Abstract

The two-dimensional regularized phase-tracking (RPT) technique is one of the most powerful approaches to demodulate a single interferogram with either open or closed fringes. However, it often fails in the cases of complex interferograms and needs well-defined scanning strategies. An improved algorithm based on the RPT is presented in this paper. We use a paraboloid phase model to approximate the phase function and modify the cost functional to search the smoothest phase solutions in the function space C2. With these modifications, the phase tracker preserves the robustness of the RPT while at the same time it is no more sensitive to stationary points and is capable of demodulating complex interferograms with arbitrary scanning schemes. Moreover, the phase reconstructed by the proposed algorithm is normally more accurate than that of the RPT both for noiseless and noisy interferograms under the same conditions. Computer simulations and experimental results are both presented.

© 2010 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693-2703 (1974).
    [CrossRef] [PubMed]
  3. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  4. L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22, 1535-1539 (1983).
    [CrossRef] [PubMed]
  5. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742-1753 (1997).
    [CrossRef]
  6. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540-4548(1997).
    [CrossRef] [PubMed]
  7. J. Villa, I. De la Rosa, G. Miramontes, and J. A. Quiroga, “Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator,” J. Opt. Soc. Am. A 22, 2766-2773 (2005).
    [CrossRef]
  8. J. C. Estrada, M. Servin, and J. L. Marroquin, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express 15, 2288-2298 (2007).
    [CrossRef] [PubMed]
  9. O. S. Dalmau-Cedeno, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single closed-fringe pattern,” J. Opt. Soc. Am. A 25, 1361-1370 (2008).
    [CrossRef]
  10. H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17, 15118-15127 (2009).
    [CrossRef] [PubMed]
  11. Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
    [CrossRef] [PubMed]
  12. H. Y. Yun, C. K. Hong, and S. W. Chang, “Least-squares phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry,” J. Opt. Soc. Am. A 20, 240-247 (2003).
    [CrossRef]
  13. E. Robin, V. Valle, and F. Bremand, “Phase demodulation method from a single fringe pattern based on correlation with a polynomial form,” Appl. Opt. 44, 7261-7269 (2005).
    [CrossRef] [PubMed]
  14. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170-1175 (2005).
    [CrossRef]
  15. 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 18, 689-695 (2001).
    [CrossRef]
  16. J. A. Quiroga, J. A. Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
    [CrossRef]
  17. J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227(2003).
    [CrossRef]
  18. M. Servin, F. J. Cuevas, D. Malacara, and J. L. Marroquin, “Direct ray aberration estimation in Hartmanngrams by use of a regularized phase-tracking system,” Appl. Opt. 38, 2862-2869 (1999).
    [CrossRef]
  19. J. A. Quiroga and A. Gonzalez-Cano, “Separation of isoclinics and isochromatics from photoelastic data with a regularized phase-tracking technique,” Appl. Opt. 39, 2931-2940 (2000).
    [CrossRef]
  20. J. Villa, J. A. Quiroga, and M. Servin, “Improved regularized phase-tracking technique for the processing of squared-grating deflectograms,” Appl. Opt. 39, 502-508 (2000).
    [CrossRef]
  21. J. C. Estrada, M. Servin, J. A. Quiroga, and J. L. Marroquin, “Path independent demodulation method for single image interferograms with closed fringes within the function space C2,” Opt. Express 14, 9687-9698 (2006).
    [CrossRef] [PubMed]

2009 (1)

2008 (1)

2007 (2)

2006 (1)

2005 (3)

2003 (2)

2001 (2)

J. A. Quiroga, J. A. Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
[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 18, 689-695 (2001).
[CrossRef]

2000 (2)

1999 (1)

1997 (2)

1992 (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

1983 (1)

1982 (1)

1974 (1)

Antonio Quiroga, J.

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227(2003).
[CrossRef]

Brangaccio, D. J.

Bremand, F.

Bruning, J. H.

Chang, S. W.

Cuevas, F. J.

Dalmau-Cedeno, O. S.

De la Rosa, I.

Estrada, J. C.

Feng, L.

Gallagher, J. E.

Garcia-Botella, A.

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

Gomez-Pedrero, J. A.

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

Gonzalez-Cano, A.

Herriott, D. R.

Hong, C. K.

Ina, H.

Kemao, Q.

Kobayashi, S.

Legarda-Saenz, R.

Malacara, D.

Marroquin, J. L.

Mertz, L.

Miramontes, G.

Nam, L. T. H.

Quiroga, J. A.

Rivera, M.

Robin, E.

Rodriguez-Vera, R.

Rosenfeld, D. P.

Servin, M.

