Revised Manuscript: August 28, 2012

Manuscript Accepted: September 14, 2012

Published: October 11, 2012

**Citation**

Lei Song, Huimin Yue, Hanshin Kim, Yuxiang Wu, Yong Liu, and Yongzhi Liu, "A study on carrier phase distortion in phase measuring deflectometry with non-telecentric imaging," Opt. Express **20**, 24505-24515 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-24505

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### References

- S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are,” Opt. Lasers Eng. 48(2), 133–140 (2010). [CrossRef]
- M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004). [CrossRef]
- D. Pérard and J. Beyerer, “Three-dimensional measurement of specular free-form surfaces with a structured-lighting reflection technique,” Proc. SPIE 3204, 74–80 (1997). [CrossRef]
- M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Optical Metrology in Production Engineering, Proc. SPIE 5457, 366–376 (2004). [CrossRef]
- R. Muhr, G. Schutte, and M. Vincze, “A triangulation method for 3D-measurement of specular surfaces,” Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci. XXXVIII(Part 5), 466–471 (2010).
- H. W. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010). [CrossRef]
- Y. Tang, X. Y. Su, Y. K. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16(19), 15090–15096 (2008). [CrossRef] [PubMed]
- C. Quan, C. J. Tay, and L. J. Chen, “A study on carrier-removal techniques in fringe projection profilometry,” Opt. Laser Technol. 39(6), 1155–1161 (2007). [CrossRef]
- H. W. Guo, M. Y. Chen, and P. Zheng, “Least-squares fitting of carrier phase distribution by using a rational function in profilometry fringe projection,” Opt. Lett. 31(24), 3588–3590 (2006). [CrossRef] [PubMed]
- B. A. Rajoub, M. J. Lalor, D. R. Burton, and S. A. Karout, “A new model for measuring object shape using non-collimated fringe-pattern projections,” J. Opt. A, Pure Appl. Opt. 9(6), S66–S75 (2007). [CrossRef]
- L. J. Chen and C. J. Tay, “Carrier phase component removal: a generalized least-squares approach,” J. Opt. Soc. Am. A 23(2), 435–443 (2006). [CrossRef] [PubMed]
- L. J. Chen and C. G. Quan, “Fringe projection profilometry with nonparallel illumination: a least-squares approach,” Opt. Lett. 30(16), 2101–2103 (2005). [CrossRef] [PubMed]
- W. S. Li, T. Bothe, C. von Kopylow, and W. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004). [CrossRef]

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