Abstract

A technique for recovering the phase from single fringe-pattern images is presented. It is based on the estimation of the local frequency of the pattern by successive decoupled estimation of the local orientation, direction, and magnitude of the frequency field. Once this field is known, the local phase is recovered from the complex output of an adaptive quadrature filter. It is shown that by the use of Gauss–Markov measure field models all these estimation steps may be implemented by solving linear systems of equations (i.e., minimizing quadratic functions), which makes the procedure robust and computationally efficient. Examples are presented of the application of this technique to the recovery of phase from single electronic speckle-pattern interferograms.

© 1998 Optical Society of America

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References

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  6. A. Davila, D. Kerr, G. H. Kaufmann, “Fast electro-optical system for pulsed ESPI carrier fringe generation,” Opt. Commun. 123, 457–464 (1996).
    [CrossRef]
  7. D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
    [CrossRef]
  8. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  9. J. L. Marroquin, M. Servin, 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]
  10. J. L. Marroquin, “Gauss–Markov measure fields for image processing,” (Centro de Investigacion en Matemáticas, Guanajuato, Mexico, 1997).
  11. J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
    [CrossRef]
  12. J. L. Marroquin, “Random measure fields and the integration of visual information,” IEEE Trans. Syst. Man Cybern. 22, 705–716 (1992).
    [CrossRef]
  13. N. I. Fisher, Statistical Analysis of Circular Data (Cambridge U. Press, Cambridge, UK, 1993).
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    [CrossRef]
  15. G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).
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    [CrossRef]
  17. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  18. M. Rivera, R. Rodriguez-Vera, J. L. Marroquin, “Robust procedure for fringe analysis,” Appl. Opt. 36, 8391–8396 (1997).
    [CrossRef]

1997 (3)

1996 (1)

A. Davila, D. Kerr, G. H. Kaufmann, “Fast electro-optical system for pulsed ESPI carrier fringe generation,” Opt. Commun. 123, 457–464 (1996).
[CrossRef]

1995 (1)

1994 (1)

1992 (1)

J. L. Marroquin, “Random measure fields and the integration of visual information,” IEEE Trans. Syst. Man Cybern. 22, 705–716 (1992).
[CrossRef]

1987 (1)

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

1986 (1)

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

1982 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

Davila, A.

A. Davila, D. Kerr, G. H. Kaufmann, “Fast electro-optical system for pulsed ESPI carrier fringe generation,” Opt. Commun. 123, 457–464 (1996).
[CrossRef]

Farrant, D. I.

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
[CrossRef]

Figueroa, J. E.

Fisher, N. I.

N. I. Fisher, Statistical Analysis of Circular Data (Cambridge U. Press, Cambridge, UK, 1993).

Gallager, J. E.

Ghiglia, D. C.

Gollub, G. H.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

Herriott, D. R.

Ina, H.

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).

Kaufmann, G. H.

A. Davila, D. Kerr, G. H. Kaufmann, “Fast electro-optical system for pulsed ESPI carrier fringe generation,” Opt. Commun. 123, 457–464 (1996).
[CrossRef]

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
[CrossRef]

Kerr, D.

A. Davila, D. Kerr, G. H. Kaufmann, “Fast electro-optical system for pulsed ESPI carrier fringe generation,” Opt. Commun. 123, 457–464 (1996).
[CrossRef]

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
[CrossRef]

Kobayashi, S.

Kreis, Th.

Marroquin, J.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Marroquin, J. L.

Mitter, S.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Oreb, B. F.

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
[CrossRef]

Petzing, J. N.

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
[CrossRef]

Poggio, T.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Rivera, M.

Rodriguez-Vera, R.

Romero, L. A.

Rosefeld, D. P.

Servin, M.

Takeda, M.

Tyrer, J. R.

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
[CrossRef]

Van Loan, C. F.

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

White, A. D.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).

Appl. Opt. (2)

IEEE Trans. Syst. Man Cybern. (1)

J. L. Marroquin, “Random measure fields and the integration of visual information,” IEEE Trans. Syst. Man Cybern. 22, 705–716 (1992).
[CrossRef]

J. Am. Stat. Assoc. (1)

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

A. Davila, D. Kerr, G. H. Kaufmann, “Fast electro-optical system for pulsed ESPI carrier fringe generation,” Opt. Commun. 123, 457–464 (1996).
[CrossRef]

Opt. Eng. (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
[CrossRef]

Other (6)

N. I. Fisher, Statistical Analysis of Circular Data (Cambridge U. Press, Cambridge, UK, 1993).

G. H. Gollub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1990).

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, D. Kerr, “Transient deformation measurement using dual-pulse addition ESPI,” in Ultrahigh- and High-Speed Photography and Image-Based Motion Measurement, D. R. Snyder, A. Davidhazy, T. Etoh, C. Johnson, J. S. Walton, eds., Proc. SPIE3173, 132–140 (1997).
[CrossRef]

J. L. Marroquin, “Gauss–Markov measure fields for image processing,” (Centro de Investigacion en Matemáticas, Guanajuato, Mexico, 1997).

