Abstract

We report a method for two-dimensional phase unwrapping based on the transport of intensity equation (TIE). Given a wrapped phase profile, we generate an auxiliary complex field and propagate it to small distances to simulate two intensity images on closely spaced planes. Using the longitudinal intensity derivative of the auxiliary field as an input, the TIE is solved by employing the regularized Fourier-transform-based approach. The resultant phase profile is automatically in the unwrapped form, as it has been obtained as a solution of a partial differential equation rather than as an argument of a complex-valued function. Our simulations and experimental results suggest that this approach is fast and accurate and provides a simple and practical solution for routine phase unwrapping tasks in interferometry and digital holography.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  8. J. M. Huntley, “Noise immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [Crossref]
  9. J. R. Buckland, J. M. Huntley, and S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum cost matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  14. M. D. Pratt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  23. G. Goldstein and K. Creath, “Quantitative phase microscopy: automated background leveling techniques and smart temporal phase unwrapping,” Appl. Opt. 54, 5175–5185 (2015).
    [Crossref]
  24. S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. 49, 4735–4745 (2010).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  30. K. Ichikawa, A. W. Lohmann, and M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
    [Crossref]
  31. J. Primot, “Three-wave lateral shearing interferometer,” Appl. Opt. 32, 6242–6249 (1993).
    [Crossref]
  32. T. Gureyev, A. Roberts, and K. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1942 (1995).
    [Crossref]
  33. T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
    [Crossref]
  34. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [Crossref]
  35. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  36. K. Ishizuka, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
    [Crossref]
  37. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
    [Crossref]
  38. D. Paganin and K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
    [Crossref]
  39. V. V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using transport of intensity equation,” Micron 33, 411–416 (2002).
    [Crossref]
  40. K. Khare, P. T. Samsheerali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Express 21, 2581–2591 (2013).
    [Crossref]
  41. P. T. Samsheerali, K. Khare, and J. Joseph, “Quantitative phase imaging using single shot digital holography,” Opt. Commun. 319, 85–89 (2014).
    [Crossref]
  42. M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement using classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
    [Crossref]

2015 (3)

2014 (4)

2013 (1)

2012 (1)

2010 (1)

2007 (1)

L. Aiello, “Green’s formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750–755 (2007).
[Crossref]

2006 (1)

2005 (1)

K. Ishizuka, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

2003 (1)

2002 (2)

D. Paganin and K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
[Crossref]

V. V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

1998 (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

1997 (1)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

1996 (3)

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

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

M. D. Pratt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[Crossref]

1995 (3)

1994 (2)

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[Crossref]

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

1993 (2)

1991 (1)

1989 (1)

1988 (3)

1983 (1)

1982 (2)

1977 (1)

Aiello, L.

L. Aiello, “Green’s formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750–755 (2007).
[Crossref]

Albertazzi, A. G.

Almoro, P.

Arsenin, V. I.

A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Awatsuji, Y.

Bai, J.

Barbastathis, G.

Bone, D. J.

Bryanston-Cross, P. J.

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

Buckland, J. R.

Chen, X.

Chen, Z.

Cheng, Z.

Creath, K.

da Silva Maciel, L.

Francos, J.

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

Fried, D. L.

Friedlander, B.

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

Galizzi, G. E.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pratt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Goldstein, G.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Graef, M. D.

V. V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Gureyev, T.

Gureyev, T. E.

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Heshmat, S.

Huang, H. Y. H.

Huang, W.

Huntley, J. M.

Ichikawa, K.

Iglesias, I.

Ina, H.

Inoue, J.

Ishizuka, K.

K. Ishizuka, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

Ito, Y.

Itoh, K.

Jha, A. K.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement using classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Joseph, J.

P. T. Samsheerali, K. Khare, and J. Joseph, “Quantitative phase imaging using single shot digital holography,” Opt. Commun. 319, 85–89 (2014).
[Crossref]

K. Khare, P. T. Samsheerali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Express 21, 2581–2591 (2013).
[Crossref]

Judge, T. R.

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

Kaufmann, G. H.

Kerr, D.

Khare, K.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement using classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

P. T. Samsheerali, K. Khare, and J. Joseph, “Quantitative phase imaging using single shot digital holography,” Opt. Commun. 319, 85–89 (2014).
[Crossref]

K. Khare, P. T. Samsheerali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Express 21, 2581–2591 (2013).
[Crossref]

Kobayashi, S.

Lee, Y.

