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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 24505–24515
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A study on carrier phase distortion in phase measuring deflectometry with non-telecentric imaging

Lei Song, Huimin Yue, Hanshin Kim, Yuxiang Wu, Yong Liu, and Yongzhi Liu  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 24505-24515 (2012)
http://dx.doi.org/10.1364/OE.20.024505


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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

It is known that the height-angle ambiguity problem is one of the difficulties in phase measuring deflectometry (PMD) and that the surface slope is coupled with object height in the phase map. It is difficult to isolate absolute height or slope variation from phase map without any further geometric information or assumptions. Solutions that proposed in the literature can be classified into analytical fitting methods [3

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]

] and geometric constraints enhancing methods [2

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

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

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]

]. Analytical models express the phase-to-height relation in accordance with the system geometry and the models are fitted to the observed data iteratively. However, accuracy of calibration parameters limited the improvement of the overall accuracy. On the other hand, Knauer 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]

] enhanced geometric constraints by employing stereo measurement with two cameras. Muhr et al. [5

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. XXXVIII(Part 5), 466–471 (2010).

] 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]

] shifted the monitor to determine the locus of incident ray for each pixel and further reconstruct the 3D shape in the least squares sense. Shifting the screen and camera respectively, Tang 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]

] calculated the slope and reconstructed the shape by the phase maps of the recorded fringe patterns. Consequently, the request of mechanical shifting and geometrical restrictions of the time-consuming calibration reduced the flexibility of these techniques.

However, the ambiguity can be neglected if a quasi-plane with very small depth variation is measured. In this situation, the phase distribution is only slope dependent and conversion of the measured phase distribution to the object slope distribution is required. Topography of the specular object is reconstructed by 2D numerical integration. Nevertheless, the phase distribution is sensitive to the geometric parameters and the local slope. The overall accuracy is limited by the accuracy of geometric parameters. The geometric constraints enhancing methods with time-consuming calibration [2

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

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]

] is not necessary. Here, conventional carrier removal methods [8

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]

] can serve as easier solutions to the phase-to-slope translation accordingly.

Carrier removal methods are usually limited by specific geometric arrangements. In fringe projection methods, system arrangement is usually adjusted carefully to help remove nonlinear carrier components and guarantee measurement accuracy. In order to eliminate restrictions on system geometries, models that describe exactly carrier phase distribution and phase-to-height relationship have been developed for arbitrary fringe projection profilometry system [9

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. 31(24), 3588–3590 (2006). [CrossRef] [PubMed]

,10

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. 9(6), S66–S75 (2007). [CrossRef]

]. Various carrier-removal techniques have been proposed to determine the carrier frequency subsequently [8

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]

,9

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. 31(24), 3588–3590 (2006). [CrossRef] [PubMed]

,11

11. 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]

,12

12. 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]

] and which might be adopted in the PMD. However, carrier phase distribution in a generalized PMD should be clarified to choose and develop carrier removal methods accordingly. To the best of authors’ knowledge, analytical description of the carrier phase distribution in PMD with an arbitrary geometric arrangement is still not available.

I(x,y)=A(x,y)+B(x,y)cos[2πf0(x,y)x+φ(x,y)].
(1)

In section 2, we demonstrate the analytical description of carrier component in 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

Figure 2
Fig. 2 Geometry of the PMD. (a) Normal view. (b) Normal view. (c) Side view.
shows the detailed geometrical arrangement of the PMD for further derivation of carrier phase distribution. Here, 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 (Xf, Zf). The optical axis of the CCD camera crosses the imaging center (X0, Z0) perpendicularly. The LCD plane is vertical to xz-plane, making an angle θ with xy-plane. P(x0, z0) is the original point on LCD, where the phase is set to zero. The light reflected by any point C on the specular surface, and received by CCD passing through the optical center, can be traced back to its source location on Bx or By to calculate the phase distribution.

A. Reference phase distribution of carrier fringe in x direction

In Fig. 2(a), the phase axis Px can be expressed by a line representation,
z=(xx0)tanθ+z0.
(2)
Without loss of a generality, assuming a pinhole model for camera operation, a light beam is reflected by a point C(xi, zi) and intersects on the CCD passing through (Xf, Zf), making an angle βx with x-axis. For planar-like surface, coordinate of any point C is set to (xi, 0). The reflected light can be described using the slope intercept form of a line representation,
0=(xiXf)tanβx+Zf.
(3)
The source location Bx is traced back by Eq. (2) and (3) according to the reflection law. The x coordinate of Bx is described as follow,

x=XftanβxZfz0+x0tanθtanθ+tanβx.
(4)

Assume that the uniform fringe pitch on the LCD is T0 and the corresponding carrier frequency is f0. According to the geometry and Eq. (2), the carrier phase can be represented as,
φ{βx}=2πxx0T0cosθ=2πf0cosθ(XftanβxZfz0+x0tanθtanθ+tanβxx0).
(5)
Furthermore, from Fig. 2(a) we can write,
tanβx=X0Xf+xccosγxcsinγ+Z0Zf.
(6)
Substituting tanβx from Eq. (6) into Eq. (5) we get the carrier phase distribution on CCD of x direction carrier fringe pattern,