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, UK, 1989).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

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Figures (4)

Fig. 1
Fig. 1

(a) ESPI fringe pattern corresponding to the thermal deformation of an aluminium plate. (b) Orientation field obtained from the gradient direction of image (a) smoothed with a membrane low-pass filter with λ=10. (c) Regularized orientation field corresponding to the modes of the distributions p(r,·), where the p field is the minimizer of Eq. (5) with M=12 and λ=1000. (d) Regularized direction field obtained from Eq. (9), where p is the minimizer of Eq. (8) with μ=1000. (e) Regularized magnitude field corresponding to the modes of the distributions p(r,·), where the p field is the minimizer of Eq. (5) with M=12 and λ=100; the fields fk for k =1,, 12, were obtained by minimizing Eq. (2) with λ=50. (f) Reconstructed (wrapped) phase obtained by applying Eq. (3) to the corresponding field f.

Fig. 2
Fig. 2

(a) Detail of the regularized orientation field of Fig. 1(c) near the critical point on the middle left; thick lines indicate direction discontinuities. (b) Corresponding detail of the regularized direction field of Fig. 1(d).

Fig. 3
Fig. 3

(a) Fringe pattern obtained by adding uniform phase noise in the interval [-1.5, 1.5] to the quadratic phase shown wrapped in (b). (c) Reconstructed phase obtained with the conventional Fourier method and panel. (d) Reconstruction with the technique presented here.

Fig. 4
Fig. 4

(a) ESPI pattern corresponding to the thermal deformation of an aluminum plate. (b) Indicator function for region L (black pixels indicate that no information is present). (c) Regularized direction field. (d) Reconstructed phase (the parameters are the same as those of Fig. 1).

Equations (31)

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I(r)=a(r)+b(r)cos[ϕ(r)]+n(r),
ϕ(r)=ω0·r+α(r),
φ(r)=A(r)cos[ω(r)·r+α(r)],
φ(s)A(r)cos[ω(r)·s+α(r)].
ψ(r)A(r)sin[ω(r)·r+α(r)].
U(f )=r,sL|f(r)-f(s)-2[I(r)-I(s)]|2+λr,sLf(r)exp-i2ω(r)·(r-s)-f(s)×exp-i2ω(s)·(s-r)2,
ϕ(r)=arctanψ(r)φ(r).
ω(r)=[ρ(r)cos θ(r), ρ(r)sin θ(r)],
pˆ(r, k)=[1/Z(r)]exp{-[I(r)-qk]2/(2σ2)},
U(p)=k=1MrL[p(r, k)-pˆ(r, k)]2+λr,sLk=1M×[p(r, k)-p(s, k)]2b(r, s, |k-k|),
b(r, s, m)=1,ifm=0,=0,otherwise,
U(p)=k=1MUk(p),
Uk(p)=rL[p(r, k)-pˆ(r, k)]2+λr,sL[p(r, k)-p(s, k)]2,
θˆ(r)=θI(r)arctan-I(r)/xI(r)/y,
pˆ(r, k)=1+cos{2[qk-θI(r)]}i=1M(1+cos{2[qi-θI(r)]}).
b(r, s, 0)={1+cos[θˆ(r)-θˆ(s)]}ξr,s,
b(r, s, 1)={1-cos[θˆ(r)-θˆ(s)]}ξr,s,
U(p)=r,sL{(1+crs)[p(r, 1)-p(s, 1)]2+(1-crs)[p(r, 1)+p(s, 1)-1]2}ξr,s+μ[p(r0, 1)-1]2,
θ(r)=θˆ(r),ifp(r, 1)<0.5=θˆ(r)+π,ifp(r, 1)0.5.
ωk(r)={qk cos[θ(r)], qk sin[θ(r)]},
Z(r)=k=1M|fk(r)|.
p(r, k)=Pr[f(r)=qk|I],rL.
U(p)=k=1MrL[p(r, k)-pˆ(r, k)]2a(r)+λr,sLk=1M[p(r, k)-p(s, k)]2×b(r, s, |k-k|),
b(r, s, m)=b(r, s, M-m)
a(r)[p(r, k)-pˆ(r, k)]+λsNrLk=1M[p(r, k)
-p(s, k)]b(r, s, |k-k|)=0,
p(r, k)=a(r)pˆ(r, k)+λsNrLk=1Mp(s, k)b(r, s, |k-k|)a(r)+λsNrLk=1Mb(r, s, |k-k|).
a(r)[A(r)-1]+λsNrL[A(r)-A(s)]B(r, s)=0,
k=1Mb(r, s, |k-k|)=m=0M-1b(r, s, m)B(r, s)>0.
U(A)=rLa(r)[A(r)-1]2+λr,sL[A(r)-A(s)]2B(r, s).
k=1Mp(r, k)=1

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