Abstract

The performance of a minimum L 0-norm unwrapping algorithm is investigated by use of synthetic digital speckle-pattern interferometry (DSPI) wrapped phase maps that simulate experimentally obtained data. This algorithm estimates its own weights to mask inconsistent pixels. Particular features usually included in DSPI wrapped phase distributions, such as shears, speckle noise, fringe cuts, object physical limits, and superimposed phase maps, are analyzed. Some adequate approaches to solving these features are discussed. Finally, it is shown that a complex case in which shears and fringe cuts coexist in the wrapped phase cannot be solved satisfactorily with the minimum L 0-norm algorithm by itself. To cope with this problem, we propose a new scheme.

© 1998 Optical Society of America

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References

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  1. O. J. Løkberg, “Recent developments in video speckle interferometry,” in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, New York, 1993), pp. 157–194.
  2. 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.
  3. 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.
  4. 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.
  5. A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
    [CrossRef]
  6. S. M. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
    [CrossRef]
  7. J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A 12, 1990–1996 (1995).
    [CrossRef]
  8. R. T. Judge, P. J. Bryanston-Cross, “A review of phase unwrapping techniques in fringe analysis,” Opt. Laser Eng. 21, 199–239 (1994).
    [CrossRef]
  9. J. A. Quiroga, E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
    [CrossRef] [PubMed]
  10. J. A. Quiroga, A. Gonzalez-Cano, E. Bernabeu, “Phase-unwrapping algorithm based on an adaptive criterion,” Appl. Opt. 34, 2560–2563 (1995).
    [CrossRef] [PubMed]
  11. R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef] [PubMed]
  12. J. L. Marroquin, M. Rivera, “Quadratic regularization functional for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  13. J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping of noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [CrossRef] [PubMed]
  14. C. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Disk-growing algorithm for phase-map unwrapping: application to speckle interferograms,” Appl. Opt. 35, 240–247 (1996).
    [CrossRef] [PubMed]
  15. P. G. Charette, I. W. Hunter, “Robust phase-unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506–3513 (1996).
    [CrossRef] [PubMed]
  16. M. Arevalillo Herraez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
    [CrossRef]
  17. K. A. Stetson, J. Wahid, P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 20, 4830–4838 (1997).
    [CrossRef]
  18. T. J. Fynn, “Two-dimensional phase unwrapping algorithm with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
    [CrossRef]
  19. M. Rivera, J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36, 8381–8390 (1997).
    [CrossRef]
  20. H. Takajo, T. Takahashi, “Least-squares phase estimation from phase differences,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  21. H. Takajo, T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  22. 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]
  23. 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]
  24. G. H. Kaufmann, G. E. Galizzi, “Unwrapping of electronic speckle pattern interferometry phase maps: evaluation of an iterative weighted algorithm,” Opt. Eng. 37, 622–628 (1998).
    [CrossRef]
  25. G. H. Kaufmann, G. E. Galizzi, P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of DSPI phase maps,” Appl. Opt. 37, 3076–3084 (1998).
    [CrossRef]
  26. D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2012 (1996).
    [CrossRef]
  27. A. Davila, G. H. Kaufmann, D. Kerr, “Digital processing of ESPI addition fringes,” in Fringe’93—Second International Workshop in Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

1998 (2)

G. H. Kaufmann, G. E. Galizzi, “Unwrapping of electronic speckle pattern interferometry phase maps: evaluation of an iterative weighted algorithm,” Opt. Eng. 37, 622–628 (1998).
[CrossRef]

G. H. Kaufmann, G. E. Galizzi, P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of DSPI phase maps,” Appl. Opt. 37, 3076–3084 (1998).
[CrossRef]

1997 (3)

1996 (5)

1995 (6)

1994 (3)

1993 (1)

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

1988 (2)

Arevalillo Herraez, M.

Bernabeu, E.

Berwart, L.

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]

Buckland, J. R.

Burton, D. R.

Charette, P. G.

Clegg, D. B.

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—Second International Workshop in Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

De Veuster, C.

Farris-Manning, P. J.

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

Fynn, T. J.

Galizzi, G. E.

Gauthier, P.