Ling, T.

Liu, D.

Liu, Y.

Lohmann, A. W.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Marrouquin, J. L.

Matoba, O.

Miao, L.

Miyamoto, N.

Nishio, K.

Nishiyama, S.

Nugent, K.

Nugent, K. A.

D. Paganin and K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

Osten, W.

Paganin, D.

D. Paganin and K. A. Nugent, “Phase measurement of waves that obey nonlinear equations,” Opt. Lett. 27, 622–624 (2002).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

Pedrini, G.

Prabhakar, S.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement using classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Pratt, M. D.

M. D. Pratt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[Crossref]

D. C. Ghiglia and M. D. Pratt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Primot, J.

Pritt, M. D.

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[Crossref]

Rivera, M.

Roberts, A.

Roddier, F.

Saldner, H.

Samsheerali, P. T.

P. T. Samsheerali, K. Khare, and J. Joseph, “Quantitative phase imaging using single shot digital holography,” Opt. Commun. 319, 85–89 (2014).
[Crossref]

K. Khare, P. T. Samsheerali, and J. Joseph, “Single shot high resolution digital holography,” Opt. Express 21, 2581–2591 (2013).
[Crossref]

Schofield, M. A.

Shen, Y.

Shipman, J. S.

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[Crossref]

Singh, M.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement using classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Singh, R. P.

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement using classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Tahara, T.

Takeda, M.

Teague, M. R.

Tian, L.

Tikhonov, A. N.

A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Tomioka, S.

Turner, S. R. E.

Ura, S.

Volkov, V. V.

V. V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Xia, P.

Yang, Y.

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Zhang, L.

Zhang, Z.

Zhu, Y.

M. A. Schofield and Y. Zhu, “Fast phase unwrapping algorithm for interferometric applications,” Opt. Lett. 28, 1194–1196 (2003).
[Crossref]

V. V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Appl. Opt. (13)

J. M. Huntley, “Noise immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[Crossref]

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

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[Crossref]

K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
[Crossref]

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

J. M. Huntley and H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[Crossref]

G. Goldstein and K. Creath, “Quantitative phase microscopy: automated background leveling techniques and smart temporal phase unwrapping,” Appl. Opt. 54, 5175–5185 (2015).
[Crossref]

S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. 49, 4735–4745 (2010).
[Crossref]

L. da Silva Maciel and A. G. Albertazzi, “Swarm-based algorithm for phase unwrapping,” Appl. Opt. 53, 5502–5509 (2014).
[Crossref]

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
[Crossref]

K. Ichikawa, A. W. Lohmann, and M. Takeda, “Phase retrieval based on the irradiance transport equation and the Fourier transform method: experiments,” Appl. Opt. 27, 3433–3436 (1988).
[Crossref]

J. Primot, “Three-wave lateral shearing interferometer,” Appl. Opt. 32, 6242–6249 (1993).
[Crossref]

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (2)

M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[Crossref]

M. D. Pratt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
[Crossref]

IEEE Trans. Signal Process. (1)

B. Friedlander and J. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[Crossref]

J. Electron Microsc. (1)

K. Ishizuka, “Phase measurement of atomic resolution image using transport of intensity equation,” J. Electron Microsc. 54, 191–197 (2005).
[Crossref]

J. Opt. Soc. Am. (3)

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

Micron (1)

V. V. Volkov, Y. Zhu, and M. D. Graef, “A new symmetrized solution for phase retrieval using transport of intensity equation,” Micron 33, 411–416 (2002).
[Crossref]

Opt. Commun. (2)

T. E. Gureyev and K. A. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
[Crossref]

P. T. Samsheerali, K. Khare, and J. Joseph, “Quantitative phase imaging using single shot digital holography,” Opt. Commun. 319, 85–89 (2014).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (2)

L. Aiello, “Green’s formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750–755 (2007).
[Crossref]

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

Opt. Lett. (3)

Phys. Rev. A (1)

M. Singh, K. Khare, A. K. Jha, S. Prabhakar, and R. P. Singh, “Accurate multipixel phase measurement using classical-light interferometry,” Phys. Rev. A 91, 021802 (2015).
[Crossref]

Phys. Rev. Lett. (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[Crossref]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[Crossref]

Other (5)

A. N. Tikhonov and V. I. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

P. Ferraro, A. Wax, and Z. Zalevsky, eds., Coherent Light Microscopy: Imaging and Quantitative Phase Analysis (Springer, 2011).

