Abstract

One of the powerful approaches to demodulate a single fringe pattern is the regularized phase tracking (RPT) technique. Here, a new improvement in the RPT technique is presented. This new improvement consists in the addition of one term that models the fringe-pattern modulation. With this new term, the RPT technique can be used for the demodulation of nonnormalized fringe patterns. The performance of the improved RPT technique is shown on examples of various fringe patterns.

© 2002 Optical Society of America

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References

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  1. D. W. Robison, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, 1993).
  2. J. L. Marroquin, M. Rivera, S. Botello, R. Rodríguez-Vera, M. Servin, “Regularization methods for processing fringe-pattern images,” Appl. Opt. 38, 788–794 (1999).
    [CrossRef]
  3. M. Servin, J. L. Marroquin, F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
    [CrossRef]
  4. M. Servin, J. L. Marroquin, 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]
  5. J. Villa, J. A. Quiroga, M. Servin, “Improved regularized phase tracking technique for the processing of squared-grating deflectograms,” Appl. Opt. 39, 502–508 (2000).
    [CrossRef]
  6. J. L. Marroquin, J. E. Figueroa, M. Servin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
    [CrossRef]
  7. J. A. Quiroga, J. A. Gómez-Pedrero, A. García-Botella, “Algorithm for fringe pattern,” Opt. Commun. 197, 43–51 (2001).
    [CrossRef]
  8. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999), Sec. 10.7.
  9. M. Servin, J. A. Quiroga, “Isochromatics demodulation from a single image using the regularized phase tracking technique,” J. Mod. Opt. 48, 521–531 (2001).

2001 (3)

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

M. Servin, J. A. Quiroga, “Isochromatics demodulation from a single image using the regularized phase tracking technique,” J. Mod. Opt. 48, 521–531 (2001).

M. Servin, J. L. Marroquin, 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 (1)

1999 (1)

1997 (2)

Botello, S.

Cuevas, F. J.

Figueroa, J. E.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999), Sec. 10.7.

García-Botella, A.

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

Marroquin, J. L.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999), Sec. 10.7.

Quiroga, J. A.

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

M. Servin, J. A. Quiroga, “Isochromatics demodulation from a single image using the regularized phase tracking technique,” J. Mod. Opt. 48, 521–531 (2001).

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

Rivera, M.

Rodríguez-Vera, R.

Servin, M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999), Sec. 10.7.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999), Sec. 10.7.

Villa, J.

Appl. Opt. (3)

J. Mod. Opt. (1)

M. Servin, J. A. Quiroga, “Isochromatics demodulation from a single image using the regularized phase tracking technique,” J. Mod. Opt. 48, 521–531 (2001).

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

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

Other (2)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, Cambridge, England, 1999), Sec. 10.7.

D. W. Robison, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, 1993).

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Figures (4)

Fig. 1
Fig. 1

(a) Fringe pattern from holographic interferometry. The line indicates the row 269. (b) Figure 1(a) without background illumination. (c) Resultant phase estimation with figure 1(b) and the basic RPT function with λ = 2, α = 0.1π, and N xy = 9. (d) Normalized fringe pattern from Fig. 1(b). (e) Resultant phase estimation with Fig. 1(d) and the basic RPT function with λ = 2, α = 0.1π, and N xy = 9. (f) Resultant phase estimation with Fig. 1(b) and the modified RPT function with λ = μ = 2, α = 0.1π, and N xy = 13. Phase estimations are wrapped for the purpose of illustration.

Fig. 2
Fig. 2

(a) Synthetic fringe pattern. The line indicates the column 132. (b) Wrapped phase used to generate the synthetic fringe pattern. (c) Resultant phase estimation with Fig. 2(a) and the basic RPT function with λ = 3, α = 0.1π, and N xy = 9. (d) Resultant phase estimation with Fig. 2(a) and the modified RPT function with λ = 3, μ = 100, α = 0.1π, and N xy = 9. (c) Absolute difference between the unwrapped phases of Figs. 2(b) and 2(c). (f) Absolute difference between the unwrapped phases of Figs. 2(b) and 2(d). (g) Absolute difference between the unwrapped phase of Fig. 2(b) and the resultant phase estimation with the modified RPT function with λ = 3, μ = 100, α = 0.1π, and N xy = 11. (h) Absolute difference between the unwrapped phase of Fig. 2(b) and the resultant phase estimation with the modified RPT function with λ = 3, μ = 100, α = 0.1π, and N xy = 13. Phase estimations are wrapped for the purpose of illustration.

Fig. 3
Fig. 3

(a) Fringe pattern from holographic interferometry. The line indicates the row 336. (b) Figure 3(a) without background illumination. (c) Normalized fringe pattern from Fig. 3(a). (d) Section of the fringe pattern used to show the phase-estimation errors with normalized fringe patterns. (e) Resultant phase estimation with Fig. 3(c) and the basic RPT function with λ = 3, α = 0.1π, and N xy = 13. (f) Resultant phase estimation with Fig. 3(d) and the modified RPT function with λ = μ = 3, α = 0.1π, and N xy = 13. Phase estimations are wrapped for the purpose of illustration.

Fig. 4
Fig. 4

(a)–(d) Sequence used to demodulate the fringe pattern shown in Fig. 3(b). (e) Resultant demodulation with the complete fringe pattern of Fig. 3(b). See the explanation in the text. Phase estimations are wrapped for the purpose of illustration.

Equations (5)

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Ix, y = ax, y + bx, ycosϕx, y + nx, y,
Uxyϕ0,ωx,ωy=ε,ηNxyLIε,η - cosϕex,y,ε,η2+Iε,η - cosϕex,y,ε,η + α2+ λϕ0ε,η - ϕex,y,ε,η2mε,η,
ϕex, y, ε, η=ϕ0x, y+ωxx, yx-ε+ωyx, yy-η
Uxyϕ0, ωx, ωy, B0, βx, βy=ε,ηNxyLIBε, η - Be cosϕex, y, ε, η2+ IBε, η - Be cosϕex, y, ε, η + α2+ λϕ0ε, η - ϕex, y, ε, η2mε, η+ μB0ε, η - Bex, y, ε, η2mε, η,
Bex, y, ε, η=B0x, y+βxx, yx-ε+ βyx, yy - η

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