Abstract

Recent studies have demonstrated that the phase recovery from a single fringe pattern with closed fringes can be properly performed if the modulo 2π fringe orientation is estimated. For example, the fringe pattern in quadrature can be efficiently obtained in terms of the orientational phase spatial operator using fast Fourier transformations and a spiral phase spectral operator in the Fourier space. The computation of the modulo 2π fringe orientation, however, is by far the most difficult task in the global process of phase recovery. For this reason we propose the demodulation of fringe patterns with closed fringes through the computation of the modulo 2π fringe orientation using an orientational vector-field-regularized estimator. As we will show, the phase recovery from a single pattern can be performed in an efficient manner using this estimator, provided that it requires one to solve locally in the fringe pattern a simple linear system to optimize a regularized cost function. We present simulated and real experiments applying the proposed methodology.

© 2005 Optical Society of America

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References

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2004

2003

2002

2001

2000

1998

1996

1982

Bone, D. J.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).

Cuevas, F.

García-Botella, A.

J. A. Quiroga, J. A. Gómez-Pedrero, A. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Gómez-Pedrero, J. A.

J. A. Quiroga, J. A. Gómez-Pedrero, A. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Ina, H.

Jüptner, W.

Kobayashi, S.

Larkin, K. G.

Legarda-Senz, R.

Malacara, D.

D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Malacara, Z.

D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Marroquín, J. L.

Oldfield, M. A.

Osten, W.

Quiroga, J. A.

Rodríguez-Vera, R.

Servín, M.

Ströbel, B.

Takeda, M.

Villa, J.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

J. A. Quiroga, J. A. Gómez-Pedrero, A. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43–51 (2001).
[CrossRef]

Other

D. Malacara, M. Servín, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, 2000).

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Figures (6)

Fig. 1
Fig. 1

(a) Synthetic fringe pattern corrupted with 5.2 dB of Gaussian additive noise. (b) Gray-level codification of the theoretical phase (shown wrapped). Black, π rad; white, π rad. (c) Quality map I ( r ) for the fringe orientation estimation. Brighter zones represent the first sites to be estimated.

Fig. 2
Fig. 2

Sequence of the fringe orientation estimation from the synthetic fringe pattern shown in Fig. 1a using the regularized estimator and Ströbel’s pixel-queuing algorithm. Black, π rad; white, π rad.

Fig. 3
Fig. 3

(a) Estimated fringe pattern in quadrature from the synthetic fringe pattern shown in Fig. 1a. (b) Gray-level codification of the computed phase using the SPQT. Black, π rad; white, π rad.

Fig. 4
Fig. 4

(a) Real fringe pattern from holographic interferometry. Gray-level codification of the fields (b) ψ x ( r ) (black, 0; white, 1) and (c) ψ y ( r ) (black, 1 ; white, 1).

Fig. 5
Fig. 5

Estimations from the fringe pattern shown in Fig. 4a. Components of the orientational vector field p ( r ) estimated with the proposed method. (a) x component (black, 0; white, 1). (b) y component (black, 1 ; white, 1). (c) Modulo 2 π fringe orientation. (d) Gray-level codification of the phase using the SPQT. Black, π rad; white, π rad.

Fig. 6
Fig. 6

(a) Real interferometric fringe pattern. (b) Estimated modulo 2 π fringe orientation. (c) Gray-level codification of the computed phase using the SPQT. Black, π rad; white, π rad.

Equations (18)

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I ( r ) = a ( r ) + b ( r ) cos [ ϕ ( r ) ] , r = ( x , y ) L ,
ϕ a ( r ) = exp ( x 2 + y 2 ) ,
ϕ b ( r ) = exp ( x 2 + y 2 ) ,
ϕ c ( r ) = { exp ( x 2 + y 2 ) if x > 0 , exp ( x 2 + y 2 ) if x 0 . } ,
ϕ d ( r ) = { exp ( x 2 + y 2 ) if x > 0 , exp ( x 2 + y 2 ) if x 0 . } .
Q n { b ( r ) cos [ ϕ ( r ) ] } = n ϕ I ( r ) I ϕ ( r ) ,
n ϕ = ϕ ( r ) ϕ ( r ) = cos [ θ 2 π ( r ) ] i + sin [ θ 2 π ( r ) ] j .
f ( x ) = cos ϕ ( x ) + i sin ϕ ( x ) = exp [ i ϕ ( x ) ] ,
sin ϕ ( x ) = H { cos ϕ ( x ) } = 1 π x cos ϕ ( x ) for ϕ ( x ) > 0 ,
sin ϕ ( x ) = F 1 { i sign ( f x ) F { cos ϕ ( x ) } } ,
i exp [ i θ ( r ) ] b ( r ) sin [ ϕ ( r ) ] F 1 { S ( f ) F { I ( r ) } } ,
S ( f ) = f x + i f y f x 2 + f y 2 .
ϕ ( r ) = ϕ ( r ) [ cos θ ( r ) , sin θ ( r ) ] .
θ π ( r ) = tan 1 [ I ( r ) y I ( r ) x ] , π 2 θ π π 2 .
Ψ ( r ) n ( r ) = 0 , ( r L ) .
U r ( n ) = r ̃ ( Γ L ) { [ Ψ ( r ̃ ) n ( r ) ] 2 + μ [ n x ( r ) n x ( r ̃ ) ] 2 s ( r ̃ ) + μ [ n y ( r ) n y ( r ̃ ) ] 2 s ( r ̃ ) } ,
n ( r 0 ) = [ ψ y ( r 0 ) , ψ x ( r 0 ) ] or n ( r 0 ) = [ ψ y ( r 0 ) , ψ x ( r 0 ) ] ,
[ r ̃ Γ { ψ x ( r ̃ ) 2 + μ s ( r ̃ ) } r ̃ Γ { ψ y ( r ̃ ) ψ x ( r ̃ ) } r ̃ Γ { ψ x ( r ̃ ) ψ y ( r ̃ ) } r ̃ Γ { ψ y ( r ̃ ) 2 + μ s ( r ̃ ) } ] ( n x ( r ) n y ( r ) ) = ( μ r ̃ Γ { n x ( r ̃ ) s ( r ̃ ) } μ r ̃ Γ { n y ( r ̃ ) s ( r ̃ ) } ) .

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