D. C. Ghiglia and M. D. Pratt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (9)

Fig. 1.
Fig. 1.

Illustration of typical wavefront phase reconstruction using the TIE-based approach. The arrow indicates the nominal direction of wave propagation, and u ( x , y , z ) denotes the scalar field in the corresponding z plane.

Fig. 2.
Fig. 2.

Flow chart showing the basic steps involved in phase unwrapping using the TIE-based approach.

Fig. 3.
Fig. 3.

Scheme used for symmetrization prior to TIE-based phase unwrapping. The top left corner, with the letters “TIE,” corresponds to the position of the wrapped phase image, which is reflected as shown. This extended phase map is used to form a complex field with a uniform amplitude. This field is propagated in both the + z and - z directions a distance of 10 μm. Using these propagated intensities, the intensity derivative is approximated, and the TIE is solved using Eq. (8). The top left corner of the resultant phase solution is taken as the required unwrapped phase map.

Fig. 4.
Fig. 4.

(a), (c) Quadratic and cubic phase maps used for illustration of phase unwrapping per Eqs. (14) and (15), respectively; (b),(d) wrapped version of the phase maps in (a),(c).

Fig. 5.
Fig. 5.

Unwrapped quadratic and cubic phase maps using (a), (c) the Goldstein et al. method and (b),(d) the TIE-based method. Arrows in (a) and (c) indicate edge row/column discontinuity artifacts.

Fig. 6.
Fig. 6.

Phase unwrapping illustration for mask object: (a) original phase function, (b) wrapped version of the phase function, (c) phase unwrapping result using the Goldstein et al. method, (d) phase unwrapping result using the TIE-based method.

Fig. 7.
Fig. 7.

Phase unwrapping illustration for the experimental interferogram data: (a) one of the four phase-shifting interferograms, (b) wrapped version of the phase function, (c) phase unwrapping result using the Goldstein et al. method, (d) phase unwrapping result using the TIE-based method.

Fig. 8.
Fig. 8.

Phase unwrapping illustration for the experimental digital holographic microscopy data: (a) digital hologram of a rat neuron sample, (b) wrapped version of the phase function, (c) phase unwrapping result using the Goldstein et al. method, (d) phase unwrapping result using the TIE-based method.

Fig. 9.
Fig. 9.

Zoomed view of phase unwrapping results in Fig. 8: (a) wrapped phase map of a region showing artifacts in the unwrapped phase map obtained by the Goldstein et al. method [this map corresponds to a region in the top left corner of Fig. 8(b)], (b) phase unwrapping result using the Goldstein et al. method, (c) binary image showing branch cuts obtained as an intermediate step in the Goldstein method, (d) phase unwrapping result using the TIE-based method.

Tables (1)

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Table 1. RMS Error and Timing Performance for the Goldstein et al. and TIE-based Methodsa

Equations (15)

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ϕ ( x , y ) = ϕ w ( x , y ) + 2 n ( x , y ) π ,
u ( x , y ; z ) = I ( x , y ; z ) exp [ i ϕ ( x , y ; z ) ] ,
( i z + 1 2 k x y 2 + k ) u ( x , y ; z ) = 0 .
x y = ( x , y ) .
k I z = x y · ( I x y ϕ ) .
ϕ ( x , y ; z ) = k x y 2 x y · [ 1 I x y x y 2 ( I z ) ] .
x y 2 g ( x , y ) = F 1 { D ( f x , f y ) F [ g ] D ( f x , f y ) 2 + ε 2 } ,
ϕ ( x , y ; z ) = k I x y 2 ( I z ) .
u 0 ( x , y ; 0 ) = exp [ i ϕ w ( x , y ) ] .
u 0 ( x , y ; ± Δ z ) = u 0 ( x , y ; 0 ) * h ( x , y ; ± Δ z ) .
h ( x , y ; z ) = exp ( i k R ) 2 π R ( i k + 1 R ) z R ,
H ( f x , f y ; z ) = F [ h ( x , y ; z ) ] = exp [ i z k 2 4 π 2 ( f x 2 + f y 2 ) ] .
I z = | u 0 ( x , y ; Δ z ) | 2 | u 0 ( x , y ; Δ z ) | 2 2 Δ z + O ( ( Δ z ) 2 ) .
ϕ 2 ( x , y ) = α ( x 2 + y 2 ) ,
ϕ 3 ( x , y ) = β ( x 3 + y 3 ) .

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