Abstract

Fringe patterns in optical metrology systems need to be demodulated to yield the desired parameters. Time-frequency analysis is a useful concept for fringe demodulation, and a windowed Fourier transform is chosen for the determination of phase and phase derivative. Two approaches are developed: the first is based on the concept of filtering the fringe patterns, and the second is based on the best match between the fringe pattern and computer-generated windowed exponential elements. I focus on the extraction of phase and phase derivatives from either phase-shifted fringe patterns or a single carrier fringe pattern. Principles as well as examples are given to show the effectiveness of the proposed methods.

© 2004 Optical Society of America

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References

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  1. D. W. Robinson, G. T. Reid(eds.), Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, England, 1993).
  2. M. Servin, M. Kujawinska, “Modern fringe analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 373–426.
  3. J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
    [CrossRef]
  4. M. Servin, J. L. Marroguin, 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. M. Servin, R. Rodriguez-Vera, J. L. Marraquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1819 (1998).
    [CrossRef]
  6. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
    [CrossRef]
  7. L. R. Watkins, S. M. Tan, T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
    [CrossRef]
  8. K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
    [CrossRef]
  9. J. Fang, C. Y. Xiong, H. J. Li, Z. L. Zhang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
    [CrossRef]
  10. K. Qian, H. S. Seah, A. Asundi, “Instantaneous frequency and its application in strain extraction in moiré interferometry,” Appl. Opt. 42, 6504–6513 (2003).
    [CrossRef]
  11. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, San Diego, 1999).
  12. K. Kröchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Boston, 2001).
  13. matlab codes for both WFF and WFR are available upon request.
  14. M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
    [CrossRef]
  15. K. Qian, H. S. Seah, A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42, 2792–2793 (2003).
    [CrossRef]
  16. M. Takeda, H. Ina, S. Kobayashi, “Fourier transform methods of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  17. N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
    [CrossRef]
  18. P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
    [CrossRef]
  19. H. Miao, “The studies on real time phase measurement technique and its application to protein crystal growth,” Ph.D Dissertation (University of Science and Technology of China, Beijing, 1999).

2003 (3)

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

K. Qian, H. S. Seah, A. Asundi, “Instantaneous frequency and its application in strain extraction in moiré interferometry,” Appl. Opt. 42, 6504–6513 (2003).
[CrossRef]

K. Qian, H. S. Seah, A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42, 2792–2793 (2003).
[CrossRef]

2001 (1)

J. Fang, C. Y. Xiong, H. J. Li, Z. L. Zhang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

1999 (2)

1998 (2)

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
[CrossRef]

M. Servin, R. Rodriguez-Vera, J. L. Marraquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1819 (1998).
[CrossRef]

1997 (1)

1996 (1)

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

1995 (1)

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

1992 (1)

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

1982 (1)

Asundi, A.

K. Qian, H. S. Seah, A. Asundi, “Instantaneous frequency and its application in strain extraction in moiré interferometry,” Appl. Opt. 42, 6504–6513 (2003).
[CrossRef]

K. Qian, H. S. Seah, A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42, 2792–2793 (2003).
[CrossRef]

Barnes, T. H.

Cuevas, F. J.

Delprat, N.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Escudié, B.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Fang, J.

J. Fang, C. Y. Xiong, H. J. Li, Z. L. Zhang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

Guillemain, P.

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Huntley, J. M.

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
[CrossRef]

Ina, H.

Kadooka, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Kobayashi, S.

Kröchenig, K.

K. Kröchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Boston, 2001).

Kronland-Martinet, R.

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Kujawinska, M.

M. Servin, M. Kujawinska, “Modern fringe analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 373–426.

Kunoo, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Li, H. J.

J. Fang, C. Y. Xiong, H. J. Li, Z. L. Zhang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

Malacara, D.

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

M. Servin, R. Rodriguez-Vera, J. L. Marraquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1819 (1998).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, San Diego, 1999).

Marraquin, J. L.

M. Servin, R. Rodriguez-Vera, J. L. Marraquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1819 (1998).
[CrossRef]

Marroguin, J. L.

Miao, H.

H. Miao, “The studies on real time phase measurement technique and its application to protein crystal growth,” Ph.D Dissertation (University of Science and Technology of China, Beijing, 1999).

Nagayasu, T.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Ono, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Qian, K.

K. Qian, H. S. Seah, A. Asundi, “Instantaneous frequency and its application in strain extraction in moiré interferometry,” Appl. Opt. 42, 6504–6513 (2003).
[CrossRef]

K. Qian, H. S. Seah, A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42, 2792–2793 (2003).
[CrossRef]

Rodriguez-Vera, R.

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

M. Servin, R. Rodriguez-Vera, J. L. Marraquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1819 (1998).
[CrossRef]

Seah, H. S.

K. Qian, H. S. Seah, A. Asundi, “Instantaneous frequency and its application in strain extraction in moiré interferometry,” Appl. Opt. 42, 6504–6513 (2003).
[CrossRef]

K. Qian, H. S. Seah, A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42, 2792–2793 (2003).
[CrossRef]

Servin, M.

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934–1941 (1999).
[CrossRef]

M. Servin, R. Rodriguez-Vera, J. L. Marraquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1819 (1998).
[CrossRef]

M. Servin, J. L. Marroguin, 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, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

M. Servin, M. Kujawinska, “Modern fringe analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 373–426.

Takeda, M.

