Abstract

The phase unwrapping problem consists in singling out an integer field whose values make the original wrapped phase field continuous. Even if in principle the problem is very simple—a direct integration of the wrapped phase field suffices—in the presence of noise and/or undersampling, the solution is no longer unique and the direct integration methods usually fail to find an acceptable solution. This work presents what is to my knowledge a new unwrapping algorithm that attempts to find the solution by iteratively merging and shifting the continuous areas until a single region is built or no further moves are possible. Unlike the tile methods, the regions can have arbitrary shape and need not be single-connected so that, by removing the predefined size and shape constraint, the algorithm is very robust. The greater freedom of the regions’ shape makes their handling more problematic, so that certain implementation aspects, critical to algorithm performance, are presented here. Some unwrapping examples are also presented and memory requirements are discussed.

© 2003 Optical Society of America

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References

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  1. K. Itho, “Analysis of the phase unwrapping problem,” Appl. Opt. 21, 2470 (1982).
    [CrossRef]
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    [CrossRef]
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  4. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK and Philadelphia, US, 1993), pp. 194–229.
  5. N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
    [CrossRef] [PubMed]
  6. M. Takeda, T. Abe, “Phase unwrapping based on maximum cross-amplitude spanning tree algorithm: a comparative study,” in Interferometry VII: Techniques and Analysis, M. Kujawińska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2544, 122–129 (1996).
    [CrossRef]
  7. M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2341 (1996).
    [CrossRef]
  8. T. J. Flynn, “Consistent 2-D phase unwrapping guided by a quality map,” in IEEE 1996 International Geoscience and Remote Sensing Symposium Proceedings (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2057–2059.
    [CrossRef]
  9. 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).
  10. K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-squares approach,” Opt. Eng. 37, 2965–2970 (1998).
    [CrossRef]
  11. A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187–1194 (2001).
    [CrossRef]
  12. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. C. W. Chen, H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
    [CrossRef]
  18. M. D. Pritt, “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]
  20. A. Baldi, F. Bertolino, F. Ginesu, “Phase unwrapping algorithms: a comparison,” in Interferometry in Speckle Light: Theory and Application. Proceedings of the International Conference, P. Jacquot, J. M. Fournier, eds., (Springer, Berlin, 2000), pp. 483–490.
    [CrossRef]
  21. A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
    [CrossRef]
  22. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
    [CrossRef]
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    [CrossRef]

2002 (1)

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

2001 (1)

2000 (1)

1998 (2)

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

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

1997 (1)

1996 (3)

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

D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
[CrossRef]

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2341 (1996).
[CrossRef]

1995 (2)

1994 (1)

1992 (1)

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

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

1987 (1)

1982 (1)

Abe, T.

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2341 (1996).
[CrossRef]

M. Takeda, T. Abe, “Phase unwrapping based on maximum cross-amplitude spanning tree algorithm: a comparative study,” in Interferometry VII: Techniques and Analysis, M. Kujawińska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2544, 122–129 (1996).
[CrossRef]

Aoki, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

Baldi, A.

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187–1194 (2001).
[CrossRef]

A. Baldi, F. Bertolino, F. Ginesu, “Phase unwrapping algorithms: a comparison,” in Interferometry in Speckle Light: Theory and Application. Proceedings of the International Conference, P. Jacquot, J. M. Fournier, eds., (Springer, Berlin, 2000), pp. 483–490.
[CrossRef]

Bertolino, F.

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

A. Baldi, F. Bertolino, F. Ginesu, “Phase unwrapping algorithms: a comparison,” in Interferometry in Speckle Light: Theory and Application. Proceedings of the International Conference, P. Jacquot, J. M. Fournier, eds., (Springer, Berlin, 2000), pp. 483–490.
[CrossRef]

Braun, M.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

Buckland, J. R.

Chen, C. W.

Ching, N. H.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

Cusack, R.

Flynn, T. J.

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
[CrossRef]

T. J. Flynn, “Consistent 2-D phase unwrapping guided by a quality map,” in IEEE 1996 International Geoscience and Remote Sensing Symposium Proceedings (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2057–2059.
[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).

Ginesu, F.

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

A. Baldi, F. Bertolino, F. Ginesu, “Phase unwrapping algorithms: a comparison,” in Interferometry in Speckle Light: Theory and Application. Proceedings of the International Conference, P. Jacquot, J. M. Fournier, eds., (Springer, Berlin, 2000), pp. 483–490.
[CrossRef]

Goldrein, H. T.

