Abstract

We present an unscented Kalman filter to identify the phase step imparted to a piezoelectric transducer in phase shifting interferometry in the presence of Gaussian noise. The advantage of the proposed algorithm lies in its ability to determine the phase step values between π and π rad without any prior calibration of the piezoelectric device. The algorithm is tested rigorously by using the simulated data in the presence of Gaussian distributed noise. Experimental validations are also performed in a holographic interferometry optical setup to verify the proposed approach. Once the phase step is identified, the interference phase can be estimated by using the least-squares fitting approach.

© 2009 Optical Society of America

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References

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2007

2004

2000

S. J. Julier, J. K. Uhlmann, and H. F. Durrant-Whyte, IEEE Trans. Autom. Control 5, 477 (2000).
[CrossRef]

1995

1984

J. E. Grievenkamp, Opt. Eng. 23, 350 (1984).

1982

1966

P. Carré, Metrologia 2, 13 (1966).
[CrossRef]

Cai, L. Z.

Carré, P.

P. Carré, Metrologia 2, 13 (1966).
[CrossRef]

Creath, K.

K. Creath, in Holographic Interferometry, P.K.Rastogi, ed., Springer Series in Optical Sciences, (Springer-Verlag, 1994), Vol. 68, pp. 109-150.

Durrant-Whyte, H. F.

S. J. Julier, J. K. Uhlmann, and H. F. Durrant-Whyte, IEEE Trans. Autom. Control 5, 477 (2000).
[CrossRef]

Grievenkamp, J. E.

J. E. Grievenkamp, Opt. Eng. 23, 350 (1984).

Han, B.

Julier, S. J.

S. J. Julier, J. K. Uhlmann, and H. F. Durrant-Whyte, IEEE Trans. Autom. Control 5, 477 (2000).
[CrossRef]

Langoju, R.

Liu, Q.

Morgan, C. J.

Patil, A.

Rastogi, P.

Rathjen, C.

Uhlmann, J. K.

S. J. Julier, J. K. Uhlmann, and H. F. Durrant-Whyte, IEEE Trans. Autom. Control 5, 477 (2000).
[CrossRef]

Wang, Z.

Yang, X. L.

Appl. Opt.

IEEE Trans. Autom. Control

S. J. Julier, J. K. Uhlmann, and H. F. Durrant-Whyte, IEEE Trans. Autom. Control 5, 477 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Metrologia

P. Carré, Metrologia 2, 13 (1966).
[CrossRef]

Opt. Eng.

J. E. Grievenkamp, Opt. Eng. 23, 350 (1984).

Opt. Lett.

Other

K. Creath, in Holographic Interferometry, P.K.Rastogi, ed., Springer Series in Optical Sciences, (Springer-Verlag, 1994), Vol. 68, pp. 109-150.

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Figures (3)

Fig. 1
Fig. 1

Plot of percentage error in the estimation of phase step for α equal to 30° (○), 45° (◇), 60° (+), 90° (▽) versus the intensity covariance matrix R n for different values of the process covariance matrix Q n , with N = 10 data frames and 1000 randomly selected pixels.

Fig. 2
Fig. 2

Plot of calculated and actual phase step α (in degrees) versus SNR (in decibels) for N = 10 data frames and 1000 randomly selected pixels, for α = 30 ° , 45 ° , 60 ° , 90 ° .

Fig. 3
Fig. 3

Plot of (a) fringe obtained in a holographic interferometry setup; (b) histogram for the phase step α = 30 ° , with 1000 randomly selected pixels and N = 10 data frames; (c) wrapped phase; and (d) unwrapped phase. Note that the axes in (a), (c), and (d) are in pixels.

Equations (12)

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I n ( x , y ) = I dc ( x , y ) { 1 + γ ( x , y ) cos [ θ ( x , y ) + n α ( x , y ) ] } + ν n ( x , y ) , for n = 1 , 2 , , N ,
I n ( x , y ) = h ( ξ n ( x , y ) ) ,
I n ( x , y ) = I dc ( x , y ) [ 1 + γ ( x , y ) cos ( ξ n ( x , y ) ) ] + ν n ( x , y ) ,
for n = 1 , 2 , , N .
ξ n + 1 ( x , y ) = f ( ξ n ( x , y ) ) = ξ n ( x , y ) + α ( x , y ) + w n ( x , y ) ,
I n ( x , y ) = I dc ( x , y ) [ 1 + γ ( x , y ) cos ( ξ n ( x , y ) ) ] + ν n ( x , y ) , for n = 1 , 2 , , N ,
X n = [ ξ ̂ n ( ξ ̂ n + ϵ P n ) ( ξ ̂ n ϵ P n ) ] ,
ξ ̂ n = k = 0 2 L W k ( m ) X k , n 1 * ,
P I ̂ n I ̂ n = k = 0 2 L W k ( c ) [ I k , n 1 I ̂ n ] [ I k , n 1 I ̂ n ] T + R n ,
P ξ ̂ n I ̂ n = k = 0 2 L W k ( c ) [ X k , n 1 ξ ̂ n ] [ I k , n 1 I ̂ n ] T ,
ξ ̂ n = ξ ̂ n + P ξ ̂ n I ̂ n P I ̂ n I ̂ n 1 ( I n I ̂ n ) ,
P n = P n P ξ ̂ n I ̂ n P I ̂ n I ̂ n 1 P ξ ̂ n I ̂ n T ,

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