Abstract

In fast phase-measuring profilometry, phase error caused by gamma distortion is the dominant error source. Previous phase-error compensation or gamma correction methods require the projector to be focused for best performance. However, in practice, as digital projectors are built with large apertures, they cannot project ideal focused fringe images. In this Letter, a thorough theoretical model of the gamma-distorted fringe image is derived from an optical perspective, and a highly accurate and easy to implement gamma correction method is presented to reduce the obstinate phase error. With the proposed method, high measuring accuracy can be achieved with the conventional three-step phase-shifting algorithm. The validity of the technique is verified by experiments.

© 2011 Optical Society of America

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References

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2010 (6)

2009 (2)

2007 (1)

2000 (1)

C. Coggrave and J. Huntley, Opt. Eng. 39, 91 (2000).
[CrossRef]

1992 (1)

X. Su, W. Zhou, and G. Bally, Opt. Commun. 94, 561 (1992).
[CrossRef]

Asundi, A.

Bally, G.

X. Su, W. Zhou, and G. Bally, Opt. Commun. 94, 561 (1992).
[CrossRef]

Barnes, J.

Z. Wang, D. Nguyen, and J. Barnes, Opt. Lasers Eng. 48, 218 (2010).
[CrossRef]

Brown, M.

M. Brown, P. Song, and T. Cham, in Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), p. 1956.
[PubMed]

Cham, T.

M. Brown, P. Song, and T. Cham, in Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), p. 1956.
[PubMed]

Coggrave, C.

C. Coggrave and J. Huntley, Opt. Eng. 39, 91 (2000).
[CrossRef]

Gorthi, S.

S. Gorthi and P. Rastogi, Opt. Lasers Eng. 48, 133 (2010).
[CrossRef]

Hao, Q.

Hassebrook, L.

Hoang, T.

Huang, L.

Huntley, J.

C. Coggrave and J. Huntley, Opt. Eng. 39, 91 (2000).
[CrossRef]

Lau, D.

Lei, S.

Liu, K.

Nguyen, D.

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, Opt. Lett. 35, 1992 (2010).
[CrossRef] [PubMed]

Z. Wang, D. Nguyen, and J. Barnes, Opt. Lasers Eng. 48, 218 (2010).
[CrossRef]

Pan, B.

Qian, K.

Rastogi, P.

S. Gorthi and P. Rastogi, Opt. Lasers Eng. 48, 133 (2010).
[CrossRef]

Song, P.

M. Brown, P. Song, and T. Cham, in Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), p. 1956.
[PubMed]

Su, X.

X. Su, W. Zhou, and G. Bally, Opt. Commun. 94, 561 (1992).
[CrossRef]

Wang, Y.

Wang, Z.

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, Opt. Lett. 35, 1992 (2010).
[CrossRef] [PubMed]

Z. Wang, D. Nguyen, and J. Barnes, Opt. Lasers Eng. 48, 218 (2010).
[CrossRef]

Yau, S. T.

Zhang, S.

Zhou, W.

X. Su, W. Zhou, and G. Bally, Opt. Commun. 94, 561 (1992).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

X. Su, W. Zhou, and G. Bally, Opt. Commun. 94, 561 (1992).
[CrossRef]

Opt. Eng. (1)

C. Coggrave and J. Huntley, Opt. Eng. 39, 91 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (3)

S. Gorthi and P. Rastogi, Opt. Lasers Eng. 48, 133 (2010).
[CrossRef]

Z. Wang, D. Nguyen, and J. Barnes, Opt. Lasers Eng. 48, 218 (2010).
[CrossRef]

S. Zhang, Opt. Lasers Eng. 48, 149 (2010).
[CrossRef]

Opt. Lett. (3)

Other (1)

M. Brown, P. Song, and T. Cham, in Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2006), p. 1956.
[PubMed]

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Figures (2)

Fig. 1
Fig. 1

Experiment results: (a) gamma calibration results and (b) phase errors without and with gamma correction.

Fig. 2
Fig. 2

Measurement result of a complex plaster model: (a) captured fringe image, measured 3D surface (b) without and (c) with our gamma correction, and (d) partial enlarged detail.

Equations (12)

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I n P ( x , y ) = { A P ( x , y ) + B P ( x , y ) cos [ ϕ ( x , y ) + δ n ] } γ = A ( x , y ) + k = 1 B k ( x , y ) cos { k [ ϕ ( x , y ) + δ n ] } ,
B k + 1 / B k = ( γ k ) / ( γ + k + 1 ) .
G ( x , y ) = 1 2 π σ 2 exp ( x 2 + y 2 2 σ 2 ) ,
I n C = α I n P * G ( x , y ) ,
T ( f ) = G ( x ) exp ( i 2 π f x ) d x = exp ( 2 π 2 σ 2 f 2 ) .
I n C = α I n P T ( f ) = A ˜ + k = 1 B ˜ k cos [ k ( ϕ + δ n ) ] ,
ϕ = arctan [ B ˜ 1 sin ( ϕ ) B ˜ 2 sin ( 2 ϕ ) B ˜ 1 cos ( ϕ ) + B ˜ 2 cos ( 2 ϕ ) ] .
Δ ϕ = arctan { [ B ˜ 2 B ˜ 1 sin ( 3 ϕ ) ] / [ 1 + B ˜ 2 B ˜ 1 cos ( 3 ϕ ) ] } .
Δ ϕ max = arctan { [ ( B ˜ 2 / B ˜ 1 ) 2 1 ( B ˜ 2 / B ˜ 1 ) 2 ] 1 / 2 } = arctan [ ( p 2 1 p 2 ) 1 / 2 ] ,
p = B ˜ 2 B ˜ 1 = T ( 2 f 0 ) T ( f 0 ) B 2 B 1 = exp ( 6 π 2 σ 2 f 0 2 ) γ 1 γ + 2 .
B ˜ k = 1 L { [ l = 1 L I l C sin ( k 2 π l L ) ] 2 + [ l = 1 L I l C cos ( k 2 π l L ) ] 2 } 1 / 2 ,
B ˜ 2 B ˜ 1 = T ( 2 f 0 ) T ( f 0 ) B 2 B 1 = exp ( 6 π 2 σ 2 f 0 2 ) γ / γ 1 γ / γ + 2 .

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