K. A. Stetson, J. Wahid, P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 20, 4830–4838 (1997).
[CrossRef]

Ghiglia, D. C.

Glover, G. H.

S. M. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Goldrein, H. T.

Gonzalez-Cano, A.

Gray, A. L.

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

Hunter, I. W.

Huntley, J. M.

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]

Kaufmann, G. H.

G. H. Kaufmann, G. E. Galizzi, P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of DSPI phase maps,” Appl. Opt. 37, 3076–3084 (1998).
[CrossRef]

G. H. Kaufmann, G. E. Galizzi, “Unwrapping of electronic speckle pattern interferometry phase maps: evaluation of an iterative weighted algorithm,” Opt. Eng. 37, 622–628 (1998).
[CrossRef]

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—Second International Workshop in Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

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—Second International Workshop in Automatic Processing of Fringe Patterns, W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1993), pp. 339–346.

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.

Lalor, M. J.

Lion, Y.

Løkberg, O. J.

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

Marroquin, J. L.

Napel, S.

S. M. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Pelc, N. J.

S. M. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Quiroga, J. A.

Renotte, Y.

Rivera, M.

Robinson, D. W.

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.

Romero, L. A.

Ruiz, P. D.

Servin, M.

Slangen, P.

Song, S. M.

S. M. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Stetson, K. A.

K. A. Stetson, J. Wahid, P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 20, 4830–4838 (1997).
[CrossRef]

Takahashi, T.

Takajo, H.

Turner, S. R. E.

Wahid, J.

K. A. Stetson, J. Wahid, P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 20, 4830–4838 (1997).
[CrossRef]

Appl. Opt. (11)

J. A. Quiroga, E. Bernabeu, “Phase-unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
[CrossRef] [PubMed]

J. A. Quiroga, A. Gonzalez-Cano, E. Bernabeu, “Phase-unwrapping algorithm based on an adaptive criterion,” Appl. Opt. 34, 2560–2563 (1995).
[CrossRef] [PubMed]

R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
[CrossRef] [PubMed]

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping of noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
[CrossRef] [PubMed]

C. De Veuster, P. Slangen, Y. Renotte, L. Berwart, Y. Lion, “Disk-growing algorithm for phase-map unwrapping: application to speckle interferograms,” Appl. Opt. 35, 240–247 (1996).
[CrossRef] [PubMed]

P. G. Charette, I. W. Hunter, “Robust phase-unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506–3513 (1996).
[CrossRef] [PubMed]

M. Arevalillo Herraez, D. R. Burton, M. J. Lalor, D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
[CrossRef]

K. A. Stetson, J. Wahid, P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 20, 4830–4838 (1997).
[CrossRef]

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

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]

G. H. Kaufmann, G. E. Galizzi, P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of DSPI phase maps,” Appl. Opt. 37, 3076–3084 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

A. L. Gray, P. J. Farris-Manning, “Repeat-pass interferometry with airborne synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens. 31, 180–191 (1993).
[CrossRef]

IEEE Trans. Image Process. (1)

S. M. Song, S. Napel, N. J. Pelc, G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Eng. (1)

G. H. Kaufmann, G. E. Galizzi, “Unwrapping of electronic speckle pattern interferometry phase maps: evaluation of an iterative weighted algorithm,” Opt. Eng. 37, 622–628 (1998).
[CrossRef]

Opt. Laser Eng. (1)

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

Other (5)

O. J. Løkberg, “Recent developments in video speckle interferometry,” in Speckle Metrology, R. S. Sirohi, ed. (Marcel 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.

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

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Figures (6)

Fig. 1
Fig. 1

Unwrapping of a circular phase map with 10 fringes and a shear for ∊0 = 0.01: (a) original wrapped phase map, (b) U 0(i, j) weights from the initial guess of ϕ(i, j) = 0, (c) V 0(i, j) weights from the initial guess of ϕ(i, j) = 0, (d) rewrapped phase after iteration l = 1, (e) U 1(i, j) weights, (f) V 1(i, j) weights, (g) rewrapped solution after iteration l = 2, (h) U 8(i, j) weights, (i) V 8(i, j) weights, (j) rewrapped phase after iteration l = 9, and (k) unwrapped phase after iteration l = 9.