Goldstein, R. M.

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

Hung, K. M.

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

Huntley, J. M.

Itho, K.

Komiyama, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

Mastin, G. A.

Miyamoto, Y.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

Ozawa, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

Pritt, M. D.

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

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

Robinson, D. W.

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

Romero, L. A.

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

Sotomaru, T.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

Takeda, M.

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2341 (1996).
[CrossRef]

M. Takeda, T. Abe, “Phase unwrapping based on maximum cross-amplitude spanning tree algorithm: a comparative study,” in Interferometry VII: Techniques and Analysis, M. Kujawińska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2544, 122–129 (1996).
[CrossRef]

Turner, S. R. E.

Werner, C. L.

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

Yamada, T.

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

Zebker, H. A.

C. W. Chen, H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
[CrossRef]

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

Appl. Opt. (5)

IEEE Trans. Geosci. Remote Sens. (1)

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

IEEE Trans. Image Process. (1)

N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (5)

Opt. Eng. (2)

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2341 (1996).
[CrossRef]

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

Opt. Lasers Eng. (1)

A. Baldi, F. Bertolino, F. Ginesu, “On the performance of some unwrapping algorithms,” Opt. Lasers Eng. 37, 313–330 (2002).
[CrossRef]

Opt. Rev. (1)

T. Aoki, T. Sotomaru, T. Ozawa, T. Komiyama, Y. Miyamoto, M. Takeda, “Two-dimensional phase unwrapping by direct elimination of rotational vector fields from phase gradients obtained by heterodyne techniques,” Opt. Rev. 5, 374–379 (1998).
[CrossRef]

Radio Sci. (1)

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

Other (6)

T. J. Flynn, “Consistent 2-D phase unwrapping guided by a quality map,” in IEEE 1996 International Geoscience and Remote Sensing Symposium Proceedings (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 2057–2059.
[CrossRef]

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).

M. Takeda, T. Abe, “Phase unwrapping based on maximum cross-amplitude spanning tree algorithm: a comparative study,” in Interferometry VII: Techniques and Analysis, M. Kujawińska, R. J. Pryputniewicz, M. Takeda, eds., Proc. SPIE2544, 122–129 (1996).
[CrossRef]

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

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

A. Baldi, F. Bertolino, F. Ginesu, “Phase unwrapping algorithms: a comparison,” in Interferometry in Speckle Light: Theory and Application. Proceedings of the International Conference, P. Jacquot, J. M. Fournier, eds., (Springer, Berlin, 2000), pp. 483–490.
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Monodimensional cellular automata: initial phase.

Fig. 2
Fig. 2

Monodimensional cellular automata: automaton state at iteration 8.

Fig. 3
Fig. 3

Identifying points belonging to a region (boundary): each element of the region (boundary) matrix contains the index of next point of the same region (boundary); the last one contains a negative index. In this way all the points of the region (boundary) are linked together.

Fig. 4
Fig. 4

Holographic interferometry measurement of a flange under internal pressure.

Fig. 5
Fig. 5

Wrapped phase of Fig. 4 obtained by fast Fourier transform technique.

Fig. 6
Fig. 6

Unwrapping process. Step 3.

Fig. 7
Fig. 7

Unwrapping process. Step 4.

Fig. 8
Fig. 8

Unwrapping process. Step 7.

Fig. 9
Fig. 9

Unwrapped phase.

Fig. 10
Fig. 10

Speckle fringe field near a hole (residual stress analysis by the hole drilling technique).

Fig. 11
Fig. 11

Unwrapped phase. Note that no mask has been used.

Fig. 12
Fig. 12

Comparison of various unwrapping algorithms: mean signal-to-noise ratio.

Fig. 13
Fig. 13

Comparison of various unwrapping algorithms: execution time. Note the significantly longer times when region splitting is enabled.

Equations (6)

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

ϕi, j=ψi, j+2πki, j,
μi=j=14int|Δϕijk|+π2πsignΔϕijk,
Su=iBAjNiĀ wiwjγij|ϕj-ϕi+2π| Sh=iBAjNiĀ wiwjγij|ϕj-ϕi| Sd=iBAjNiĀ wiwjγij|ϕj-ϕi-2π|,
αk=iBAjNik wiwjπ-|Δϕij|
μij=int|Δϕij|+π2πsignΔϕij,
k=r mod n,

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