Abstract

In the last few years, works have been published about demodulating Single Fringe Pattern Images (SFPI) with closed fringes. The two best known methods are the regularized phase tracker (RPT), and the two-dimensional Hilbert Transform method (2D-HT). In both cases, the demodulation success depends strongly on the path followed to obtain the expected estimation. Therefore, both RPT and 2D-HT are path dependent methods. In this paper, we show a novel method to demodulate SFPI with closed fringes which follow arbitrary sequential paths. Through the work presented here, we introduce a new technique to demodulate SFPI with estimations within the function space C 2; in other words, estimations where the phase curvature is continuous. The technique developed here, uses a frequency estimator which searches into a frequency discrete set. It uses a second order potential regularizer to force the demodulation system to look into the function space C 2. The obtained estimator is a fast demodulator system for normalized SFPI with closed fringes. Some tests to demodulate SFPI with closed fringes using this technique following arbitrary paths are presented. The results are compared to those from RPT technique. Finally, an experimental normalized interferogram is demodulated with the herein suggested technique.

© 2006 Optical Society of America

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References

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  2. J. N. Butters and J. A. Leendertz, "A double exposure technique for speckle pattern interferometry," J. Phys. E: Sci. Instrum. 4277-279 (1971).
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  3. J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt Commun. 224221- 227 (2003).
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  4. J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett. 30318-320 (2005).
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    [CrossRef]
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  13. J. Villa, I. De la Rosa and G. Miramontes, "Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator," J. Opt. Soc. Am. A 222766-2773 (2005).
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  14. M. Servin and R. Rodriguez-Vera, "Two-dimensional phase locked loop demodulation of interferograms," J. Mod. Opt. 402087-2094 (1993).
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2006 (1)

A. Mujeeb, V. U. Nayar and V. R. Ravindran, "Electronic Speckle Pattern Interferometry techniques for nondestructive evaluation: a review," INSIGHT 45275-281 (2006).
[CrossRef]

2005 (3)

2003 (1)

J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt Commun. 224221- 227 (2003).
[CrossRef]

2002 (1)

2001 (3)

1993 (2)

M. Servin and R. Rodriguez-Vera, "Two-dimensional phase locked loop demodulation of interferograms," J. Mod. Opt. 402087-2094 (1993).
[CrossRef]

MuntherA. Gdeisat, David R. Burton and Michael J. Lalor, "Real-Time Fringe Pattern Demodulation with a Second-Order Digital Phase-Locked Loop," Appl. Opt. 395326-5336 (1993).
[CrossRef]

1988 (1)

1982 (1)

1974 (1)

1971 (1)

J. N. Butters and J. A. Leendertz, "A double exposure technique for speckle pattern interferometry," J. Phys. E: Sci. Instrum. 4277-279 (1971).
[CrossRef]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J.

Butters, J. N.

J. N. Butters and J. A. Leendertz, "A double exposure technique for speckle pattern interferometry," J. Phys. E: Sci. Instrum. 4277-279 (1971).
[CrossRef]

Cuevas, F. J.

De la Rosa, I.

Donald, K. G.

Gallagher, 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. 30318-320 (2005).
[CrossRef]

Herriott, D. R.

Ina, H.

Kobayashi, S.

Larkin, K.

Larkin, K. G.

Leendertz, J. A.

J. N. Butters and J. A. Leendertz, "A double exposure technique for speckle pattern interferometry," J. Phys. E: Sci. Instrum. 4277-279 (1971).
[CrossRef]

Marroquin, J. L.

J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett. 30318-320 (2005).
[CrossRef]

M. Servin, J. L. Marroquin and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A 18689-695 (2001).
[CrossRef]

Miramontes, G.

Mujeeb, A.

A. Mujeeb, V. U. Nayar and V. R. Ravindran, "Electronic Speckle Pattern Interferometry techniques for nondestructive evaluation: a review," INSIGHT 45275-281 (2006).
[CrossRef]

Munther,

Nayar, V. U.

A. Mujeeb, V. U. Nayar and V. R. Ravindran, "Electronic Speckle Pattern Interferometry techniques for nondestructive evaluation: a review," INSIGHT 45275-281 (2006).
[CrossRef]

Quiroga, J. A.

