Abstract

A new phase unwrapping algorithm is described that uses local phase information to mask out those parts of the field that cause inconsistencies in the unwrapping. Unlike earlier techniques, which produce only a consistent unwrapping of the phase in the presence of discontinuities, this technique can produce an approximately correct unwrapping. The technique is tolerant of discontinuities and noise in the phase and is fast, efficient, and simple to implement. In the absence of discontinuities an rms signal-to-noise ratio in the wrapped phase of <2:1 can be tolerated.

© 1991 Optical Society of America

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References

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  1. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
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  3. 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]
  5. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. G. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).
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    [CrossRef] [PubMed]
  7. K. Creath, Y. Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength, phase shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
    [CrossRef]
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    [CrossRef] [PubMed]

1989

1988

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

1987

1985

K. Creath, Y. Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength, phase shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

1983

1982

Cheng, Y. Y.

K. Creath, Y. Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength, phase shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Creath, K.

K. Creath, Y. Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength, phase shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[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. G. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).

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]

Greivenkamp, J. E.

Huntley, J. M.

Ina, H.

Kobayashi, S.

Macy, W. W.

Mastin, G. A.

Romero, L. A.

Takeda, M.

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]

Wyant, J. C.

K. Creath, Y. Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength, phase shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Zebker, H. A.

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.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

K. Creath, Y. Y. Cheng, J. C. Wyant, “Contouring aspheric surfaces using two-wavelength, phase shifting interferometry,” Opt. Acta 32, 1455–1464 (1985).
[CrossRef]

Radio Sci.

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

Other

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. G. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).

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Figures (3)

Fig. 1
Fig. 1

a, A helical-ramp phase pattern used to simulate a discontinuity in the phase field. Black to white indicates a range of 1.5 fringes, b, The helical-ramp phase with a linear carrier phase having a gradient of 7 fringes in both the vertical and horizontal directions added. The phase is wrapped into the range from −π to π. c, The positions of residues: black indicates a negative residue; white indicates a positive residue. The residue is associated with the group of four points above and to the right of the indicated pixel, d, The bit-map mask based on second differences with a threshold of 0.08 fringes, e, The unwrapped fringe shift using the Goldstein et al.3 algorithm for the branch-cut construction, f, The unwrapped phase using the current algorithm.

Fig. 2
Fig. 2

a, A noisy phase field. The rms noise level is 0.14 fringes with peak-to-peak noise of 0.86 fringes. The noise has an approximate probability distribution, b, The cosine of the phase at a sample density of 100 × 100 points to give an idea of the level of noise in the data, c, The wrapped phase with a sampling density of 50 × 50 points, d, Tho residues, e, The mask with a threshold second difference of 0.5 fringes, f, The recovered phase.

Fig. 3
Fig. 3

a, Interferometric fringe-shift data derived by taking the difference in the recovered phase from a test and reference interferogram. The areas with a highly convoluted phase surface actually contain no fringes, b, The residues, c, The mask with a threshold second difference of 0.08 fringes, d, The unwrapped phase.

Equations (10)

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Δ = k = 1 N ( { Φ [ x ( k ) , y ( k ) ] Φ [ x ( k 1 ) , y ( k 1 ) ] } / 2 π ) ,
· ( i , j + 1 ) · ( i + 1 , j + 1 ) · ( i 1 , j ) · ( i , j ) · ( i + 1 , j ) · ( i , j 1 ) .
Δ x y ( i , j ) = Δ x ( i , j + 1 ) Δ x ( i , j ) , Δ x 2 ( i , j ) = Δ x ( i + 1 , j ) Δ x ( i 1 , j ) , Δ y 2 ( i , j ) = Δ x ( i , j + 1 ) Δ x ( i , j 1 ) ,
Δ x ( i , j ) = ϕ ( i + 1 , j ) ϕ ( i , j ) , Δ y ( i , j ) = ϕ ( i , j + 1 ) ϕ ( i , j ) .
ϕ 1 , ϕ 2 , ϕ 0 , ϕ 3 ,
Δ ϕ i = ϕ ( i + 1 ) mod 4 ϕ i ( i = 1 4 ) .
1 Δ ϕ i / 2 π 1 ( i = 0 3 ) .
0.5 f i 0.5 ( i = 0 3 ) ,
[ 0.5 ] = [ 0.5 ] = 0 .
Δ = i I i = i f i .

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