Abstract

We present a new Fourier-based exact solution for deterministic phase unwrapping from experimental maps of wrapped phase in the presence of noise and phase vortices. This single-step approach has superior performance for images with high phase gradients or insufficient digital sampling approaching 2π/pixel and therefore performs as a fast and practical solution for the phase-unwrapping problem for experimental applications in applied optics, physics, and medicine.

© 2003 Optical Society of America

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  1. A. V. Oppenheim and R. W. Schafer, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  2. H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
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  4. G. H. Kaufmann and G. E. Galizzi, Appl. Opt. 41, 7254 (2002).
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  5. D. L. Fried, Opt. Commun. 200, 43 (2001).
    [CrossRef]
  6. P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher, Science 293, 2470 (2001).
    [CrossRef] [PubMed]
  7. R. Cusack and N. Papadakis, Neuroimage 16, 754 (2002).
    [CrossRef] [PubMed]
  8. V. Pascazio and G. Schirinzi, IEEE Trans. Image Process. 11, 1478 (2002).
    [CrossRef]
  9. E. Volkl, L. F. Allard, and D. C. Joy, eds., Introduction to Electron Holography (Plenum, New York, 1999).
    [CrossRef]
  10. S. Stramaglia, A. Refice, and L. Guerriero, Physica A 276, 521 (2000).
    [CrossRef]
  11. C. F. Lo, X. Peng, and L. L. Cai, Optik 113, 439 (2002).
    [CrossRef]
  12. P. Soille, Opt. Lasers Eng. 32, 339 (2000).
    [CrossRef]
  13. S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, IEEE Trans. Image Process. 4, 667 (1995).
    [CrossRef]
  14. M. A. Schofield and Y. Zhu, Microsc. Microanal. Microstruct. 8(Suppl. 2), 532 CD (2002).
  15. V. V. Volkov and Y. Zhu, Phys. Rev. Lett. 91, 043904 (2003).
    [CrossRef]
  16. V. V. Volkov, Y. Zhu, and M. De Graef, Micron 33, 411 (2002).
    [CrossRef]

2003 (1)

V. V. Volkov and Y. Zhu, Phys. Rev. Lett. 91, 043904 (2003).
[CrossRef]

2002 (6)

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

C. F. Lo, X. Peng, and L. L. Cai, Optik 113, 439 (2002).
[CrossRef]

M. A. Schofield and Y. Zhu, Microsc. Microanal. Microstruct. 8(Suppl. 2), 532 CD (2002).

G. H. Kaufmann and G. E. Galizzi, Appl. Opt. 41, 7254 (2002).
[CrossRef] [PubMed]

R. Cusack and N. Papadakis, Neuroimage 16, 754 (2002).
[CrossRef] [PubMed]

V. Pascazio and G. Schirinzi, IEEE Trans. Image Process. 11, 1478 (2002).
[CrossRef]

2001 (2)

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

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

2000 (2)

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

P. Soille, Opt. Lasers Eng. 32, 339 (2000).
[CrossRef]

1999 (1)

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

1995 (1)

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, IEEE Trans. Image Process. 4, 667 (1995).
[CrossRef]

1985 (1)

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
[CrossRef]

1979 (1)

1975 (1)

A. V. Oppenheim and R. W. Schafer, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Cai, L. L.

C. F. Lo, X. Peng, and L. L. Cai, Optik 113, 439 (2002).
[CrossRef]

Cusack, R.

R. Cusack and N. Papadakis, Neuroimage 16, 754 (2002).
[CrossRef] [PubMed]

De Graef, M.

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

Downing, P. E.

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

Fried, D. L.

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

Galizzi, G. E.

Ghiglia, D. C.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
[CrossRef]

Glover, G. H.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, IEEE Trans. Image Process. 4, 667 (1995).
[CrossRef]

Guerriero, L.

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

Hjalmarson, H. P.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
[CrossRef]

Jiang, Y.

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

Jones, E. D.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
[CrossRef]

Kanwisher, N.

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

Kaufmann, G. H.

Lo, C. F.

C. F. Lo, X. Peng, and L. L. Cai, Optik 113, 439 (2002).
[CrossRef]

Mertz, L. N.

Napel, S.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, IEEE Trans. Image Process. 4, 667 (1995).
[CrossRef]

Norris, C. B.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Papadakis, N.

R. Cusack and N. Papadakis, Neuroimage 16, 754 (2002).
[CrossRef] [PubMed]

Pascazio, V.