Tan, S. M.

Tchamitchian, P.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Torrésani, B.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Uda, N.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Watkins, L. R.

Xiong, C. Y.

J. Fang, C. Y. Xiong, H. J. Li, Z. L. Zhang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

Zhang, Z. L.

J. Fang, C. Y. Xiong, H. J. Li, Z. L. Zhang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

Appl. Opt. (3)

Exp. Mech. (1)

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

IEEE Trans. Inf. Theory (1)

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

J. Mod. Opt. (3)

J. Fang, C. Y. Xiong, H. J. Li, Z. L. Zhang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

M. Servin, R. Rodriguez-Vera, J. L. Marraquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1819 (1998).
[CrossRef]

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Strain Anal. (1)

J. M. Huntley, “Automated fringe pattern analysis in experimental mechanics: a review,” J. Strain Anal. 33, 105–125 (1998).
[CrossRef]

Opt. Eng. (1)

K. Qian, H. S. Seah, A. Asundi, “Filtering the complex field in phase shifting interferometry,” Opt. Eng. 42, 2792–2793 (2003).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

Other (6)

H. Miao, “The studies on real time phase measurement technique and its application to protein crystal growth,” Ph.D Dissertation (University of Science and Technology of China, Beijing, 1999).

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

M. Servin, M. Kujawinska, “Modern fringe analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson, eds. (Marcel Dekker, New York, 2001), pp. 373–426.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, San Diego, 1999).

K. Kröchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Boston, 2001).

matlab codes for both WFF and WFR are available upon request.

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Figures (8)

Fig. 1
Fig. 1

Simulated data: (a) phase, (b) wrapped phase, (c) phase derivative in horizontal.

Fig. 2
Fig. 2

WFF with phase-shifting technique: (a)–(d) Phase-shifted fringe patterns, (e) phase by directly using phase-shifting algorithm, (f) phase by the proposed WFF, (g) phase derivative from (f) by numerical differentiation.

Fig. 3
Fig. 3

WFF with carrier technique: (a) carrier fringe pattern, (b) phase by traditional Fourier transform method, (c) phase by the proposed WFF, (d) phase derivative from (c) by numerical differentiation.

Fig. 4
Fig. 4

WFR with phase-shifting technique: (a) phase derivative, (b) phase from ridges, (c) phase by integration; the result is wrapped for comparison.

Fig. 5
Fig. 5

Path for phase integration: (a) phase of central row is first integrated from the central point; (b) phase of all columns is then integrated from the central row.

Fig. 6
Fig. 6

WFR with carrier technique: (a) phase derivative, (b) phase from ridges, (c) phase by integration; the result is wrapped for comparison. (d) Phase derivative by one-dimensional wavelet ridges.

Fig. 7
Fig. 7

Roles of WFF and WFR in fringe demodulation.

Fig. 8
Fig. 8

Real application: (a) fringe pattern, (b) phase map obtained using the proposed WFF, (c) unwrapped phase map.

Tables (1)

Tables Icon

Table 1 Accuracies of the WFF and WFR

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

fx=ax+bxcosφx+nx,
Ffξ=- fxexp-jξxdx,
fx=12π- Ffξexpjξxdξ,
Sfu, ξ=- fxgx-uexp-jξxdx,
fx=12π-- Sfu, ξgx-u×expjξxdξdu,
gx=exp-x2/2σ2,
fix=ax+bxcosφx+i-1α, i=1, 2, 3,.
fpsex=12f1x-f3x+jf4x-jf2x=bxexpjφx,
fcx=ax+bxcosωcx+φx,
fcx=ax+12 bxexpjωcx+jφx+12 bxexp-jωcx-jφx =ax+fcex+fce*x,
fx=12π-fxhx, ξhx, ξdξ,
f¯x=12πabfxhx, ξ¯hx, ξdξ,
φx=anglef¯x.
φu=φu+1-φu.
φ0x, y=0.002×ρ2ρ2=x2+y212720.002×1272ρ2=x2+y2>1272
fix=1+cosφ0x+i-1π/2+nix, y, i=1, 2, 3, 4,
fcx, y=1+cosπx/2+φ0x, y+nx, y.
Sfpseu, ξ=- fpsexgx-uexp-jξxdx=- bxgx-uexpjφx-jξxdx.
Sfpseu, ξ=buexpjφu-ξu- gtexp-jξ-φutdt.
Sfpseu, ξ=buexpjφu-ξuGξ-φu.
φu=arg maxξ|Sfpseu, ξ|,
fpsexhx, ξ=expjξuSfpseu, ξ =buexpjφuGξ-φu;
φu=arg maxξ|fpseuhu, ξ|.
φu+ωc=arg maxξ|Sfcu, ξ|
φu+ωc=arg maxξ|fcuhu, ξ|.
φu=anglefpseuhu, φu.
φu=φu0+u0u φtdt
φu=φu-1+φu-1.
Δφ=mean|φx-φ0x|,
Δφ=mean|φx-φ0x|,
-12arctanσ2φuuu, v-12arctanσ2φvvu, v,
φuuu, v=2φu2,φvvu, v=2φv2,φuvu, v=2φuv.

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