Ravindran, V. R.

A. Mujeeb, V. U. Nayar and V. R. Ravindran, "Electronic Speckle Pattern Interferometry techniques for nondestructive evaluation: a review," INSIGHT 45275-281 (2006).
[CrossRef]

Rivera, M.

J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett. 30318-320 (2005).
[CrossRef]

Rodriguez-Vera, R.

M. Servin and R. Rodriguez-Vera, "Two-dimensional phase locked loop demodulation of interferograms," J. Mod. Opt. 402087-2094 (1993).
[CrossRef]

Rosenfel, D. P.

Servin, M.

J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt Commun. 224221- 227 (2003).
[CrossRef]

M. Servin, J. L. Marroquin and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A 18689-695 (2001).
[CrossRef]

M. Servin and R. Rodriguez-Vera, "Two-dimensional phase locked loop demodulation of interferograms," J. Mod. Opt. 402087-2094 (1993).
[CrossRef]

Stetson, K. A.

Takeda, M.

Villa, J.

White, A. D.

Appl. Opt. (2)

INSIGHT (1)

A. Mujeeb, V. U. Nayar and V. R. Ravindran, "Electronic Speckle Pattern Interferometry techniques for nondestructive evaluation: a review," INSIGHT 45275-281 (2006).
[CrossRef]

J. Mod. Opt. (1)

M. Servin and R. Rodriguez-Vera, "Two-dimensional phase locked loop demodulation of interferograms," J. Mod. Opt. 402087-2094 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

J. Phys. E: Sci. Instrum. (1)

J. N. Butters and J. A. Leendertz, "A double exposure technique for speckle pattern interferometry," J. Phys. E: Sci. Instrum. 4277-279 (1971).
[CrossRef]

Opt Commun. (1)

J. A. Quiroga and M. Servin, "Isotropic n-dimensional fringe pattern normalization," Opt Commun. 224221- 227 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

J. A. Guerrero, J. L. Marroquin, and M. Rivera, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett. 30318-320 (2005).
[CrossRef]

Other (1)

Jorge Nocedal and Stephen J. Wright, Numerical Optimization. Springer, 1999.
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Two estimated phases using the RPT from the same SFPI given. Panel labeled Wrong phase shows the obtained phase by following an arbitrary path, while panel labeled Expected phase is obtained using a fringe following path. The SFPI given is in panel labeled SFPI.

Fig. 2.
Fig. 2.

In this figure, (a) is the graph of the cost function (14) and (b) is the graph of the regularizer potential (20). The graph (a) was generated using the parameters I′(x)=0.66 and ϕ ^ (x)=20. We also show the series S 1 and S 2, which corresponds to the minimum values of the cost function (14). In graph (b) we see that exists only one frequency ω(x)∈Ω={S 1,S 2} that minimizes Eq. (20).

Fig. 3.
Fig. 3.

In graph 1) a signal is shown which phase is modeled as ϕ ( x ) = 1 2 a x 2 . In graph 2) we show the obtained phase using the FCT and in graph 4) we show its curvature. In graph 3) we show the obtained phase using the RPT technique and in 5) we show its curvature. As we can see, the phase curvature presented in 4) is a continuous line while the phase curvature in 5) has one abrupt variation.

Fig. 4.
Fig. 4.

Steps for the row by row scanning strategy to demodulate a frame in the image interferogram lattice. Step 1 shows the initial seed with a circle, Steep 2 shows the column demodulation, Step 3 shows the row demodulation and Step 4 the row by row scanning to demodulate each row using the values of the column as an initial seed for each row.

Fig. 5.
Fig. 5.

In this figure we illustrate a path that is followed with square frames. Each site of the path, represted with a dark point, is the center of its frame and each frame is such that intersects the frame of the neighbor site and the neighbor site belong to this intersection.

Fig. 6.
Fig. 6.

(a) is the simulated image interferogram I′(x,y), (b) is the path to follow, (c) is the demodulated phase using the FCT in two dimensions and (d) is the demodulated phase using the RPT.

