Abstract

An algorithm for unwrapping noisy phase maps has recently been proposed, based on the identification of discontinuity sources that mark the start or end of a 2π phase discontinuity. Branch cuts between sources act as barriers to unwrapping, resulting in a unique phase map that is independent of the unwrapping route. We investigate four methods for optimizing the placement of the cuts. A modified nearest neighbor approach is found to be the most successful and can reliably unwrap unfiltered speckle-interferometry phase maps with discontinuity source densities of 0.05 sources pixel−1.

© 1995 Optical Society of America

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References

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  1. G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
    [CrossRef]
  2. J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [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. D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69, 9–14 (1962).
    [CrossRef]
  7. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  8. E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines (Wiley, New York, (1989).
  9. J. M. Huntley, L. Benckert, “Measurement of dynamic crack tip displacement field by speckle photography and interferometry,” Opt. Lasers Eng. 19, 299–312 (1993).
    [CrossRef]

1993 (2)

J. M. Huntley, H. Saldner, “Temporal phase unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

J. M. Huntley, L. Benckert, “Measurement of dynamic crack tip displacement field by speckle photography and interferometry,” Opt. Lasers Eng. 19, 299–312 (1993).
[CrossRef]

1991 (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]

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]

1986 (1)

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1962 (1)

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69, 9–14 (1962).
[CrossRef]

Aarts, E.

E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines (Wiley, New York, (1989).

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]

Benckert, L.

J. M. Huntley, L. Benckert, “Measurement of dynamic crack tip displacement field by speckle photography and interferometry,” Opt. Lasers Eng. 19, 299–312 (1993).
[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]

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]

Gale, D.

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69, 9–14 (1962).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

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.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Korst, J.

E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines (Wiley, New York, (1989).

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]

Reid, G. T.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[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]

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]

Saldner, H.

Shapley, L. S.

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69, 9–14 (1962).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

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]

Am. Math. Mon. (1)

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69, 9–14 (1962).
[CrossRef]

Appl. Opt. (2)

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]

Opt. Lasers Eng. (2)

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[CrossRef]

J. M. Huntley, L. Benckert, “Measurement of dynamic crack tip displacement field by speckle photography and interferometry,” Opt. Lasers Eng. 19, 299–312 (1993).
[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]

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimisation by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (1)

E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines (Wiley, New York, (1989).

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Figures (11)

Fig. 1
Fig. 1

(a) Computer-generated sine fringe pattern, with superimposed disk to simulate a region of low SNR. (b) Wrapped phase map from application of Fourier transform method to (a). Black represents −π, and white represents +π.

Fig. 2
Fig. 2

(a) Portion of wrapped phase map obtained by phase-stepping speckle interferometry. Black represents −π, and white represents +π. (b) Wrapped phase map from the same data as in (a) but after smoothing the sine and cosine correlation fringes with a 3 × 3 filter. (c), (d) Discontinuity sources in (a) and (b), respectively. Filled circles represent negative sources; open ones represent positive sources.

Fig. 3
Fig. 3

Variation of negative and positive source densities against distance averaged over many negative sources in one speckle-interferometry phase map.

Fig. 4
Fig. 4

Branch-cut arrangements: (a) correct one for three dipoles and a monopole, (b) branch cuts positioned by the nearest neighbor algorithm, starting from the left and working right, (c) correct branch cuts for sources in (b).

Fig. 5
Fig. 5

Branch-cut pattern that is (a) incorrect but a set of stable marriages, (b) correct but not a set of stable marriages (A and B prefer each other to their current partners).

Fig. 6
Fig. 6

Simulated annealing move generator: (a) swap, in which cuts AB and CD are broken and replaced by cuts AC and BD; (b) recombine, in which two monopoles, joined to the boundary, form a dipole; (c) split, in which a dipole is split into two monopoles connected to the boundary.

Fig. 7
Fig. 7

Distribution of cuts at four stages of a simulated annealing run, decreasing in temperature from (a) to (d); (d) represents the final cut distribution.

Fig. 8
Fig. 8

(a) Typical branch cut situation after each source has been joined to its nearest neighbor, or to the boundary if this was closer. (b) Commonly occurring charged fragments. (c) The source marked with an asterisk is the one being considered. It has two partners with one connection. The closest second nearest neighbor of its partners is joined to that partner. (d) A disconnection stage. The source marked with an asterisk has one partner with one connection and another with two. The join to the partner with two is broken. (e) Similar to (d). The joins to the sources with more than one connection are broken. (f) Sources with a connection to the boundary and one to a source with just one connection are disconnected from the boundary. (g) The source marked with an asterisk has two connections, both to sources with more than one connection. The longest join is broken.

Fig. 9
Fig. 9

Phase maps from an unfiltered speckle interferogram with seven fringes (top row) and 30 fringes (bottom row) across the field of view unwrapped, using a, e, no branch cuts; b, f, nearest neighbor branch cuts; c, g, stable marriage algorithm branch cuts; d, h, modified nearest neighbor branch cuts.

Fig. 10
Fig. 10

rms phase error obtained by unwrapping with three branch-cut placement algorithms on speckle-interferometry data with three different discontinuity source densities.

Fig. 11
Fig. 11

a, Wrapped phase map obtained by Fourier transform analysis of a speckle interferogram showing the interaction between a stress wave and horizontal stationary crack; b, map in a, unwrapped by use of the nearest neighbor algorithm; c, map in a, unwrapped by use of the modified nearest neighbor algorithm.

Equations (6)

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d ( i ) = [ { Φ ( i ) Φ ( i 1 ) } / 2 π ] ,
ν = i = 1 N d ( i ) .
Δ Φ = tan 1 [ Δ I 42 ( t 2 ) Δ I 13 ( t 1 ) Δ I 13 ( t 2 ) Δ I 42 ( t 1 ) Δ I 13 ( t 2 ) Δ I 13 ( t 1 ) + Δ I 42 ( t 2 ) Δ I 42 ( t 1 ) ] ,
Δ I i j ( t ) = I i ( t ) I j ( t ) .
c r + 1 = c r 1 + [ c r ln ( 1 + δ ) / 3 σ ] .
u x = Ω y + u 0 ,

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