Abstract

A robust algorithm for phase recovery from multi-phase-stepping images is presented. This algorithm is based on the minimization of an energy (cost) functional and is equivalent to the simultaneous application of a fixed temporal quadrature filter and a spatial adaptive quadrature filter to the phase-stepping pattern ensemble. The algorithm, believed to be new, is specially suited for those applications in which a large number of phase-stepping images may be obtained, e.g., profilometry with a computer-controlled fringe projector. We discuss the selection of parameter values and present examples of its performance in both synthetic and real image sequences.

© 2000 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonsinuosoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A. 12, 761–768 (1995).
    [CrossRef]
  3. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A. 14, 918–930 (1997).
    [CrossRef]
  4. K. Hibino, “Susceptibility of error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, (1997).
  5. Y. Surrel, “Design of algorithms for phase measurements by use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  6. Y. Surrel, “Extended averanging and data windowing techniques in phase-stepping measurements: an approach using the characteristics polynomial theory,” Opt. Eng. 37, 2314–2319 (1998).
    [CrossRef]
  7. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  8. P. de Groot, “101-frame algorithm for phase-shifting interferometry,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 283–292 (1997).
    [CrossRef]
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  12. M. Rivera, J. L. Marroquin, M. Servín, R. Rodriguez-Vera, “Fast algorithm for intergrating inconsistent gradient fields,” Appl. Opt. 36, 8381–8390 (1997).
    [CrossRef]
  13. J. L. Marroquin, M. Servín, R. Rodriguez-Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23, 238–240 (1998).
    [CrossRef]
  14. J. L. Marroquin, M. Servín, 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]
  15. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).
  16. A. L. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 11.
  17. J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
    [CrossRef]
  18. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 13.
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1998 (2)

Y. Surrel, “Extended averanging and data windowing techniques in phase-stepping measurements: an approach using the characteristics polynomial theory,” Opt. Eng. 37, 2314–2319 (1998).
[CrossRef]

J. L. Marroquin, M. Servín, R. Rodriguez-Vera, “Adaptive quadrature filters for multiple phase-stepping images,” Opt. Lett. 23, 238–240 (1998).
[CrossRef]

1997 (5)

J. L. Marroquin, M. Servín, 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]

M. Rivera, J. L. Marroquin, M. Servín, R. Rodriguez-Vera, “Fast algorithm for intergrating inconsistent gradient fields,” Appl. Opt. 36, 8381–8390 (1997).
[CrossRef]

J. L. Marroquin, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A. 14, 918–930 (1997).
[CrossRef]

K. Hibino, “Susceptibility of error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, (1997).

1996 (2)

1995 (3)

1993 (1)

1982 (1)

1974 (1)

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Botello, S.

J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

de Groot, P.

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[CrossRef]

P. de Groot, “101-frame algorithm for phase-shifting interferometry,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 283–292 (1997).
[CrossRef]

Farrant, D. I.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A. 14, 918–930 (1997).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonsinuosoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A. 12, 761–768 (1995).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 13.

Freischlad, K.

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Gallegher, J. E.

Herriot, D. R.

Hibino, K.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A. 14, 918–930 (1997).
[CrossRef]

K. Hibino, “Susceptibility of error-compensating algorithms to random noise in phase-shifting interferometry,” Appl. Opt. 36, (1997).

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonsinuosoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A. 12, 761–768 (1995).
[CrossRef]

Huntley, J. M.

Ina, H.

Kobayashi, S.

Koliopulos, C. L.

Larkin, K. G.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A. 14, 918–930 (1997).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonsinuosoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A. 12, 761–768 (1995).
[CrossRef]

Marroquin, J. L.

Oppenheim, A. L.

A. L. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 11.

Oreb, B. F.

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A. 14, 918–930 (1997).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonsinuosoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A. 12, 761–768 (1995).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 13.

Rivera, M.

M. Rivera, J. L. Marroquin, M. Servín, R. Rodriguez-Vera, “Fast algorithm for intergrating inconsistent gradient fields,” Appl. Opt. 36, 8381–8390 (1997).
[CrossRef]

J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
[CrossRef]

Rodriguez-Vera, R.

Rosenfeld, D. P.

Sander, H.

Schafer, R. W.

A. L. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 11.

Schmit, J.

Servín, M.

Surrel, Y.

Y. Surrel, “Extended averanging and data windowing techniques in phase-stepping measurements: an approach using the characteristics polynomial theory,” Opt. Eng. 37, 2314–2319 (1998).
[CrossRef]

Y. Surrel, “Design of algorithms for phase measurements by use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

Takeda, M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 13.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 13.

White, A. D.