Fig. 7.
Fig. 7.

A more complicated structure of fringes. (a) and (b) are simulated fringe patterns and (c) and (d) shows its estimated phase with the FCT using a row by row scanning strategy. The phase is shown wrapped for illustration purposes.

Fig. 8.
Fig. 8.

(a) is a real experimental SFPI, (b) is its normalized version, (c) is the obtained phase with the FCT using a row by row scanning strategy and (d) is the wrapped phase showed for illustration purposes.

Equations (28)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ] ,
I ' ( x , y ) = cos [ ϕ ( x , y ) ] .
ϕ ( x , y ) = ϕ 0 ( x , y ) + ω x ,
𝒱 ( I ) = i exp ( i β 2 π ) 𝓕 1 { exp ( i ψ ) 𝓕 { I } } ,
ϕ ̂ ( x , y ) = arctan [ 𝒱 [ I ( x , y ) ] I ( x , y ) ] .
β 2 π ( x , y ) = arctan [ ϕ ( x , y ) y ϕ ( x , y ) x ] .
exp [ i ψ ( u , v ) ] = u + i v u 2 + v 2
U [ n ( x , y ) ] = ( η , ξ ) Γ x , y { I ' ( η , ξ ) · n ( η , ξ ) 2 n ( η , ξ ) n 0 ( η , ξ ) 2 } ,
β ̂ 2 π ( x , y ) = arctan [ n y ( x , y ) n x ( x , y ) ] .
U [ ϕ ̂ ( x , y ) , ω x , ω y ] = ( η , ξ ) Γ x , y { [ cos [ P ( x , y , η , ξ ) ] I ' ( η , ξ ) ] 2
+ λ [ ϕ ( η , ξ ) P ( x , y , η , ξ ) ] 2 } ,
I ' ( x ) = cos [ ϕ ( x ) ] ,
cos [ ϕ ̂ ( x ) + ω ( x ) ( x x + ) ] = I ' ( x + ) ,
x + = { x 1 left x + 1 right
U [ ω ( x ) ] = { cos [ ϕ ̂ ( x ) + ω ( x ) ( x x + ) ] I ' ( x + ) } 2 .
S 1 = ( W [ ϕ ̂ ( x ) ] arccos [ I ' ( x + ) ] ) ( x x + ) ± 2 n π
S 2 = ( W [ ϕ ̂ ( x ) ] + arccos [ I ' ( x + ) ] ) ( x x + ) ± 2 n π
ω 1 = { ω ( x ) if ω ( x ) π ω ( x ) 2 π if ω ( x ) > π ω ( x ) + 2 π if ω ( x ) < π
ω 2 = { ω ( x ) if ω ( x ) π ω ( x ) 2 π if ω ( x ) > π ω ( x ) + 2 π if ω ( x ) < π
V [ ϕ ̂ ( x ) ] = [ C ϕ ̂ ( x ) C ϕ ̂ ( x + 1 ) ] 2
= [ ϕ ̂ ( x 3 ) 2 ϕ ̂ ( x 2 ) + ϕ ̂ ( x )
ϕ ̂ ( x 2 ) + 2 ϕ ̂ ( x ) ϕ ̂ ( x + 1 ) ] 2 ,
V [ ω ( x ) ] = [ ω ( x 2 ) 2 ω ( x 1 ) + ω ( x ) ] 2 .
V [ ω ( x ) ] = [ ω ( x ) 2 ω ( x + 1 ) + ω ( x + 2 ) ] 2 .
ω ̂ ( x ) = min ω ( x ) Ω V [ ω ( x ) ] ,
ϕ ̂ ( x + ) = ϕ ̂ ( x ) + ω ̂ ( x ) ( x x + ) ,
ϕ ̂ ( x + 1 ) = ϕ ̂ ( x ) τ [ I ' ( x ) I ' ( x 1 ) ] sin [ ϕ ( x ) + ω 0 x ] ,
[ ω ̂ ( x 1 ) , ω ̂ ( x + 1 ) ] = min ω ( x 1 ) Ω l , ω ( x + 1 ) Ω r [ ω ( x 1 ) ω ( x + 1 ) ] 2 .

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