Abstract

We propose a new approach to demodulate a single fringe pattern with closed fringes by using Local Adaptable Quadrature Filters (LAQF). Quadrature filters have been widely used to demodulate complete image interferograms with carrier frequency. However, in this paper, we propose the use of quadrature filters locally, assuming that the phase is locally quasimonochromatic, since quadrature filters are not capable to demodulate image interferograms with closed fringes. The idea, in this paper, is to demodulate the fringe pattern with closed fringes sequentially, using a fringe following scanning strategy. In particular we use linear robust quadrature filters to obtain a fast and robust demodulation method for single fringe pattern images with closed fringes. The proposed LAQF method does not require a previous fringe pattern normalization. Some tests with experimental interferograms are shown to see the performance of the method along with comparisons to its closest competitor, which is the Regularized Phase Tracker (RPT), and we will see that this method is tolerant to higher levels of noise.

© 2007 Optical Society of America

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References

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    [CrossRef]
  6. M. Servin and J. L. Marroquin and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A  18, 689-695 (2001).
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  7. J. A. Quiroga, J. A G’omez-Pedrero, and A. Garc’ıa-Botella, "Algorithm for fringe pattern normalization," Opt. Commun.  19, 743-51 (2001).
    [CrossRef]
  8. J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt. Commun.  22, 4221- 227 (2003).
    [CrossRef]
  9. J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett.  30, 318-320 (2005).
    [CrossRef]
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  12. R. Legarda-S’aenz and W. Osten and W. J¨uptner, "Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns," Appl. Opt.  41, 5519-5526 (2002).Q1
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  13. M. Rivera, "Robust phase demodulation of interferograms with open or closed fringes," J. Opt. Soc. Am. A  22, 1170-1175 (2005).
    [CrossRef]
  14. R. Legarda-Saenz and M. Rivera, "Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns," J. Opt. Soc. Am. A  23, 2724-2731 (2006).
    [CrossRef]
  15. J. L. Marroquin and J. E. Figueroa and M. Servin, "Robust quadrature filters," J. Opt. Soc. Am. A 14, 779-791 (1997).
    [CrossRef]
  16. Jorge Nocedal and Stephen J. Wright, Numerical Optimization. Springer (1999).

2006 (1)

2005 (2)

M. Rivera, "Robust phase demodulation of interferograms with open or closed fringes," J. Opt. Soc. Am. A  22, 1170-1175 (2005).
[CrossRef]

J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett.  30, 318-320 (2005).
[CrossRef]

2003 (2)

2002 (1)

2001 (4)

1997 (1)

1996 (1)

1986 (1)

1982 (1)

Bone, D. J.

Cuevas, F. J.

Figueroa, J. E.

Guerrero, J. A.

J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett.  30, 318-320 (2005).
[CrossRef]

Ina, H.

Kobayashi, S.

Kreis, T.

Larkin, K. G.

Legarda-Saenz, R.

Marroquin, J. L.

Oldfield, M. A.

Quiroga, J. A.

M. Servin, J. A. Quiroga, and J. L. Marroquin, "General n-dimensional quadrature transform and its application to interferogram demodulation," J. Opt. Soc. Am. A  20, 925-934 (2003).
[CrossRef]

J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt. Commun.  22, 4221- 227 (2003).
[CrossRef]

J. A. Quiroga, J. A G’omez-Pedrero, and A. Garc’ıa-Botella, "Algorithm for fringe pattern normalization," Opt. Commun.  19, 743-51 (2001).
[CrossRef]

Rivera, M.

Servin, M.

Strobel, B.

Takeda, M.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Commun. (2)

J. A. Quiroga, J. A G’omez-Pedrero, and A. Garc’ıa-Botella, "Algorithm for fringe pattern normalization," Opt. Commun.  19, 743-51 (2001).
[CrossRef]

J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt. Commun.  22, 4221- 227 (2003).
[CrossRef]

Opt. Lett. (1)

J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett.  30, 318-320 (2005).
[CrossRef]

Other (2)

E. O. Brigham, The fast fourier tranform. (Prentice-Hall, 1974).

Jorge Nocedal and Stephen J. Wright, Numerical Optimization. Springer (1999).

