Abstract

One of the most successful phase-unwrapping algorithms uses branch cuts to join discontinuity sources that mark the beginning or the end of a 2π phase discontinuity. Here, using phase-stepping speckle interferometry, we verify that these sources coincide with points of very low or zero modulus and that the displacement of sources as a result of speckle decorrelation between measurements of two phase maps leads to closely spaced dipole pairs of sources in the phase-difference map. By measuring the movement of sources at high magnification, we find that the length distribution of correct branch cuts needed to unwrap a phase-difference map is approximately Gaussian. This provides a theoretical justification for unwrapping with the set of branch cuts that minimizes the sum of squares of cut lengths.

© 1995 Optical Society of America

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References

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  1. D. W. Robinson, “Phase unwrapping methods,” Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.
  2. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  3. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  4. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  5. L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
    [CrossRef]
  6. N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).
  7. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).
  8. A. W. F. Edwards, Likelihood (Cambridge U. Press, Cambridge, 1972).
  9. J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 36, 5100–5108 (1995).
    [CrossRef]

1995 (1)

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 36, 5100–5108 (1995).
[CrossRef]

1991 (2)

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[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]

1982 (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).

1981 (1)

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Baranova, N. B.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Barr, L. D.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Bone, D. J.

Buckland, J. R.

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 36, 5100–5108 (1995).
[CrossRef]

Coudé du Foresto, V.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Edwards, A. W. F.

A. W. F. Edwards, Likelihood (Cambridge U. Press, Cambridge, 1972).

Fox, J.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[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]

Huntley, J. M.

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 36, 5100–5108 (1995).
[CrossRef]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

Mameav, A. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).

Pilipetskii, N. F.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).

Poczulp, G. A.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Richardson, J.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

Roddier, C.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Roddier, F.

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Shkunov, V. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).

Turner, S. R. E.

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 36, 5100–5108 (1995).
[CrossRef]

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]

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]

Zel’dovich, B. Ya.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Appl. Opt. (3)

Opt. Eng. (1)

L. D. Barr, V. Coudé du Foresto, J. Fox, G. A. Poczulp, J. Richardson, C. Roddier, F. Roddier, “Large-mirror testing facility at the National Optical Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[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]

Sov. Phys. JETP (2)

N. B. Baranova, B. Ya. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mameav, N. F. Pilipetskii, V. V. Shkunov, “Dislocation density on wavefront of a speckle-structure light field,” Sov. Phys. JETP 56, 983–988 (1982).

Other (2)

A. W. F. Edwards, Likelihood (Cambridge U. Press, Cambridge, 1972).

D. W. Robinson, “Phase unwrapping methods,” Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.

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Figures (9)

Fig. 1
Fig. 1

Wrapped phase map containing discontinuity sources 1 and 2. The phase at Q relative to P is obtained from the number of 2π discontinuities crossed by any path linking the two points and varies according to the path taken.

Fig. 2
Fig. 2

a, six pixels from phase map ϕ1 containing a −1 discontinuity source at pixel (m, n); phase values are in units of π (−1 to +1 represents −π to +π); b, the source has moved to pixel (m + 1, n) in phase map ϕ2; c, the phase-difference map ϕ2ϕ1 contains a dipole pair of sources at (m, n) and (m + 1, n).

Fig. 3
Fig. 3

Optical arrangement for measuring speckle pattern dislocations.

Fig. 4
Fig. 4

a, wrapped phase map measured from two interfering speckle patterns; b, phase-difference map showing the change in phase distribution between part a and a similar, but slightly decorrelated, second speckle pattern; c, discontinuity source distribution for part b; d, discontinuity source distribution for phase-difference map between part a and a more highly decorrelated speckle pattern. Open and filled circles represent +1 and −1 sources, respectively.

Fig. 5
Fig. 5

Distribution of source displacements (in units of speckle diameter) for increasing degrees of decorrelation (correlation coefficients are given in the top-left corner of each graph). Parts a–d and e–h show x and y distributions, respectively.

Fig. 6
Fig. 6

Standard deviations of the distributions in Fig. 5 as functions of speckle correlation. sx and sy are represented by filled and open circles, respectively.

Fig. 7
Fig. 7

Combined distributions of source displacement, normalized by the standard deviations: a, x distribution; b, y distribution.

Fig. 8
Fig. 8

a, enlarged region of Fig. 4a, showing a discontinuity source; b, the same region in a slightly decorrelated speckle pattern; c, phase-difference map between parts a and b; d, e, and f, modulus of the interference pattern corresponding to a, b, and c, respectively.

Fig. 9
Fig. 9

Fraction of dipoles correctly paired by minimizing the sum of squares of the cut lengths, both with (open circles) and without (filled circles) compensation for the speckle drift.

Equations (11)

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d ( i ) = [ { ϕ ( i ) - ϕ ( i - 1 ) } / 2 π ] ,
ν = i = 1 M d ( i ) ,
Δ I 42 ( m , n ) = R ( m , n ) sin ϕ ( m , n ) ,
Δ I 13 ( m , n ) = R ( m , n ) cos ϕ ( m , n ) ,
Δ I i j ( m , n ) = I i ( m , n ) - I j ( m , n ) ,
ϕ ( m , n ) = tan - 1 [ Δ I 42 ( m , n ) Δ I 13 ( m , n ) ] ,
R ( m , n ) = { [ Δ I 42 ( m , n ) ] 2 + [ Δ I 13 ( m , n ) ] 2 } 1 / 2 .
Δ ϕ = tan - 1 ( Δ I 42 ( 2 ) Δ I 13 ( 1 ) - Δ I 13 ( 2 ) Δ I 42 ( 1 ) Δ I 13 ( 2 ) Δ I 13 ( 1 ) + Δ I 42 ( 2 ) Δ I 42 ( 1 ) ) ,
R = [ ( Δ I 42 ( 2 ) Δ I 13 ( 1 ) - Δ I 13 ( 2 ) Δ I 42 ( 1 ) ) 2 + ( Δ I 13 ( 2 ) Δ I 13 ( 1 ) + Δ I 42 ( 2 ) Δ I 42 ( 1 ) ) 2 ] 1 / 4 ,
P = exp { - [ ( m j + - m j - ) 2 + ( n j + - n j - ) 2 ] / 2 s 2 } .
L = exp { - j = 1 N [ ( m J + - m j - ) 2 + ( n j + - n j - ) 2 ] / 2 s 2 } .

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