Abstract

The nonlinear intensity response of a digital fringe projection profilometry (FPP) system causes the captured fringe patterns to be nonsinusoidal waveforms and leads to an additional phase measurement error for commonly used three- and four-step phase-shifting algorithms. We perform theoretical analysis of the phase error owing to the nonsinusoidal waveforms. Based on the derived theoretical model, a novel and simple iterative phase compensation algorithm is proposed to compensate the nonsinusoidal phase error. Experiments show that the proposed algorithm can be used for effective phase error compensation in practical phase-shifting FPP.

© 2009 Optical Society of America

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References

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  1. Z. Y. Wang, H. Du, and B. Han, Opt. Express 14, 12122 (2006).
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2007 (1)

2006 (1)

2004 (1)

2003 (1)

P. S. Huang, C. Zhang, and F.-P. Chiang, Opt. Eng. (Bellingham) 42, 163 (2003).
[CrossRef]

1996 (1)

1994 (2)

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 31 (1994).
[CrossRef]

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 3 (1994).
[CrossRef]

Asundi, A.

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 31 (1994).
[CrossRef]

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 3 (1994).
[CrossRef]

Baker, M. J.

M. J. Baker, J. Xi, and J. F. Chicharo, presented at the 4th IEEE International Symposium on Electronic Design, Test and Applications, Hong Kong, 23-25, January 2008.

Chan, C. S.

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 3 (1994).
[CrossRef]

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 31 (1994).
[CrossRef]

Chen, M.

Chiang, F.-P.

P. S. Huang, C. Zhang, and F.-P. Chiang, Opt. Eng. (Bellingham) 42, 163 (2003).
[CrossRef]

Chicharo, J. F.

M. J. Baker, J. Xi, and J. F. Chicharo, presented at the 4th IEEE International Symposium on Electronic Design, Test and Applications, Hong Kong, 23-25, January 2008.

Du, H.

Guo, H.

Han, B.

He, H.

Huang, P. S.

P. S. Huang, C. Zhang, and F.-P. Chiang, Opt. Eng. (Bellingham) 42, 163 (2003).
[CrossRef]

Surrel, Y.

Wang, Z. Y.

Xi, J.

M. J. Baker, J. Xi, and J. F. Chicharo, presented at the 4th IEEE International Symposium on Electronic Design, Test and Applications, Hong Kong, 23-25, January 2008.

Yau, S.

Zhang, C.

P. S. Huang, C. Zhang, and F.-P. Chiang, Opt. Eng. (Bellingham) 42, 163 (2003).
[CrossRef]

Zhang, S.

Appl. Opt. (3)

Opt. Eng. (Bellingham) (1)

P. S. Huang, C. Zhang, and F.-P. Chiang, Opt. Eng. (Bellingham) 42, 163 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (2)

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 31 (1994).
[CrossRef]

A. Asundi and C. S. Chan, Opt. Lasers Eng. 21, 3 (1994).
[CrossRef]

Other (1)

M. J. Baker, J. Xi, and J. F. Chicharo, presented at the 4th IEEE International Symposium on Electronic Design, Test and Applications, Hong Kong, 23-25, January 2008.

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Figures (5)

Fig. 1
Fig. 1

Intensity of the actual fringe images is expressed as a function of the ideal sinusoidal fringes.

Fig. 2
Fig. 2

Fringe pattern projected onto (a) reference plane, (b) test plaster model, and (c) the frequency spectrum of the 512th column of (a).

Fig. 3
Fig. 3

(a) Phase error owing to nonsinusoidal waveforms. (b) Phase error for 101 column points. The thin curve shows the average phase error of the 101 column points.

Fig. 4
Fig. 4

(a) Compensated phase error. (b) Compensated phase error for 101 column points. The thin curve shows the average phase error of the 101 column points.

Fig. 5
Fig. 5

Phase distribution of a sculpture (a) before and (b) after compensation (unit: radians).

Equations (12)

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I i c ( x , y ) = f [ I i ( x , y ) ] ,
I i ( x , y ) = I b ( x , y ) + I m ( x , y ) cos [ ϕ ( x , y ) + δ i ] ,
ϕ ( x , y ) = arctan [ i = 0 N 1 I i c ( x , y ) sin ( δ i ) i = 0 N 1 I i c ( x , y ) cos ( δ i ) ] .
I i c ( x , y ) = f [ I i ( x , y ) ] a 0 + k = 1 5 a k cos [ k ( ϕ ( x , y ) + δ i ) ] ,
ϕ ( x , y ) = arctan [ i = 0 N 1 { a 0 + k = 1 5 a k cos [ k ( ϕ ( x , y ) + δ i ) ] } sin ( δ i ) i = 0 N 1 { a 0 + k = 1 5 a k cos [ k ( ϕ ( x , y ) + δ i ) ] } cos ( δ i ) ] .
ϕ ( x , y ) = arctan [ a 1 sin [ ϕ ( x , y ) ] a 2 sin [ 2 ϕ ( x , y ) ] + a 4 sin [ 4 ϕ ( x , y ) ] a 5 sin [ 5 ϕ ( x , y ) ] a 1 cos [ ϕ ( x , y ) ] + a 2 cos [ 2 ϕ ( x , y ) ] + a 4 cos [ 4 ϕ ( x , y ) ] + a 5 cos [ 5 ϕ ( x , y ) ] ] .
ϕ ( x , y ) = ϕ ( x , y ) + Δ ϕ ( x , y ) .
Δ ϕ ( x , y ) = arctan [ ( a 2 a 4 ) sin [ 3 ϕ ( x , y ) ] a 5 sin [ 6 ϕ ( x , y ) ] a 1 + ( a 2 + a 4 ) cos [ 3 ϕ ( x , y ) ] + a 5 cos [ 6 ϕ ( x , y ) ] ] c 1 sin [ 3 ϕ ( x , y ) ] c 2 sin [ 6 ϕ ( x , y ) ] ,
ϕ ( x , y ) = arctan [ a 1 sin [ ϕ ( x , y ) ] a 3 sin [ 3 ϕ ( x , y ) ] + a 5 sin [ 5 ϕ ( x , y ) ] a 1 cos [ ϕ ( x , y ) ] + a 3 cos [ 3 ϕ ( x , y ) ] + a 5 cos [ 5 ϕ ( x , y ) ] ] .
Δ ϕ ( x , y ) = arctan [ ( a 3 a 5 ) sin [ 4 ϕ ( x , y ) ] a 1 + ( a 3 + a 5 ) cos [ 4 ϕ ( x , y ) ] ] c sin [ 4 ϕ ( x , y ) ] ,
Δ ϕ ( x , y ) = arctan [ a 4 sin [ 5 ϕ ( x , y ) ] a 1 + a 4 cos [ 5 ϕ ( x , y ) ] ] c sin [ 5 ϕ ( x , y ) ] ,
ϕ k + 1 ( x , y ) = ϕ ( x , y ) + c sin [ 4 ϕ k ( x , y ) ] ,

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