Appl. Opt. (7)

J. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonsinuosoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A. 12, 761–768 (1995).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A. 14, 918–930 (1997).
[CrossRef]

Opt. Eng. (1)

Y. Surrel, “Extended averanging and data windowing techniques in phase-stepping measurements: an approach using the characteristics polynomial theory,” Opt. Eng. 37, 2314–2319 (1998).
[CrossRef]

Opt. Lett. (1)

Other (4)

A. L. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Chap. 11.

J. L. Marroquin, M. Rivera, S. Botello, “Adaptive quantization and filtering using Gauss–Markov measure field models,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 238–249 (1998).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 13.

P. de Groot, “101-frame algorithm for phase-shifting interferometry,” in Optical Inspection and Micromeasurements II, C. Gorecki, ed., Proc. SPIE3098, 283–292 (1997).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Original and recovered phase when the fringe pattern is temporally modulated (with k = 1 and M = 256). (b) Temporal modulation m(t).

Fig. 2
Fig. 2

(a) Samples (with arrows) of a square waveform. (b) The phase shift Δϕ in the signal is not detected in the samples.

Fig. 3
Fig. 3

MSE for different temporal frequencies. Solid curve with triangles, sinusoidal waveform with temporal modulation. Dotted curve, nonsinusoidal waveform with constant temporal modulation. Solid curve with asterisks, nonsinusoidal waveform with variable temporal modulation.

Fig. 4
Fig. 4

MSE versus number of phase steps for fringe patterns with additive Gaussian noise: STQF (solid curve) and without spatial filtering (λ = 0 and μ = 0; dashed curve).

Fig. 5
Fig. 5

(a) Synthetic phase. (b) Noisy fringe pattern. (c) Phase computed with the presented algorithm for N = 50, λ = 1, μ = 5. (d) Computed MSE map.

Fig. 6
Fig. 6

Application of the STQF to real objects. (a) Magnitude of ı̂. (b) Real part of ı̂. (c) phase computed with the STQF.

Fig. 7
Fig. 7

Synthetic test phase (smooth curve) and computed phase (wavy curve) with the algorithm (a) proposed, (b) Bruning’s, (c) windowed-DFT, and (d) Hibino’s.

Tables (1)

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Table 1 Summary of Experimental Results

Equations (24)

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ix, t=mtax+bxcosωs0·x+ϕx+νt+ηx, t,
ix, t0=Km0ax+Km0bxcosωs0·x+ϕx+Kt0+ ηx, t0,
ϕx=arctanIm fxRe fx-ωs0·x.
Uf, ω=Udf+Urf, ω,
Udf=t=0N-1xL |fxexpjνt-2gtx|2.
gtx=ix, t-Hix, t,
Urf, ω=λ x, yLfxexpj 12 ωx·y-x-fyexpj 12 ωy·x-y2+μ xL Vcωx,
x, y=x, y:yx1+1, x2, x1, x2+1,x1-1, x2, x1, x2-1.
t=0N-1 |fxexpjνt-2gtx|2=N|fx-ı˜x|2+Kx,
Kx=1Nt=0N-1 |2gtx|2-1Nt=0N-1 2gtxexp-jνt2.
ı˜x=2ax1Nt=0N-1 mtexp-jνt+expjKsxbx1Nt=0N-1 mt+expjKsxbx1Nt=0N-1 mtexp-j2νt+2 1Nt=0N-1 ηx, texp-jνt,
ϕx=arctanIm ı˜xRe ı˜x-ωs0·x.
ıˆx=1Nt 2ix, tht-t0exp-iνt.
ht=exp-tN/22
Uf, ω=N xLfx-ıˆx|ıˆx|2+λ x, yLfxexpj 12 ωx·y-x-fyexpj 12 ωy·x-y2+μ xL Vcωx,
αx=1Nt=0N-1 ηx, tht-t0exp-jνt
Eα=0,  limN varα=0,
hˆ=1Nt=0N-1 |ht-t0|2.
ϕsx=t ix, tht-t0sin-iνtt ix, tht-t0cos-iνt,
Rfx=t=0N-1 |fx-2gtxexpjνt|2.
|fx-2gtxexpjνt|2=fx-2gtxexpjνt×fx-2gtxexpjνt*=|fx|2-4fxgtxexpjνt+|2gtx|2,
Rfx=t=0N-1 |fx|2-t=0N-1 4fxgtxexpjνt+t=0N-1 |2gtx|2+N1Nt=0N-1 2gtxexp-jνt2-N1Nt=0N-1 2gtxexp-jνt2,
Rfx=N|fx|2-2Nfx1Nt=0N-1 2gtxexpjνt+N1Nt=0N-1 2gtxexp-jνt2+1Nt=0N-1 |2gtx|2-N1Nt=0N-1 2gtxexpjνt2.
Rfx=Nfx-1Nt=0N-1 2gt exp-jνt2+1Nt=0N-1 |2gtx|2-1Nt=0N-1 2gtxexp-jνt2.

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