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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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]

2010

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]

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

2008

2007

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]

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]

2006

2005

2004

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]

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]

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

1997

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]

Beyerer, J.

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]

Bothe, T.

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

Burton, D. R.

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]

Chen, L. J.

Chen, M. Y.

Feng, P.

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]

Gorthi, S. S.

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

Guo, H. W.

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]

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]

Häusler, G.

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]

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]

Jing, H.

Jüptner, W.

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

Kaminski, J.

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]

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]

Karout, S. A.

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]

Knauer, M. C.

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]

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]

Lalor, M. J.

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]

Li, W. S.

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

Liu, Y. K.

Muhr, R.

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).

Pérard, D.

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]

Quan, C.

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]

Quan, C. G.

Rajoub, B. A.

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]

Rastogi, P.

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

Schutte, G.

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).

Su, X. Y.

Tang, Y.

Tao, T.

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]

Tay, C. J.

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]

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]

Vincze, M.

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).

von Kopylow, C.

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

Zheng, P.

Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.

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).

J. Opt. A, Pure Appl. Opt.

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]

J. Opt. Soc. Am. A

Opt. Express

Opt. Laser Technol.

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]

Opt. Lasers Eng.

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]

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

Opt. Lett.

Optical Metrology in Production Engineering, Proc. SPIE

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]

Proc. SPIE

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Schematic setup of the PMD.

Fig. 2
Fig. 2

Geometry of the PMD. (a) Normal view. (b) Normal view. (c) Side view.

Fig. 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.

Fig. 4
Fig. 4

The extracted nonlinear carrier phase components by simulations and experiments. (a) Nonlinear carrier phase component in x direction by computer simulation. (b) Experimental result of nonlinear carrier phase component in x direction. (c) Nonlinear carrier phase component in y direction by computer simulation. (d) Experimental result of nonlinear carrier phase component in y direction.

Fig. 5
Fig. 5

Phase uncertainty of series-expansion method and reference subtraction method

Fig. 6
Fig. 6

3D reconstruction of a spherical convex mirror. (a) One frame of deformed fringe pattern in x direction. (b) Slope-related phase in y direction extracted by plane-fitting technique. (c) Slope-related phase in y direction extracted by reference subtraction technique. (d) Reconstructed 3D shape by reference subtraction technique. (e) One line of the best fitted spherical surface by least squares method.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

I(x,y)=A( x,y )+B( x,y )cos[ 2π f 0 ( x,y )x+φ( x,y ) ].
z=(x x 0 )tanθ+ z 0 .
0=( x i X f )tan β x + Z f .
x = X f tan β x Z f z 0 + x 0 tanθ tanθ+tan β x .
φ{ β x }=2π x x 0 T 0 cosθ = 2π f 0 cosθ ( X f tan β x Z f z 0 + x 0 tanθ tanθ+tan β x x 0 ).
tan β x = X 0 X f + x c cosγ x c sinγ+ Z 0 Z f .
φ{ x c }= 2π f 0 cosθ { ( x c sinγ Z 0 + Z f )[ ( X f x 0 )tanθ+ Z f + z 0 ] x c ( cosγtanθsinγ )+( Z 0 Z f )tanθ+ X 0 X f + X f x 0 }.
l Lf = f 2 + x c 2 f x c tanγ .
l+ l = Z f + z sin β x =l[ ( X f x 0 )tan β x Z f z 0 tanθ+tan β x tanθ Z f + z 0 + Z f Z f ].
d i =lsin( β x +γ )+f.
d i + d i =( l+ l )sin( β x +γ )+f =[ ( X f x 0 )tan β x Z f z 0 tanθ+tan β x tanθ Z f + z 0 + Z f Z f ] ( Lf )f f x c tanγ +f.
sin( β x +γ )= f f 2 + x c 2 .
Y = Y 0 tan β y +f( d i + d i ) tan β y = Y 0 +cot β y [ f( d i + d i ) ].
φ{ β x , β y }=2π Y Y 0 T 0 =2π f 0 cot β y [ f( d i + d i ) ].
φ{ x c , y c }= 2π f 0 Z f ( fL ) y c f x c tanγ { { ( X f x 0 )+ [ ( X f x 0 )tanθ+ Z f + z 0 ]( x c sinγ Z 0 + Z f ) ( cosγsinγtanθ ) x c +( Z 0 Z f )tanθ+ X 0 X f }tanθ+ Z 0 + Z f }.
R=[ cosϕ sinϕcosρ sinϕsinρ sinϕ cosϕcosρ cosϕsinρ 0 sinρ cosρ ]
t=[ fsinϕsinρ fcosϕsinρ f( 1cosρ ) ]
[ X Y Z ]=R[ X Y Z ]+t
X =Xcosϕ+Ysinϕcosρ+( Zf )sinϕsinρ Y =Xsinϕ+Ycosϕcosρ+( Zf )cosϕsinρ Z =Ysinρ+( Zf )cosρ+f

Metrics