V. Pascazio and G. Schirinzi, IEEE Trans. Image Process. 11, 1478 (2002).
[CrossRef]

Pelc, N. J.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, IEEE Trans. Image Process. 4, 667 (1995).
[CrossRef]

Peng, X.

C. F. Lo, X. Peng, and L. L. Cai, Optik 113, 439 (2002).
[CrossRef]

Refice, A.

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

Romero, L. A.

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schirinzi, G.

V. Pascazio and G. Schirinzi, IEEE Trans. Image Process. 11, 1478 (2002).
[CrossRef]

Schofield, M. A.

M. A. Schofield and Y. Zhu, Microsc. Microanal. Microstruct. 8(Suppl. 2), 532 CD (2002).

Shuman, M.

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

Soille, P.

P. Soille, Opt. Lasers Eng. 32, 339 (2000).
[CrossRef]

Song, S. M.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, IEEE Trans. Image Process. 4, 667 (1995).
[CrossRef]

Stramaglia, S.

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

Volkov, V. V.

V. V. Volkov and Y. Zhu, Phys. Rev. Lett. 91, 043904 (2003).
[CrossRef]

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

Zhu, Y.

V. V. Volkov and Y. Zhu, Phys. Rev. Lett. 91, 043904 (2003).
[CrossRef]

M. A. Schofield and Y. Zhu, Microsc. Microanal. Microstruct. 8(Suppl. 2), 532 CD (2002).

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

Appl. Opt. (2)

IEEE Trans. Image Process. (2)

V. Pascazio and G. Schirinzi, IEEE Trans. Image Process. 11, 1478 (2002).
[CrossRef]

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, IEEE Trans. Image Process. 4, 667 (1995).
[CrossRef]

Micron (1)

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

Microsc. Microanal. Microstruct. (1)

M. A. Schofield and Y. Zhu, Microsc. Microanal. Microstruct. 8(Suppl. 2), 532 CD (2002).

Neuroimage (1)

R. Cusack and N. Papadakis, Neuroimage 16, 754 (2002).
[CrossRef] [PubMed]

Opt. Commun. (1)

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

Opt. Lasers Eng. (1)

P. Soille, Opt. Lasers Eng. 32, 339 (2000).
[CrossRef]

Optik (1)

C. F. Lo, X. Peng, and L. L. Cai, Optik 113, 439 (2002).
[CrossRef]

Phys. Rev. B (1)

H. P. Hjalmarson, L. A. Romero, D. C. Ghiglia, E. D. Jones, and C. B. Norris, Phys. Rev. B 32, 4300 (1985).
[CrossRef]

Phys. Rev. Lett. (1)

V. V. Volkov and Y. Zhu, Phys. Rev. Lett. 91, 043904 (2003).
[CrossRef]

Physica A (1)

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

Science (1)

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

Other (2)

A. V. Oppenheim and R. W. Schafer, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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

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Figures (3)

Fig. 1
Fig. 1

Image of sculpture of Aristotle, illustrating several aspects of the phase-unwrapping process. (a) Synthetic wrapped phase of 512×512 pixels obtained from an image identical to the image shown in (c). (b) Best phase unwrapping by the conventional path-dependent approach, creating artifacts marked by arrows. (c) Exact and (d) approximate solutions with Fourier transform via Eq. (3). Note that the maximal phase variation in (c) is 14.4 rad with local phase gradients as high as 4.07 rad/pixel. The unwrapped phase (c) is pixel to pixel identical to the original image.

Fig. 2
Fig. 2

(a) Example of wrapped phase with a high density of vortices and noise jumps, as shown by the line scan at the bottom. (b) Unwrapped phase map of 512×512 pixels obtained by the solution of Eq. (3). The unwrapped noisy phase, measured in radians and shown below by the same line scan, is pixel to pixel identical to the original noisy phase. Inset in (a), enlarged pixel structure in the boxed area.

Fig. 3
Fig. 3

(a) Amplitude and (b) wrapped phase components of the complex magnetic resonance image recorded with a sampling of 185×210 pixels. Results of the exact solution for (c) an unwrapped phase map and (d) the phase gradients. The temperature color bar in (c) is calibrated in radians. Inset in (d), amplitude and direction of phase gradients in magnetic settings.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

φwr+2πkr=φr+nr,
exp2πiq·r=2πiqexp2πiq·r,
ψr=2πiF-1Fψrq.
ψr=Re12πiF-1Fxψqx+Fyψqyq2.
Zr=expiφwr+2πkr=expiφwr.

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