Abstract

Inasmuch as current fringe analysis techniques used in digital speckle-pattern interferometry (DSPI) yield a phase map modulo 2π, phase unwrapping is the final step of any data evaluation process. The performance of a recently published algorithm used to unwrap DSPI phase maps is investigated. The algorithm is based on a least-squares minimization technique that is solvable by the discrete cosine transform. When phase inconsistencies are present, they are handled by exclusion of invalid pixels from the unwrapping process through the assignment of zero-valued weights. Then the weighted unwrapping problem is solved in an iterative manner by a preconditioned conjugate-gradient method. The evaluation is carried out with computer-simulated DSPI phase maps, an approach that permits the generation of phase fields without inconsistencies, which are then used to calculate phase deviations as a function of the iteration number. Real data are also used to illustrate the performance of the algorithm.

© 1998 Optical Society of America

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References

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    [CrossRef]
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1997

M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” J. Opt. Soc. Am. A 12, 8381–8390 (1997).

1996

1995

1994

R. T. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser Eng. 21, 199–239 (1994).
[CrossRef]

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
[CrossRef]

1993

1991

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14, 25–37 (1991).
[CrossRef]

H. A. Vrooman, A. A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[CrossRef] [PubMed]

1989

1982

Bryanston-Cross, P. J.

R. T. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser Eng. 21, 199–239 (1994).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

Buckland, J. R.

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

Cusack, R.

Davila, A.

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Fringe’93, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

Díaz, F. V.

G. H. Kaufmann, F. V. Díaz, “TV holography applied to the measurement of residual deformations in ion implanted steel plates,” in Fringe’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 322–324.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Galizzi, G. E.

Ghiglia, D. C.

Goldrein, H. T.

Huntley, J. M.

Itoh, K.

Judge, R. T.

R. T. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser Eng. 21, 199–239 (1994).
[CrossRef]

Judge, T. R.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

Kaufmann, G. H.

D. Kerr, G. H. Kaufmann, G. E. Galizzi, “Unwrapping of interferometric phase-fringe maps by the discrete cosine transform,” Appl. Opt. 35, 810–816 (1996).
[CrossRef] [PubMed]

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Fringe’93, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

G. H. Kaufmann, F. V. Díaz, “TV holography applied to the measurement of residual deformations in ion implanted steel plates,” in Fringe’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 322–324.

Kerr, D.

D. Kerr, G. H. Kaufmann, G. E. Galizzi, “Unwrapping of interferometric phase-fringe maps by the discrete cosine transform,” Appl. Opt. 35, 810–816 (1996).
[CrossRef] [PubMed]

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Fringe’93, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

Kreis, Th.

Kujawinska, M.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 141–193.

Løkberg, O. J.

O. J. Løkberg, “Recent developments in video speckle interferometry,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 157–194.

Maas, A. A. M.

Marroquin, J. L.

M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” J. Opt. Soc. Am. A 12, 8381–8390 (1997).

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Rastogi, P. K.

P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 41–98.

Rivera, M.

M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” J. Opt. Soc. Am. A 12, 8381–8390 (1997).

J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

Robinson, D. W.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14, 25–37 (1991).
[CrossRef]

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

Rodriguez-Vera, R.

M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” J. Opt. Soc. Am. A 12, 8381–8390 (1997).

Romero, L. A.

Saldner, H.

Servin, M.

M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” J. Opt. Soc. Am. A 12, 8381–8390 (1997).

Sirohi, R. S.

R. S. Sirohi, “Speckle methods in experimental mechanics,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 99–155.

Spik, A.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14, 25–37 (1991).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Towers, D. P.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

Turner, S. R. E.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

Vrooman, H. A.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Laser Eng.

R. T. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser Eng. 21, 199–239 (1994).
[CrossRef]

Opt. Lasers Eng.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI,” Opt. Lasers Eng. 14, 239–282 (1991).
[CrossRef]

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14, 25–37 (1991).
[CrossRef]

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), pp. 514–521.

A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Fringe’93, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 41–98.

R. S. Sirohi, “Speckle methods in experimental mechanics,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 99–155.

O. J. Løkberg, “Recent developments in video speckle interferometry,” in Speckle Metrology, R. S. Sirohi, ed. (Dekker, New York, 1993), pp. 157–194.

K. Creath, “Temporal phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 141–193.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

G. H. Kaufmann, F. V. Díaz, “TV holography applied to the measurement of residual deformations in ion implanted steel plates,” in Fringe’97, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 322–324.

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Figures (11)

Fig. 1
Fig. 1

Wrapped-phase map determined by the Fourier-transform method for modulated DSPI carrier fringes.

Fig. 2
Fig. 2

Weighting array used to unwrap the phase map shown in Fig. 1.

Fig. 3
Fig. 3

Plot of the rms phase deviation versus the iteration number with the weighting array shown in Fig. 2: solid curve, PCG algorithm; dashed curve, Picard iterative method.

Fig. 4
Fig. 4

Unwrapped-phase distribution of Fig. 1 after three iterations with the PCG algorithm.

Fig. 5
Fig. 5

Weighting array for a vertical rectangular region of missing data.

Fig. 6
Fig. 6

Phase rms deviation versus the iteration number with the weighting array shown in Fig. 5 obtained across (c) consistent pixels and (i) a region of missing data: solid curves, PCG algorithm; dashed curves, Picard iterative method.

Fig. 7
Fig. 7

Wrapped-phase map determined by the Fourier transform method for a circular DSPI fringe pattern.

Fig. 8
Fig. 8

rms phase deviation versus the iteration number for the wrapped-phase map shown in Fig. 7: solid curves, PCG algorithm; dashed curve, Picard iterative method.

Fig. 9
Fig. 9

Unwrapped-phase distribution of Fig. 7 with the PCG algorithm.

Fig. 10
Fig. 10

Unwrapping of a phase map presenting multiple isolated regions: (a) original phase map; (b) weighting array; (c) unwrapped-phase map.

Fig. 11
Fig. 11

Unwrapping of a real phase map: (a) experimental DSPI fringes generated upon a steel plate subjected to pulsed ion implantation; (b) original phase map; (c) weighting array; (d) unwrapped-phase map.

Equations (13)

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ϕ m + 1 , n - 2 ϕ m , n + ϕ m - 1 , n + ϕ m , n + 1 - 2 ϕ m , n + ϕ m , n - 1 = ρ m , n ,
ρ m , n = Δ m , n x - Δ m - 1 , n x + Δ m , n y - Δ m , n - 1 y ,
Δ m , n x = W ψ m + 1 , n - ψ m , n ,     m = 0 , ,   N - 2 , n = 0 , ,   N - 1 ,
Δ m , n y = W ψ m , n + 1 - ψ m , n ,     m = 0 , ,   N - 1 , n = 0 , ,   N - 2 ,
P ϕ = ρ ,
Q ϕ = c ,
β k = r k - 1 T z k - 1 / r k - 2 T z k - 2 , p k = z k - 1 + β k p k - 1 .
α k = r k - 1 T z k - 1 / p k T Qp k , ϕ k = ϕ k - 1 + α k p k , r k = r k - 1 + α k Qp k .
I m , n = | Ae i θ + A m , n | 2 = | Ae i θ + FT - 1 H u , v FT exp i α m , n U m , n | 2 ,
a = N / 2 r .
I m , n 1 = | Ae i θ + 1 - δ m , n A m , n + δ m , n A m , n i | 2 ,
σ = 1 M - 1   ϕ m , n - ϕ m , n 2 1 / 2 ,
= 1 M     | ϕ m , n - ϕ m , n | ϕ m , n   100 % ,

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