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Figures (6)

Fig. 1.
Fig. 1.

In this figure, we graphically illustrate how the phase in region Γ around site (32,32) is obtained. The left image is the given interferogram where we mark the neighborhood in red. The intermediate images are the real part and imaginary part of the local quadrature filter, or LAQF, obtained after minimizing Eq. (10). Finally, the right image is the local phase obtained from the RQF.

Fig. 2.
Fig. 2.

In this figure: the variance column shows the variance of the Gaussian noise added to the ground truth phase; the fringe pattern column shows the generated fringe pattern; the RPT column shows the obtained phase using the RPT method and the LAQF column the phase obtained using the LAQF.

Fig. 3.
Fig. 3.

Ground truth phase used to generate the fringe patterns shown in Fig. 2.

Fig. 4.
Fig. 4.

(a) image interferogram with closed fringes obtained by means of a moireé technique. (b) expected obtained phase using LAQF. The image interferogram dimensions are 488 × 500.

Fig. 5.
Fig. 5.

(a) image interferogram with very low frequency zones. The image interferogram was generated with a moiré technique. (b) phase obtained using the LAQF.

Fig. 6.
Fig. 6.

(a) is the interferogram to demodulate, and (b) its demodulated phase using the LAQF. In this example, we see that the fringes exist just in a zone of the image.

Equations (19)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ] ,
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) + ωx ] ,
U [ ϕ ̂ ( x , y ) , u ( x , y ) , v ( x , y ) ] = ( η , ξ ) Γ { [ cos p ( η , ξ ) I ( η , ξ ) ] 2 + λ [ ϕ ̂ ( η , ξ ) p ( η , ξ ) 2 ] } ,
I ( x , y ) = cos [ ϕ ( x , y ) ]
g σ ( x , y ) = exp [ ( x 2 + y 2 ) σ 2 ] ,
I ( x , y ) = [ I I * g σ H ( x , y ) ] * g σ L ( x , y ) ,
U [ ϕ ̂ , u , v , b ̂ , b x , b y ] = ( η , ξ ) Γ { [ β η ξ cos p η ξ I η ξ ] 2 + λ [ ϕ ̂ η ξ p η ξ ] 2 + μ [ b ̂ η ξ β η ξ ] 2 } ,
I ( x , y ) = b ( x , y ) cos [ ϕ ( x , y ) ] ,
I 0 η ξ = b η ξ cos [ ϕ x y + u 0 ( x η ) + v 0 ( y ξ ) ] ,
U [ f ] = R Γ [ f , I ] + λ V Γ [ f ] ,
R Γ [ f , I ] = η ξ Γ f x η ξ 2 I x η ξ 2 + f y η ξ 2 I y η ξ 2 ,
V Γ ( f ) = ( η , ξ ) Γ f η ξ f ( η 1 , ξ ) e i u 0 2 + f η ξ 1 e i v 0 2 } ,
ϕ ̂ η ξ = atc tan [ ψ η ξ φ η ξ ] .
u 0 = arctan [ sin [ ϕ ̂ x y ϕ ̂ x + y ) ( x x + ) ] cos [ ϕ ̂ x y ϕ ̂ x + y ) ( x x + ) ] ]
v 0 = arctan [ sin [ ϕ ̂ x y ϕ ̂ x + y ) ( y y + ) ] cos [ ϕ ̂ x y ϕ ̂ ( x , y + ) ) ( y y + ) ] ] ,
x + = { x + 1 If previous site is to the right x 1 If previous site is to the left
y + = { y + 1 If previous site is down y 1 If previous site is up ,
g u 0 , v 0 x y = exp [ ( x 2 + y 2 ) σ 2 ] e i ( u 0 x + v 0 y ) ,
x y = arg max x i y i I ˜ x i y i ,

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