Abstract

We propose a robust procedure based on the regularized phase-tracking (RPT) technique to demodulate squared-grating deflectograms. The use of squared gratings, already reported, lets us multiplex the information of the deflections in two orthogonal directions in a single image, thus avoiding the necessity of rotating the gratings. The good noise-rejection characteristics of the RPT technique are improved by use of a quasi-Newton optimization algorithm and a quality-map-based algorithm for the crystal-growing process.

© 2000 Optical Society of America

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References

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  1. A. W. Lohmann, D. E. Silva, “An interferometer based on talbot effect,” Opt. Commun. 2, 413–415 (1971).
    [CrossRef]
  2. O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).
  3. O. Kafri, A. Livnat, “Reflective surface analysis using moiré deflectometry,” Appl. Opt. 26, 2507–2508 (1987).
  4. Z. Karny, O. Kafri, “Refractive-index measurement by moiré deflectometry,” Appl. Opt. 21, 3326–3328 (1982).
    [CrossRef] [PubMed]
  5. E. Keren, E. Bar-Ziv, “Measurements of temperature distribution flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
    [CrossRef] [PubMed]
  6. M. Servín, R. Rodriguez-Vera, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266–3270 (1990).
    [CrossRef]
  7. H. Canabal, J. A. Quiroga, “Automatic processing in moiré deflectometry by local fringe direction calculation,” Appl. Opt. 37, 5894–5901 (1998).
    [CrossRef]
  8. T. Pfeifer, B. Wang, “Phase-shifting moiré deflectometry,” Optik 98, 158–162 (1995).
  9. H. Canabal, J. A. Quiroga, E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
    [CrossRef]
  10. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]
  11. E. Bar-Ziv, “Effect of diffraction on the moiré image,” J. Opt. Soc. Am. A 2, 371–379 (1985).
    [CrossRef]
  12. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).
  13. M. Takeda, S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
    [CrossRef] [PubMed]
  14. J. A. Quiroga, D. Crespo, E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
    [CrossRef]
  15. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  16. J. Gu, F. Chen, “Fast Fourier transform, iteration, and least-squares-fit demodulation image processing of single-carrier fringe pattern,” J. Opt. Soc. Am. A 12, 2159–2164 (1995).
    [CrossRef]
  17. M. Servín, J. L. Marroquín, 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]
  18. J. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2060 (1973).
    [CrossRef] [PubMed]
  19. J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
    [CrossRef]
  20. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998), Chap. 5.
  21. Matlab Optimization Toolbox, User’s Guide, Version 5 (MathWorks Inc., 24 Prime Park Way, Natick, Mass., 01760–1500).
  22. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [CrossRef] [PubMed]
  23. R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (International Thomson Publishing, Stamford, Conn., 1997).
  24. J. Alonso, J. A. Gomez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Ophthalmic Physiol. Opt. 17, 522–529 (1997).
    [CrossRef]

1999

J. A. Quiroga, D. Crespo, E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

1998

1997

M. Servín, J. L. Marroquín, 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. Alonso, J. A. Gomez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Ophthalmic Physiol. Opt. 17, 522–529 (1997).
[CrossRef]

1996

1995

1990

1989

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

1987

1985

1984

1982

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

Z. Karny, O. Kafri, “Refractive-index measurement by moiré deflectometry,” Appl. Opt. 21, 3326–3328 (1982).
[CrossRef] [PubMed]

1981

1973

1971

A. W. Lohmann, D. E. Silva, “An interferometer based on talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Alonso, J.

J. Alonso, J. A. Gomez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Ophthalmic Physiol. Opt. 17, 522–529 (1997).
[CrossRef]

Bar-Ziv, E.

Bernabeu, E.

J. A. Quiroga, D. Crespo, E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

H. Canabal, J. A. Quiroga, E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
[CrossRef]

J. Alonso, J. A. Gomez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Ophthalmic Physiol. Opt. 17, 522–529 (1997).
[CrossRef]

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998), Chap. 5.

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998), Chap. 5.

Burden, R. L.

R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (International Thomson Publishing, Stamford, Conn., 1997).

Canabal, H.

Chen, F.

Crespo, D.

J. A. Quiroga, D. Crespo, E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

Cuevas, F. J.

Faires, J. D.

R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (International Thomson Publishing, Stamford, Conn., 1997).

Field, J. E.

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Glatt, I.

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

Gomez-Pedrero, J. A.

J. Alonso, J. A. Gomez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Ophthalmic Physiol. Opt. 17, 522–529 (1997).
[CrossRef]

Gu, J.

Huntley, J. M.

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Ina, H.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

Kafri, O.

Karny, Z.

Keren, E.

Kobayashi, S.

M. Takeda, S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
[CrossRef] [PubMed]

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

Livnat, A.

Lohmann, A. W.

A. W. Lohmann, D. E. Silva, “An interferometer based on talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Marroquín, J. L.

Pfeifer, T.

T. Pfeifer, B. Wang, “Phase-shifting moiré deflectometry,” Optik 98, 158–162 (1995).

Quiroga, J. A.

Roddier, C.

Roddier, F.

Rodriguez-Vera, R.

Servín, M.

Silva, D. E.

