Abstract

One problem to be tackled when interferometric phase-shifting techniques are used is the method in which the phase can be reconstructed. Because an inverse trigonometric function appears in the formulation, the final data are not the phase, but the phase modulo 2π. A new phase-unwrapping algorithm based on a two-step procedure is presented. In the first step, the digital image to be analyzed is divided into continuous patches by a quad-tree-like recursive procedure; in the second step, the same level patches are joined together by an error-norm-minimizing approach to obtain larger, almost continuous ones. The basic idea of the procedure is to simplify the problem by factoring the complete image into square, variable-size, homogeneous areas (i.e., regions with no internal phase jump) so that only interfaces need to be dealt with. By hierarchically recombining the so-obtained subimages, an unwrapped phase field can be obtained. After a complete description of the algorithm, some examples of its use on synthesized digital images are illustrated. As the algorithm can be used with and without quality masks and the error-minimizing step can use different norms, a full class of unwrapping algorithms can be implemented by this approach.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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2000 (1)

1998 (3)

1997 (1)

1996 (4)

D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1–15 (1996).
[CrossRef]

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

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

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

1995 (2)

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

R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 28, 781–789 (1995).
[CrossRef]

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 (2)

1982 (1)

1979 (1)

Abe, T.

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]

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.

Burton, D. 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]

Clegg, D. B.

Cusack, R.

R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 28, 781–789 (1995).
[CrossRef]

Flynn, T. J.

Fornaro, G.

Franceschetti, G.

Gallizzi, G. E.

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).

Goldrein, H. T.

R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 28, 781–789 (1995).
[CrossRef]

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]

Guerriero, L.

Herráez, M. A.

Hung, K. M.

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

Hunt, B. R.

Huntley, J. M.

Itho, K.

Kaufmann, G. H.

Lalor, M. J.

Lanari, R.

Nico, G.

Pasquariello, G.

Pritt, M. D.

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

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, U.K., 1993), pp. 192–229.

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]

Ruiz, P. D.

Sansosti, E.

Stramaglia, S.

Takahashi, T.

Takajo, H.

Takeda, M.

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-square 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. (7)

IEEE Trans. Geoscience Remote Sens. (1)

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geoscience 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. (1)

J. Opt. Soc. Am. A (6)

Opt. Eng. (1)

K. M. Hung, T. Yamada, “Phase unwrapping by regions using least-square approach,” Opt. Eng. 37, 2965–2970 (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 (3)

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, U.K., 1993), pp. 192–229.

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]

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).

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Figures (13)

Fig. 1
Fig. 1

Quad tree of the dipole surface.

Fig. 2
Fig. 2

Assembly process.

Fig. 3
Fig. 3

Phase-jump encoding.

Fig. 4
Fig. 4

Wrapped noisy plane.

Fig. 5
Fig. 5

Residues of the wrapped plane fringe field: black, negative; white, positive.

Fig. 6
Fig. 6

Unwrapped plane.

Fig. 7
Fig. 7

Unwrapped plane by algorithm of Herráez et al.12

Fig. 8
Fig. 8

Wrapped fringe field of a dipolelike function.

Fig. 9
Fig. 9

Weight array: black = 0, white = 1.

Fig. 10
Fig. 10

Residues of the wrapped dipole fringe field.

Fig. 11
Fig. 11

Unwrapped dipole function with a masked-out area.

Fig. 12
Fig. 12

Wrapped shear example.

Fig. 13
Fig. 13

Unwrapped shear example. All the diagonal squares are incorrectly unwrapped.

Equations (6)

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

ψi, j=ϕi, j+2πni, j,
δx=0, 1, 0, -1, -1, 1, 1, -1,δy=-1, 0, 1, 0, -1, -1, 1, 1,=i=1Nj=1Mk=14 |ψi, j-ψi+δxk, j+δyk|,
E=k=1Nθk, N2-θk, N2+1+θN2, k-θN2+1, k+i=1N-1θi,N2-θi+1, N2+1+θi+1, N2-θi, N2+1+j=1N-1θN2, j-θN2+1, j+1+θN2, j+1-θN2+1, j,
Wi, j=γi, j1-|θi, j-18k=18θi+δxk, j+δyk|2π.
Wi=1-Eiπni,
fr, θ=k1-rsinθ,

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