Abstract

A wide range of interferometric techniques recover phase information that is mathematically wrapped on the interval (-π,π]. Obtaining the true unwrapped phase is a longstanding problem. We present an algorithm that solves the phase unwrapping problem, using a combination of Fourier techniques. The execution time for our algorithm is equivalent to the computation time required for performing eight fast Fourier transforms and is stable against noise and residues present in the wrapped phase. We have extended the algorithm to handle data of arbitrary size. We expect the state of the art of existing interferometric applications, including the possibility for real-time phase recovery, to benefit from our algorithm.

© 2003 Optical Society of America

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References

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  1. R. Bamler and P. Hartl, Inverse Probl. 14, R1 (1998).
    [CrossRef]
  2. E. S. Kasischke, J. M. Melack, and M. C. Dobson, Remote Sens. Environ. 59, 141 (1997).
    [CrossRef]
  3. D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
    [CrossRef] [PubMed]
  4. R. F. Hanssen, T. M. Weckwerth, H. A. Zebker, and R. Klees, Science 283, 1297 (1999).
    [CrossRef] [PubMed]
  5. R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
    [CrossRef]
  6. G. Jones III and B. Bhanu, Pattern Recogn. 34, 469 (2001).
    [CrossRef]
  7. P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
    [CrossRef] [PubMed]
  8. X. Su and W. Chen, Opt. Lasers Eng. 35, 263 (2001), and references therein.
    [CrossRef]
  9. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
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    [CrossRef] [PubMed]
  11. E. Volkl, L. F. Allard, and D. C. Joy, eds., Introduction to Electron Holography (Plenum, New York, 1999).
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  13. D. L. Fried, Opt. Commun. 200, 43 (2001).
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  14. L. Xue and X. Su, Appl. Opt. 40, 1207 (2001).
    [CrossRef]
  15. A. Baldi, Appl. Opt. 40, 1187 (2001).
    [CrossRef]
  16. I. Lyuboshenko, Appl. Opt. 39, 4817 (2000).
    [CrossRef]
  17. B. Gutmann and H. Weber, Appl. Opt. 39, 4802 (2000).
    [CrossRef]
  18. S. Stramaglia, A. Refice, and L. Guerriero, Physica A 276, 521 (2000).
    [CrossRef]
  19. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, Meas. Sci. Technol. 11, 1480 (2000).
    [CrossRef]
  20. J. Strand and T. Taxt, Appl. Opt. 38, 4333 (1999).
    [CrossRef]
  21. H. A. Aebischer and S. Waldner, Opt. Commun. 162, 205 (1999).
    [CrossRef]
  22. R. C. Hardie, M. I. Younus, and J. Blackshire, Appl. Opt. 37, 4468 (1998).
    [CrossRef]
  23. L. Guerriero, G. Nico, G. Pasquariello, and S. Stramaglia, Appl. Opt. 37, 3053 (1998).
    [CrossRef]
  24. X. Xie, M. J. Lalor, D. R. Burton, and M. M. Shaw, Opt. Lasers Eng. 29, 49 (1998).
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    [CrossRef]
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    [CrossRef]
  29. V. V. Volkov, Y. Zhu, and M. De Graef, Micron. 33, 411 (2002).
    [CrossRef]
  30. M. D. Pritt and J. S. Shipman, IEEE Trans. Geosci. Remote Sens. 32, 706 (1994).
    [CrossRef]
  31. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, New York, 1989).
  32. L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
    [CrossRef] [PubMed]

2002 (1)

V. V. Volkov, Y. Zhu, and M. De Graef, Micron. 33, 411 (2002).
[CrossRef]

2001 (6)

G. Jones III and B. Bhanu, Pattern Recogn. 34, 469 (2001).
[CrossRef]

P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
[CrossRef] [PubMed]

X. Su and W. Chen, Opt. Lasers Eng. 35, 263 (2001), and references therein.
[CrossRef]

D. L. Fried, Opt. Commun. 200, 43 (2001).
[CrossRef]

L. Xue and X. Su, Appl. Opt. 40, 1207 (2001).
[CrossRef]

A. Baldi, Appl. Opt. 40, 1187 (2001).
[CrossRef]

2000 (6)

I. Lyuboshenko, Appl. Opt. 39, 4817 (2000).
[CrossRef]

B. Gutmann and H. Weber, Appl. Opt. 39, 4802 (2000).
[CrossRef]

S. Stramaglia, A. Refice, and L. Guerriero, Physica A 276, 521 (2000).
[CrossRef]

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, Meas. Sci. Technol. 11, 1480 (2000).
[CrossRef]