A. W. Lohmann, D. E. Silva, “An interferometer based on talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Ströbel, B.

Takeda, M.

M. Takeda, S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
[CrossRef] [PubMed]

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

Wang, B.

T. Pfeifer, B. Wang, “Phase-shifting moiré deflectometry,” Optik 98, 158–162 (1995).

Wyant, J. C.

Appl. Opt.

O. Kafri, A. Livnat, “Reflective surface analysis using moiré deflectometry,” Appl. Opt. 26, 2507–2508 (1987).

Z. Karny, O. Kafri, “Refractive-index measurement by moiré deflectometry,” Appl. Opt. 21, 3326–3328 (1982).
[CrossRef] [PubMed]

E. Keren, E. Bar-Ziv, “Measurements of temperature distribution flames by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
[CrossRef] [PubMed]

M. Servín, R. Rodriguez-Vera, “Automatic fringe detection algorithm used for moiré deflectometry,” Appl. Opt. 29, 3266–3270 (1990).
[CrossRef]

H. Canabal, J. A. Quiroga, “Automatic processing in moiré deflectometry by local fringe direction calculation,” Appl. Opt. 37, 5894–5901 (1998).
[CrossRef]

H. Canabal, J. A. Quiroga, E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
[CrossRef]

M. Takeda, S. Kobayashi, “Lateral aberration measurements with a digital Talbot interferometer,” Appl. Opt. 23, 1760–1764 (1984).
[CrossRef] [PubMed]

M. Servín, J. L. Marroquín, 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. C. Wyant, “Double frequency grating lateral shear interferometer,” Appl. Opt. 12, 2057–2060 (1973).
[CrossRef] [PubMed]

C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
[CrossRef] [PubMed]

B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Ophthalmic Physiol. Opt.

J. Alonso, J. A. Gomez-Pedrero, E. Bernabeu, “Local dioptric power matrix in a progressive addition lens,” Ophthalmic Physiol. Opt. 17, 522–529 (1997).
[CrossRef]

Opt. Commun.

A. W. Lohmann, D. E. Silva, “An interferometer based on talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Opt. Eng.

J. A. Quiroga, D. Crespo, E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng. 38, 974–982 (1999).
[CrossRef]

J. M. Huntley, J. E. Field, “High resolution moiré photography: application to dynamic stress analysis,” Opt. Eng. 28, 926–933 (1989).
[CrossRef]

Optik

T. Pfeifer, B. Wang, “Phase-shifting moiré deflectometry,” Optik 98, 158–162 (1995).

Other

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, London, 1998), Chap. 5.

Matlab Optimization Toolbox, User’s Guide, Version 5 (MathWorks Inc., 24 Prime Park Way, Natick, Mass., 01760–1500).

O. Kafri, I. Glatt, The Physics of Moiré Metrology (Wiley, New York, 1989).

R. L. Burden, J. D. Faires, Numerical Analysis, 6th ed. (International Thomson Publishing, Stamford, Conn., 1997).

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Figures (6)

Fig. 1
Fig. 1

Typical experimental setup used in moiré deflectometry. Gratings G1 and G2 are placed at a Talbot distance Z from each other to form a moiré fringe pattern. The fringe image is observed in the screen S.

Fig. 2
Fig. 2

(a) Reference and (b) distorted moiré deflectograms for a progressive addition lens obtained with squared gratings.

Fig. 3
Fig. 3

Phase map corresponding to the deflection in the x direction obtained with (a) the Fourier method and (b) the RPT technique.

Fig. 4
Fig. 4

Contour maps (scaled in dioptries) corresponding to (a) ω xx , (b) ω xy , (c) ω yx , and (d) ω yy .

Fig. 5
Fig. 5

(a) Cylindrical and (b) spherical power computed with the RPT technique.

Fig. 6
Fig. 6

Profiles of (a) cylindrical power along line AB in Fig. 6(a) and (b) spherical power along line CD in Fig. 6(b), together with the measurements made with a commercial focimeter along the same lines.

Equations (10)

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

UT=(x,y)L Ux,y(ϕ, ωx, ωy),
Ux,y(ϕ, ωx, ωy)=(x˜,y˜)(Nx,yL){|gh(x˜, y˜)-cos[ϕe(x, y, x˜, y˜)+u0x+v0y]|2+λ|ϕ(x˜, y˜)-ϕe(x, y, x˜, y˜)|2m(x˜, y˜)},
ϕe(x, y, x˜, y˜)=ϕ(x, y)+ωx(x, y)(x-x˜)+ωy(x, y)(y-y˜),
g(x, y)=n=-an2 cos[nΦx(x, y)]m=-cos[mqΦy(x, y)],
Φx(x, y)=Zqϕx(x, y)+u0x+v0y,
Φy(x, y)=Zqϕy(x, y)+v0x-u0y,
gh(x, y)=cos[Φx(x, y)]cos[Φy(x, y)],
ωxx=ϕxx,  ωxy=ϕxy,  ωyy=ϕyy,  ωyx=ϕyx.
C=[(ωxx+ωyy)2-4(ωxxωyy-ωxyωyx)]1/2,
S=1/2(ωxx+ωyy-C),

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