J. C. Estrada, M. Servin, and J. L. Marroquin, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express 15, 2288-2298 (2007).
[CrossRef] [PubMed]

J. C. Estrada, M. Servin, J. A. Quiroga, and J. L. Marroquin, “Path independent demodulation method for single image interferograms with closed fringes within the function space C2,” Opt. Express 14, 9687-9698 (2006).
[CrossRef] [PubMed]

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227(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 18, 689-695 (2001).
[CrossRef]

J. Villa, J. A. Quiroga, and M. Servin, “Improved regularized phase-tracking technique for the processing of squared-grating deflectograms,” Appl. Opt. 39, 502-508 (2000).
[CrossRef]

M. Servin, F. J. Cuevas, D. Malacara, and J. L. Marroquin, “Direct ray aberration estimation in Hartmanngrams by use of a regularized phase-tracking system,” Appl. Opt. 38, 2862-2869 (1999).
[CrossRef]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540-4548(1997).
[CrossRef] [PubMed]

J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742-1753 (1997).
[CrossRef]

Soon, S. H.

Takeda, M.

Valle, V.

Villa, J.

Wang, H.

White, A. D.

Yun, H. Y.

Appl. Opt. (8)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693-2703 (1974).
[CrossRef] [PubMed]

L. Mertz, “Real-time fringe-pattern analysis,” Appl. Opt. 22, 1535-1539 (1983).
[CrossRef] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540-4548(1997).
[CrossRef] [PubMed]

M. Servin, F. J. Cuevas, D. Malacara, and J. L. Marroquin, “Direct ray aberration estimation in Hartmanngrams by use of a regularized phase-tracking system,” Appl. Opt. 38, 2862-2869 (1999).
[CrossRef]

J. Villa, J. A. Quiroga, and M. Servin, “Improved regularized phase-tracking technique for the processing of squared-grating deflectograms,” Appl. Opt. 39, 502-508 (2000).
[CrossRef]

J. A. Quiroga and A. Gonzalez-Cano, “Separation of isoclinics and isochromatics from photoelastic data with a regularized phase-tracking technique,” Appl. Opt. 39, 2931-2940 (2000).
[CrossRef]

E. Robin, V. Valle, and F. Bremand, “Phase demodulation method from a single fringe pattern based on correlation with a polynomial form,” Appl. Opt. 44, 7261-7269 (2005).
[CrossRef] [PubMed]

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

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

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227(2003).
[CrossRef]

Opt. Express (3)

Other (1)

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

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Figures (9)

Fig. 1
Fig. 1

Linear and quadratic approximation: (a) one-dimensional interferogram (dashed line) and its real phase (solid line), (b) sketch of the fitting situation of the linear (dashed line) and quadratic (short dashed line) approximation in a small region (dashed rectangle) around point B.

Fig. 2
Fig. 2

Three representative path scanners: (a) row by row scanner, (b) stairway scanner, (c) squareway scanner. See the text for details.

Fig. 3
Fig. 3

Demodulation of computer-generated noiseless interferogram by use of FFRPT: (a) original interferogram, (b) obtained erroneous phase. The phase was rewrapped for the purpose of illustration.

Fig. 4
Fig. 4

Demodulation of computer-generated noiseless interferogram by use of the PIRPT: (a) phase profile of row 128, (b)–(c) two snapshots of the demodulation process with a row by row scanner, (d) fully reconstructed phase. The phase was rewrapped for the purpose of illustration.

Fig. 5
Fig. 5

Demodulation of computer-generated noisy interferogram by the PIRPT: (a) original interferogram, (b)–(c) zigzag path of the stairway scanner, (d) fully reconstructed phase. The phase was rewrapped for the purpose of illustration.

Fig. 6
Fig. 6

Demodulation of computer-generated noisy interferogram by use of the PIRPT: (a) original interferogram, (b)–(c) middle processes of the demodulation with a squareway scanner, (d) fully reconstructed phase. The phase was rewrapped for the purpose of illustration.

Fig. 7
Fig. 7

Demodulation of the experimentally obtained interferogram by use of the PIRPT: (a) original noisy interferogram with noise, background, and modulation variation; (b)–(c) two middle processes of the demodulation with a row by row scanner; (d) fully reconstructed phase. The phase was rewrapped for the purpose of illustration.

Fig. 8
Fig. 8

Comparison of the phase reconstruction accuracy of the two methods: (a) original fringe pattern; (b), (c), and (e) real phase and resultant phase by use of the RPT and PIRPT, respectively; (d) and (f) absolute error of the RPT and PIRPT, respectively.

Fig. 9
Fig. 9

Comparison of the phase reconstruction accuracy of the two methods: (a) noisy version of Fig. 8a; (b), (c), and (e) real phase and resultant phase by use of the RPT and PIRPT, respectively; (d) and (f) absolute error of the RPT and PIRPT, respectively.