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

1999 (3)

R. F. Hanssen, T. M. Weckwerth, H. A. Zebker, and R. Klees, Science 283, 1297 (1999).
[CrossRef] [PubMed]

J. Strand and T. Taxt, Appl. Opt. 38, 4333 (1999).
[CrossRef]

H. A. Aebischer and S. Waldner, Opt. Commun. 162, 205 (1999).
[CrossRef]

1998 (7)

R. C. Hardie, M. I. Younus, and J. Blackshire, Appl. Opt. 37, 4468 (1998).
[CrossRef]

L. Guerriero, G. Nico, G. Pasquariello, and S. Stramaglia, Appl. Opt. 37, 3053 (1998).
[CrossRef]

X. Xie, M. J. Lalor, D. R. Burton, and M. M. Shaw, Opt. Lasers Eng. 29, 49 (1998).
[CrossRef]

D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
[CrossRef] [PubMed]

R. Bamler and P. Hartl, Inverse Probl. 14, R1 (1998).
[CrossRef]

M. J. Hÿtch, E. Snoeck, and R. Kilaas, Ultramicroscopy 74, 131 (1998).
[CrossRef]

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

1997 (3)

E. S. Kasischke, J. M. Melack, and M. C. Dobson, Remote Sens. Environ. 59, 141 (1997).
[CrossRef]

K. A. Stetson, J. Wahid, and P. Gauthier, Appl. Opt. 36, 4830 (1997).
[CrossRef] [PubMed]

H. Kadono, H. Takei, and S. Toyooka, Opt. Lasers Eng. 26, 151 (1997).
[CrossRef]

1996 (1)

1994 (1)

M. D. Pritt and J. S. Shipman, IEEE Trans. Geosci. Remote Sens. 32, 706 (1994).
[CrossRef]

1959 (1)

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[CrossRef]

Aebischer, H. A.

H. A. Aebischer and S. Waldner, Opt. Commun. 162, 205 (1999).
[CrossRef]

Aharonov, Y.

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[CrossRef]

Akella, A.

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

Arthern, R. J.

D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
[CrossRef] [PubMed]

Baldi, A.

Bamler, R.

R. Bamler and P. Hartl, Inverse Probl. 14, R1 (1998).
[CrossRef]

Bhanu, B.

G. Jones III and B. Bhanu, Pattern Recogn. 34, 469 (2001).
[CrossRef]

Blackshire, J.

Bohm, D.

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[CrossRef]

Buelow, S. J.

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

Bürgmann, R.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

Burton, D. R.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, Meas. Sci. Technol. 11, 1480 (2000).
[CrossRef]

X. Xie, M. J. Lalor, D. R. Burton, and M. M. Shaw, Opt. Lasers Eng. 29, 49 (1998).
[CrossRef]

Chen, W.

X. Su and W. Chen, Opt. Lasers Eng. 35, 263 (2001), and references therein.
[CrossRef]

d'Alessio, M.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

De Graef, M.

V. V. Volkov, Y. Zhu, and M. De Graef, Micron. 33, 411 (2002).
[CrossRef]

Dobson, M. C.

E. S. Kasischke, J. M. Melack, and M. C. Dobson, Remote Sens. Environ. 59, 141 (1997).
[CrossRef]

Downing, P. E.

P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
[CrossRef] [PubMed]

Fielding, E.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, New York, 1989).

Fried, D. L.

D. L. Fried, Opt. Commun. 200, 43 (2001).
[CrossRef]

Funk, D. J.

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

Gahagan, K. T.

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

Gauthier, P.

Gdeisat, M. A.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, Meas. Sci. Technol. 11, 1480 (2000).
[CrossRef]

Guerriero, L.

Gureyev, T. E.

Gutmann, B.

Hanssen, R. F.

R. F. Hanssen, T. M. Weckwerth, H. A. Zebker, and R. Klees, Science 283, 1297 (1999).
[CrossRef] [PubMed]

Hardie, R. C.

Hartl, P.

R. Bamler and P. Hartl, Inverse Probl. 14, R1 (1998).
[CrossRef]

Hesselink, L.

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

Hÿtch, M. J.

M. J. Hÿtch, E. Snoeck, and R. Kilaas, Ultramicroscopy 74, 131 (1998).
[CrossRef]

Jiang, Y.