φ{xc}=2πf0cosθ{(xcsinγZ0+Zf)[(Xfx0)tanθ+Zf+z0]xc(cosγtanθsinγ)+(Z0Zf)tanθ+X0Xf+Xfx0}.
(7)

B. Reference phase distribution of carrier fringe in y direction

The carrier phase in y direction is calculated according to the geometry shown in Fig. 2(b) and (c). From similar triangles we get,
lLf=f2+xc2fxctanγ.
(8)
l+l=Zf+zsinβx=l[(Xfx0)tanβxZfz0tanθ+tanβxtanθZf+z0+ZfZf].
(9)
From the geometry in Fig. 2(b) we can write,
di=lsin(βx+γ)+f.
(10)
di+di=(l+l)sin(βx+γ)+f=[(Xfx0)tanβxZfz0tanθ+tanβxtanθZf+z0+ZfZf](Lf)ffxctanγ+f.
(11)
where

sin(βx+γ)=ff2+xc2.
(12)

The side view of PMD is shown in Fig. 2(c). A light beam is traced back to intersect at By on the virtual image of LCD screen, which can be described according to the system parameters as follow,

Y=Y0tanβy+f(di+di)tanβy=Y0+cotβy[f(di+di)].
(13)

We can write the carrier phase of By,

φ{βx,βy}=2πYY0T0=2πf0cotβy[f(di+di)].
(14)

Substituting tanβx and di + di from Eq. (6) and Eq. (11) into Eq. (14), the final carrier phase on CCD plane of y direction fringe pattern is determined by Eq. (15),

φ{xc,yc}=2πf0Zf(fL)ycfxctanγ{{(Xfx0)+[(Xfx0)tanθ+Zf+z0](xcsinγZ0+Zf)(cosγsinγtanθ)xc+(Z0Zf)tanθ+X0Xf}tanθ+Z0+Zf}.
(15)

Consequently, Eq. (7) and (15) demonstrate the fact that the carrier phase distribution is spatially-varying. Comparing 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

The PMD was simulated based on light tracing and topography of a specimen was measured to verify the theoretical analysis. Figure 3
Fig. 3 A computer generated simulation model of PMD. Vectors indicate the inverse direction of the incident light on CCD camera and the reflection light on the specular surface, respectively. Marks on the LCD screen indicate original positions of light sources on LCD screen.
shows the established simulation model of PMD which consists of a LCD screen displaying phase-shifting fringe patterns, a CCD camera with an imaging lens and a specular surface. According to the system arrangement shown in Fig. 2 and reflection law, each incident light on CCD camera was traced back to its original source location on the LCD screen for phase quantification.

Reference phase of a plane in 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.

The proposed carrier description in PMD with generalized imaging process was emphasized to give us guidance on how to choose or develop adequate carrier-removal techniques. Furthermore, experiments were conducted to study the elimination of restrictions on PMD schemes by a robust carrier-removal approach. The well developed carrier-removal techniques [8

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]

] in fringe projection profilometry might serve as alternative methods in this case, such as reference subtraction technique, least-squares approach, etc.

In the experimental work, we consider a well known reference subtraction technique for nonlinear carrier-removal. The undesirable side-effect of this approach is the increase of phase measurement uncertainty. In order to evaluate the increase of phase uncertainty, the carrier of an unwrapped phase map was removed by the series-expansion method [8

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]

] and reference subtraction method, respectively. A series function was fitted to the phase map on a plane object to estimate the carrier distribution. Subtraction of a series function will leave the phase uncertainty unaffected. On the other hand, the plane object was measured twice at the same position and the two unwrapped phase maps were subtracted to estimate the phase error. Phase uncertainty calculated from the carrier removal results of the two methods is plotted in Fig. 5
Fig. 5 Phase uncertainty of series-expansion method and reference subtraction method
. It is shown that instead of being eliminated in the subtraction, the phase uncertainty was magnified by reference subtraction method. However, the reference subtraction method is straight forward and no matter whatever features of a carrier can be removed automatically.

4. Conclusion

Acknowledgment

The authors wish to acknowledge the support by the Fundamental Research Funds for the Central Universities (No. ZYGX2011J053) and the National Nature Science Foundation of China (No. 60925019).

References and links

1.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are,” Opt. Lasers Eng. 48(2), 133–140 (2010). [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]

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]

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]

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. XXXVIII(Part 5), 466–471 (2010).

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]

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]

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]

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. 31(24), 3588–3590 (2006). [CrossRef] [PubMed]

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. 9(6), S66–S75 (2007). [CrossRef]

11.

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]

12.

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]

13.

W. S. Li, T. Bothe, C. von Kopylow, and W. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004). [CrossRef]

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

  1. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are,” Opt. Lasers Eng. 48(2), 133–140 (2010). [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]
  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]
  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]
  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. XXXVIII(Part 5), 466–471 (2010).
  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]
  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]
  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]
  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. 31(24), 3588–3590 (2006). [CrossRef] [PubMed]
  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. 9(6), S66–S75 (2007). [CrossRef]
  11. 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]
  12. 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]
  13. 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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