Fig. 2
Fig. 2

Unwrapping of a circular phase map with three fringes and a shear: (a) original wrapped phase map, (b) rewrapped phase after iteration l = 6 and for ∊0 = 0.01, (c) rewrapped phase after iteration l = 6 and for ∊0 = 0.001, (d) rewrapped phase after iteration l = 15 and for ∊0 = 0.1, and (e) plot of U(i, j) as a function of f i,j for different values of ∊0.

Fig. 3
Fig. 3

Unwrapping of a circular phase map generated from unfiltered DSPI fringes: (a) original wrapped phase map, (b) distribution of inconsistent clusters obtained from (a), (c) unwrapped phase after 30 iterations of the PCG algorithm by use of the weights shown in (b), (d) unwrapped phase after iteration l = 8 of the minimum L 0-norm algorithm with ∊0 = 0.001, and (e) plot of the rms phase deviation as a function of the iteration number for the minimum L 0-norm algorithm for ∊0 = 0.1, 0.01, 0.001.

Fig. 4
Fig. 4

Unwrapping of a phase map generated from modulated carrier DSPI fringes presenting discontinuities resulting from defficient filtering: (a) original wrapped phase map, (b) weighting array for the PCG algorithm, (c) rewrapped phase map obtained with the PCG algorithm, and (d) rewrapped phase map obtained with the minimum L 0-norm algorithm after iteration l = 1 and for ∊0 = 0.01.

Fig. 5
Fig. 5

Unwrapping of a phase map presenting two superimposed fringe patterns: (a) original wrapped phase map, (b) U 20(i, j) weights, (c) V 20(i, j) weights, (d) rewrapped phase map obtained with the minimum L 0-norm algorithm after iteration l = 20 and for ∊0 = 0.005, and (e) unwrapped phase map.

Fig. 6
Fig. 6

Unwrapping of a composite phase map presenting two superimposed fringe patterns, fringe discontinuities, and object edges by use of the minimum L 0-norm algorithm with modified weights: (a) original wrapped phase map, (b) U 0(i, j) weights, (c) binary mask, (d) rewrapped phase map, and (e) unwrapped phase map.

Equations (12)

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ψ i , j = ϕ i , j + 2 π k i , j ,
i = 0 N - 2 j = 0 N - 1   | ϕ i + 1 , j - ϕ i , j - f i , j | p + i = 0 N - 1 j = 0 N - 2   | ϕ i , j + 1 - ϕ i , j - g i , j | p ,
f i , j = W ψ i + 1 , j - ψ i , j ,     i = 0 , ,   N - 2 , j = 0 , ,   N - 1 ,
g i , j = W ψ i , j + 1 - ψ i , j ,     i = 0 , ,   N - 1 , j = 0 , ,   N - 2 ,
ϕ i + 1 , j - ϕ i , j U i ,   j + ϕ i , j + 1 - ϕ i , j V i ,   j - ϕ i , j - ϕ i - 1 , j U i - 1 ,   j - ϕ i , j - ϕ i , j - 1 V i ,   j - 1 = c i ,   j ,
c i ,   j = f i , j U i ,   j - f i - 1 , j U i - 1 ,   j + g i , j V i ,   j - g i , j - 1 V i ,   j - 1 ,
U i ,   j = | ϕ i + 1 , j - ϕ i , j - f i , j | p - 2 i = 0 , ,   N - 2 , j = 0 , ,   N - 1 0 otherwise ,
V i ,   j = | ϕ i + 1 , j - ϕ i , j - g i , j | p - 2 i = 0 , ,   N - 1 , j = 0 , ,   N - 2 0 otherwise .
U i ,   j = 0 | ϕ i + 1 , j - ϕ i , j - f i , j | 2 + 0 ,
V i ,   j = 0 | ϕ i , j + 1 - ϕ i , j - g i , j | 2 + 0 ,
R i ,   j = W ψ i ,   j - ϕ l i ,   j .
ϕ i ,   j = ϕ l i ,   j + W - 1 R i ,   j ,

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