P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
[CrossRef] [PubMed]

Jones III, G.

G. Jones III and B. Bhanu, Pattern Recogn. 34, 469 (2001).
[CrossRef]

Kadono, H.

H. Kadono, H. Takei, and S. Toyooka, Opt. Lasers Eng. 26, 151 (1997).
[CrossRef]

Kanwisher, N.

P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
[CrossRef] [PubMed]

Kasischke, E. S.

E. S. Kasischke, J. M. Melack, and M. C. Dobson, Remote Sens. Environ. 59, 141 (1997).
[CrossRef]

Kilaas, R.

M. J. Hÿtch, E. Snoeck, and R. Kilaas, Ultramicroscopy 74, 131 (1998).
[CrossRef]

Klees, R.

R. F. Hanssen, T. M. Weckwerth, H. A. Zebker, and R. Klees, Science 283, 1297 (1999).
[CrossRef] [PubMed]

Lalor, M. J.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, Meas. Sci. Technol. 11, 1480 (2000).
[CrossRef]

X. Xie, M. J. Lalor, D. R. Burton, and M. M. Shaw, Opt. Lasers Eng. 29, 49 (1998).
[CrossRef]

Lande, D.

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

Liu, A.

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

Lyuboshenko, I.

Manaker, D.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

McEvilly, T. V.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

Melack, J. M.

E. S. Kasischke, J. M. Melack, and M. C. Dobson, Remote Sens. Environ. 59, 141 (1997).
[CrossRef]

Moore, D. S.

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

Murray, M. H.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

Nadeau, R. M.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

Neurgaonkar, R. R.

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

Nicholson, J. W.

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

Nico, G.

Nugent, K. A.

Orlov, S. S.

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

Pasquariello, G.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, New York, 1989).

Pritt, M. D.

M. D. Pritt and J. S. Shipman, IEEE Trans. Geosci. Remote Sens. 32, 706 (1994).
[CrossRef]

Rabie, R. L.

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

Refice, A.

S. Stramaglia, A. Refice, and L. Guerriero, Physica A 276, 521 (2000).
[CrossRef]

Ridout, A. J.

D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
[CrossRef] [PubMed]

Scharroo, R.

D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
[CrossRef] [PubMed]

Schmidt, D.

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

Shaw, M. M.

X. Xie, M. J. Lalor, D. R. Burton, and M. M. Shaw, Opt. Lasers Eng. 29, 49 (1998).
[CrossRef]

Shipman, J. S.

M. D. Pritt and J. S. Shipman, IEEE Trans. Geosci. Remote Sens. 32, 706 (1994).
[CrossRef]

Shum, C. K.

D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
[CrossRef] [PubMed]

Shuman, M.

P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
[CrossRef] [PubMed]

Snoeck, E.

M. J. Hÿtch, E. Snoeck, and R. Kilaas, Ultramicroscopy 74, 131 (1998).
[CrossRef]

Stetson, K. A.

Stramaglia, S.

Strand, J.

Su, X.

L. Xue and X. Su, Appl. Opt. 40, 1207 (2001).
[CrossRef]

X. Su and W. Chen, Opt. Lasers Eng. 35, 263 (2001), and references therein.
[CrossRef]

Takei, H.

H. Kadono, H. Takei, and S. Toyooka, Opt. Lasers Eng. 26, 151 (1997).
[CrossRef]

Taxt, T.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, New York, 1989).

Toyooka, S.

H. Kadono, H. Takei, and S. Toyooka, Opt. Lasers Eng. 26, 151 (1997).
[CrossRef]

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, New York, 1989).

Volkov, V. V.

V. V. Volkov, Y. Zhu, and M. De Graef, Micron. 33, 411 (2002).
[CrossRef]

Wahid, J.

Waldner, S.

H. A. Aebischer and S. Waldner, Opt. Commun. 162, 205 (1999).
[CrossRef]

Weber, H.

Weckwerth, T. M.

R. F. Hanssen, T. M. Weckwerth, H. A. Zebker, and R. Klees, Science 283, 1297 (1999).
[CrossRef] [PubMed]

Wingham, D. J.

D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
[CrossRef] [PubMed]

Xie, X.

X. Xie, M. J. Lalor, D. R. Burton, and M. M. Shaw, Opt. Lasers Eng. 29, 49 (1998).
[CrossRef]

Xue, L.

Younus, M. I.

Zebker, H. A.

