Abstract

We describe what is to our knowledge a novel technique for phase unwrapping. Several algorithms based on unwrapping the most-reliable pixels first have been proposed. These were restricted to continuous paths and were subject to difficulties in defining a starting pixel. The technique described here uses a different type of reliability function and does not follow a continuous path to perform the unwrapping operation. The technique is explained in detail and illustrated with a number of examples.

© 2002 Optical Society of America

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References

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  1. D. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson, G. T. Reid, eds., (Institute of Physics Publishing, Bristol and Philadelphia, 1993), pp. 194–229.
  2. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  3. A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187–1194 (2001).
    [CrossRef]
  4. 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]
  5. K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
    [CrossRef]
  6. M. Arevallilo Herráez, D. R. Burton, D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847–5852 (1996).
    [CrossRef]
  7. P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
    [CrossRef]
  8. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).
  9. 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]
  10. R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.
  11. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  12. O. Y. Kwon, D. M. Shough, R. A. Williams, “Stroboscopic phase-shifting interferometry,” Opt. Lett. 12, 855–857 (1987).
    [CrossRef] [PubMed]
  13. J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
    [CrossRef]
  14. H. A. Vrooman, A. M. Maas, “Image processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [CrossRef] [PubMed]
  15. J. A. Quiroga, E. Bernabeu, “Phase unwrapping algorithm for noisy phase-map processing,” Appl. Opt. 33, 6725–6731 (1994).
    [CrossRef] [PubMed]
  16. J. A. Quiroga, A. Gonzalez-Cano, E. Bernabeu, “Phase unwrapping algorithm based on adaptive criterion,” Appl. Opt. 34, 2560–2563 (1995).
    [CrossRef] [PubMed]
  17. Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis, Proc. SPIE30, 254–263 (1991).
  18. H. Lim, W. Xu, X. Huang, “Two new practical methods for phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9533, Firenze, Italy, 196–198 (1995).
  19. W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9634, Lincoln, Nebraska, USA, 2044–2046 (1996).

2001 (1)

1998 (1)

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

1996 (2)

1995 (1)

1994 (3)

1991 (2)

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

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

1989 (1)

1987 (1)

Ai, C.

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis, Proc. SPIE30, 254–263 (1991).

Arevallilo Herráez, M.

Baldi, A.

Bernabeu, E.

Burton, D. R.

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

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Charette, P. G.

Clegg, D. B.

Cumming, I.

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9634, Lincoln, Nebraska, USA, 2044–2046 (1996).

Ettemeyer, A.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

Ghiglia, D. C.

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

Gonzalez-Cano, A.

Huang, X.

H. Lim, W. Xu, X. Huang, “Two new practical methods for phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9533, Firenze, Italy, 196–198 (1995).

Hung, K. M.

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

Hunter, I. W.

Huntley, J. M.

Kwon, O. Y.

Lalor, M. J.

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Lim, H.

H. Lim, W. Xu, X. Huang, “Two new practical methods for phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9533, Firenze, Italy, 196–198 (1995).

Maas, A. M.

Neupert, U.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

Obermeier, P.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

Oppenheim, A. V.

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.

Pritt, M. D.

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

Quiroga, J. A.

Robinson, D.

D. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson, G. T. Reid, eds., (Institute of Physics Publishing, Bristol and Philadelphia, 1993), pp. 194–229.

Romero, L. A.

Rottenkolber, H.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

Schafer, R. W.

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.

Schörner, J.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

Shough, D. M.

Stephenson, P.

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Vrooman, H. A.

Williams, R. A.

Winter, C.

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

Xu, W.

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9634, Lincoln, Nebraska, USA, 2044–2046 (1996).

H. Lim, W. Xu, X. Huang, “Two new practical methods for phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9533, Firenze, Italy, 196–198 (1995).

Xu, Y.

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis, Proc. SPIE30, 254–263 (1991).

Yamada, T.

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

Appl. Opt. (7)

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

Opt. Eng. (2)

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
[CrossRef]

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Opt. Lasers Eng. (1)

J. Schörner, A. Ettemeyer, U. Neupert, H. Rottenkolber, C. Winter, P. Obermeier, “New approaches in interpreting holographic images,” Opt. Lasers Eng. 14, 283–291 (1991).
[CrossRef]

Opt. Lett. (1)

Other (7)

D. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, W. R. Robinson, G. T. Reid, eds., (Institute of Physics Publishing, Bristol and Philadelphia, 1993), pp. 194–229.

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

Y. Xu, C. Ai, “Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis, Proc. SPIE30, 254–263 (1991).

H. Lim, W. Xu, X. Huang, “Two new practical methods for phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9533, Firenze, Italy, 196–198 (1995).

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. IGARSS’9634, Lincoln, Nebraska, USA, 2044–2046 (1996).

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

R. W. Schafer, A. V. Oppenheim, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), pp. 507–511.

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Figures (8)

Fig. 1
Fig. 1

Calculation of the second differences in an image.

Fig. 2
Fig. 2

Numerical example of the proposed algorithm. (Continues on next page.)

Fig. 3
Fig. 3

Flow chart of the proposed algorithm.

Fig. 4
Fig. 4

Unwrapping of a simple simulated wrapped phase distribution: (a) wrapped phase distribution (b) unwrapped phase distribution.

Fig. 5
Fig. 5

Unwrapping of a simple simulated wrapped phase distribution with a noisy area: (a) wrapped phase distribution, (b) unwrapped phase distribution.

Fig. 6
Fig. 6

Unwrapping of a wrapped phase distribution obtained from a real fringe pattern. The region crossing the image is a product of its shadow: (a) wrapped phase distribution. (b) unwrapped phase distribution.

Fig. 7
Fig. 7

Unwrapping of a wrapped phase distribution obtained from the measurement of a human face: (a) wrapped phase distribution, (b) unwrapped phase distribution, (c) three dimensional representation of the resulting phase, (d) result obtained with the linear scanning procedure.

Fig. 8
Fig. 8

Rewrapped phase map.

Equations (3)

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

Di, j=H2i, j+V2i, j+D12i, j+D22i, j1/2,
Hi, j=γφi-1, j-φi, j-γφi, j-φi+1, j Vi, j=γφi, j-1-φi, j-γφi, j-φi, j+1 D1i, j=γφi-1, j-1-φi, j-γφi, j-φi+1, j+1 D2i, j=γφi-1, j+1-φi, j-γφi, j-φi+1, j-1,
R=1D.

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