Equations (16)

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I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ϕ ( x , y ) ] + n ( x , y ) ,
U T = ( x , y ) L U x , y ( ϕ ^ 0 , ω ^ x , ω ^ y ) ,
U x , y ( ϕ ^ 0 , ω ^ x , ω ^ y ) = ( ε , η ) ( N x , y L ) ( { [ I ^ n ( ε , η ) cos [ ϕ ^ ( ε , η , x , y ) ] } 2 + β 1 [ ϕ ^ 0 ( ε , η ) ϕ ^ ( ε , η , x , y ) ] 2 m ( ε , η ) ) ,
I ^ n ( ε , η ) cos [ ϕ ( ε , η ) ] ,
ϕ ^ ( ε , η , x , y ) = ϕ ^ 0 ( x , y ) + [ ω ^ ε ( x , y ) ( ε x ) + ω ^ η ( x , y ) ( η y ) ] ,
ϕ ^ ( ε , η , x , y ) = ϕ ^ 0 ( x , y ) + ϕ ^ 1 ( ε , η , x , y ) + ϕ ^ 2 ( ε , η , x , y ) ,
ϕ ^ 1 ( ε , η , x , y ) = [ ϕ ^ ε ( x , y ) ( ε x ) + ϕ ^ η ( x , y ) ( η y ) ] , ϕ ^ 2 ( ε , η , x , y ) = [ ϕ ^ ε ε ( x , y ) ( ε x ) 2 + 2 ϕ ^ ε η ( x , y ) ( ε x ) ( η y ) + ϕ ^ η η ( x , y ) ( η y ) 2 ] / 2 ,
χ ^ ( x , y ) = [ ϕ ^ 0 ( x , y ) , ϕ ^ ε ( x , y ) , ϕ ^ η ( x , y ) , ϕ ^ ε ε ( x , y ) , ϕ ^ ε η ( x , y ) , ϕ ^ η η ( x , y ) ] T ,
U x , y ( χ ^ ( x , y ) ) = ( ε , η ) ( N x , y L ) ( { I ^ n ( ε , η ) cos [ ϕ ^ ( ε , η , x , y ) ] } 2 + β 1 [ ϕ ^ 0 ( ε , η ) ϕ ^ ( ε , η , x , y ) ] 2 m ( ε , η ) + R ( ε , η , x , y ) ) ,
R ( ε , η , x , y ) = β 2 { [ ϕ ^ ε ε ( ε , η ) ϕ ^ ε ε ( x , y ) ] 2 + [ ϕ ^ ε η ( ε , η ) ϕ ^ ε η ( x , y ) ] 2 + [ ϕ ^ η η ( ε , η ) ϕ ^ η η ( x , y ) ] 2 } m ( ε , η ) ,
ϕ ^ 0 k + 1 ( x , y ) = ϕ ^ 0 k ( x , y ) τ U x , y ( χ ^ ( x , y ) ) ϕ ^ 0 ( x , y ) , ϕ ^ ε k + 1 ( x , y ) = ϕ ^ ε k ( x , y ) τ U x , y ( χ ^ ( x , y ) ) ϕ ^ ε ( x , y ) , ϕ ^ η k + 1 ( x , y ) = ϕ ^ η k ( x , y ) τ U x , y ( χ ^ ( x , y ) ) ϕ ^ η ( x , y ) , ϕ ^ ε ε k + 1 ( x , y ) = ϕ ^ ε ε k ( x , y ) τ U x , y ( χ ^ ( x , y ) ) ϕ ^ ε ε ( x , y ) , ϕ ^ ε η k + 1 ( x , y ) = ϕ ^ ε η k ( x , y ) τ U x , y ( χ ^ ( x , y ) ) ϕ ^ ε η ( x , y ) , ϕ ^ η η k + 1 ( x , y ) = ϕ ^ η η k ( x , y ) τ U x , y ( χ ^ ( x , y ) ) ϕ ^ η η ( x , y ) ,
U x , y ( χ ^ ( x , y ) ) = ( ε , η ) ( N x , y L ) ( { I ^ n ( ε , η ) cos [ ϕ ^ ( ε , η , x , y ) ] } 2 + β 1 [ ϕ ^ 0 ( ε , η ) ϕ ^ ( ε , η , x , y ) ] 2 m ( ε , η ) + P ( χ ^ ( x , y ) ) + R ( ε , η , x , y ) ) ,
P ( χ ^ ( x , y ) ) = { I ^ n ( ε , η ) cos [ ϕ ^ ( ε , η , x , y ) + α ] } 2 ,
I ( x , y ) = cos [ 2 π × ( 21.6 x y / N 2 { 125.3 [ ( x 2 + y 2 ) / N 2 ] 2 57.8 ( x 2 + y 2 ) / N 2 + 6 } + 0.6 ) ] × Mask ,
I ( x , y ) = [ 0.2 + 2 cos ( 2 π { 0.81 ( x 2 + y 2 ) / N 2 2.5 [ cos ( 3.6 π x / N ) + cos ( 3.6 π y / N ) ] + 0.3 × ( Rand 0.5 ) } ) ] × Mask ,
ε PV = max ( | ϕ ( m , n ) ϕ ^ ( m , n ) | ) , ε MSE = 1 M m , n ( | ϕ ( m , n ) ϕ ^ ( m , n ) | ) 2 ,

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