R. F. Hanssen, T. M. Weckwerth, H. A. Zebker, and R. Klees, Science 283, 1297 (1999).
[CrossRef] [PubMed]

Zhu, Y.

V. V. Volkov, Y. Zhu, and M. De Graef, Micron. 33, 411 (2002).
[CrossRef]

Appl. Opt. (8)

IEEE Trans. Geosci. Remote Sens. (1)

M. D. Pritt and J. S. Shipman, IEEE Trans. Geosci. Remote Sens. 32, 706 (1994).
[CrossRef]

Inverse Probl. (1)

R. Bamler and P. Hartl, Inverse Probl. 14, R1 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, Meas. Sci. Technol. 11, 1480 (2000).
[CrossRef]

Micron. (1)

V. V. Volkov, Y. Zhu, and M. De Graef, Micron. 33, 411 (2002).
[CrossRef]

Opt. Commun. (2)

H. A. Aebischer and S. Waldner, Opt. Commun. 162, 205 (1999).
[CrossRef]

D. L. Fried, Opt. Commun. 200, 43 (2001).
[CrossRef]

Opt. Lasers Eng. (3)

X. Su and W. Chen, Opt. Lasers Eng. 35, 263 (2001), and references therein.
[CrossRef]

X. Xie, M. J. Lalor, D. R. Burton, and M. M. Shaw, Opt. Lasers Eng. 29, 49 (1998).
[CrossRef]

H. Kadono, H. Takei, and S. Toyooka, Opt. Lasers Eng. 26, 151 (1997).
[CrossRef]

Pattern Recogn. (1)

G. Jones III and B. Bhanu, Pattern Recogn. 34, 469 (2001).
[CrossRef]

Phys. Rev. (1)

Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[CrossRef]

Phys. Rev. Lett. (1)

K. T. Gahagan, D. S. Moore, D. J. Funk, R. L. Rabie, S. J. Buelow, and J. W. Nicholson, Phys. Rev. Lett. 85, 3205 (2000).
[CrossRef] [PubMed]

Physica A (1)

S. Stramaglia, A. Refice, and L. Guerriero, Physica A 276, 521 (2000).
[CrossRef]

Remote Sens. Environ. (1)

E. S. Kasischke, J. M. Melack, and M. C. Dobson, Remote Sens. Environ. 59, 141 (1997).
[CrossRef]

Science (5)

D. J. Wingham, A. J. Ridout, R. Scharroo, R. J. Arthern, and C. K. Shum, Science 282, 456 (1998).
[CrossRef] [PubMed]

R. F. Hanssen, T. M. Weckwerth, H. A. Zebker, and R. Klees, Science 283, 1297 (1999).
[CrossRef] [PubMed]

R. Bürgmann, D. Schmidt, R. M. Nadeau, M. d’Alessio, E. Fielding, D. Manaker, T. V. McEvilly, and M. H. Murray, Science 289, 1178 (2000).
[CrossRef]

P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
[CrossRef] [PubMed]

L. Hesselink, S. S. Orlov, A. Liu, A. Akella, D. Lande, and R. R. Neurgaonkar, Science 282, 1089 (1998).
[CrossRef] [PubMed]

Ultramicroscopy (1)

M. J. Hÿtch, E. Snoeck, and R. Kilaas, Ultramicroscopy 74, 131 (1998).
[CrossRef]

Other (3)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, New York, 1989).

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

E. Volkl, L. F. Allard, and D. C. Joy, eds., Introduction to Electron Holography (Plenum, New York, 1999).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Experimental wrapped phase (a) from holography experiments carried out on Nd2Fe14B hard magnets. The unwrapped phase obtained by (b) our algorithm and (c) a path-dependent method.

Fig. 2
Fig. 2

Wrapped phase (a) from holography experiments carried out on Nd2Fe14B hard magnets. The unwrapped phase (b) obtained by our algorithm (three iterations).

Equations (5)

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nr=12π-22ϕr-2ϕwr,
2fx,y=-4π2N2FFT-1p2+q2FFTfx,y,
-2gx,y=-N24π2FFT-1FFTgx,yp2+q2,
ϕj+1x,y=ϕjx,y+2π roundϕx,y-ϕjx,y2π,
ϕx,y=FFT-1FFTcos ϕw FFT-1p2+q2FFTsin ϕwp2+q2-FFT-1FFTsin ϕw FFT-1p2+q2FFTcos ϕwp2+q2.

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