Abstract

An algorithm for unwrapping noisy phase maps by means of branch cuts has been proposed recently. These cuts join discontinuity sources that mark the beginning or end of a 2π phase discontinuity. After the placement of branch cuts, the unwrapped phase map is unique and independent of the unwrapping route. We show how a minimum-cost-matching graph-theory method can be used to find the set of cuts that has the global minimum of total cut length, in time approximately proportional to the square of the number of sources. The method enables one to unwrap unfiltered speckle-interferometry phase maps at higher source densities (0.1 sources pixel−1) than any previous branch-cut placement algorithm.

© 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. D. W. Robinson, “Phase unwrapping methods,” Interferogram Analysis, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 194–229.
  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. J. M. Huntley, “New methods for unwrapping noisy phase maps,” presented at the SPIE International Conference on Interferometry, Warsaw, Poland, 16–20 May, 1994.
  7. J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A (to be published).
  8. D. R. Wilson, “Minimax theorems in graph theory,” Selected Topics in Graph Theory, L. W. Beineke, R. J. Wilson, eds. (Academic, New York, 1978), pp. 250–252.
  9. M. S. Bazaraa, J. J. Jarvis, H. D. Sherali, Linear Programming and Network Flows, 2nd ed. (Wiley, New York, 1990).
  10. A. Gibbons, Algorithmic Graph Theory, (Cambridge U. Press, Cambridge, UK, 1985).
  11. R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef] [PubMed]

1995

1991

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

1988

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

1986

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

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]

Bazaraa, M. S.

M. S. Bazaraa, J. J. Jarvis, H. D. Sherali, Linear Programming and Network Flows, 2nd ed. (Wiley, New York, 1990).

Buckland, J. R.

J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A (to be published).

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]

Cusack, R.

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]

Gibbons, A.

A. Gibbons, Algorithmic Graph Theory, (Cambridge U. Press, Cambridge, UK, 1985).

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]

Huntley, J. M.

R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
[CrossRef] [PubMed]

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

J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A (to be published).

J. M. Huntley, “New methods for unwrapping noisy phase maps,” presented at the SPIE International Conference on Interferometry, Warsaw, Poland, 16–20 May, 1994.

Jarvis, J. J.

M. S. Bazaraa, J. J. Jarvis, H. D. Sherali, Linear Programming and Network Flows, 2nd ed. (Wiley, New York, 1990).

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]

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]

Sherali, H. D.

M. S. Bazaraa, J. J. Jarvis, H. D. Sherali, Linear Programming and Network Flows, 2nd ed. (Wiley, New York, 1990).

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]

Wilson, D. R.

D. R. Wilson, “Minimax theorems in graph theory,” Selected Topics in Graph Theory, L. W. Beineke, R. J. Wilson, eds. (Academic, New York, 1978), pp. 250–252.

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.

Opt. Eng.

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.

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

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

J. M. Huntley, “New methods for unwrapping noisy phase maps,” presented at the SPIE International Conference on Interferometry, Warsaw, Poland, 16–20 May, 1994.

J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A (to be published).

D. R. Wilson, “Minimax theorems in graph theory,” Selected Topics in Graph Theory, L. W. Beineke, R. J. Wilson, eds. (Academic, New York, 1978), pp. 250–252.

M. S. Bazaraa, J. J. Jarvis, H. D. Sherali, Linear Programming and Network Flows, 2nd ed. (Wiley, New York, 1990).

A. Gibbons, Algorithmic Graph Theory, (Cambridge U. Press, Cambridge, UK, 1985).

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Figures (10)

Fig. 1
Fig. 1

Phase-unwrapping method. The phase at Q relative to P is obtained from the number of 2π discontinuities crossed by any path; noise results in discontinuity sources at points 1 and 2, causing unwrapping errors that are dependent on the path taken.

Fig. 2
Fig. 2

(a) Computer-generated cosine fringe pattern with super-imposed disks D1 and D2 to simulate regions of low SNR. (b) Wrapped phase map from the application of the Fourier-transform method to (a). (c) Source distribution for (b); the hollow and the filled circles are used to represent +1 and −1 sources, respectively.

Fig. 3
Fig. 3

Example of a bipartite graph. Vertices V1 and V2 represent positive and negative sources, respectively. Edges between vertices are shown by lines (both dashed and solid); matchings are shown by solid lines: (a) matching of two; (b) matching of three.

Fig. 4
Fig. 4

(a) Distribution of sources to be matched, showing cut lengths, with an elliptical boundary. (b) Optimal matching.

Fig. 5
Fig. 5

(a) Matrix representing distances between sources. Columns and rows represent +1 and −1 sources, respectively. Distances taken as infinite are represented by dashes. (b) Reduced distance matrix obtained from (a) by subtraction of the minimum entries in each column and then in each row. Independent zeros are shown in bold; six lines are needed to cross out all the zeros. (c) Final matrix. The minimum uncrossed entry in (b), 1, is subtracted from all the uncrossed entries in (b) to obtain (c). The seven independent zeros represent the optimal matching.

Fig. 6
Fig. 6

(a) Graph representing the zeros in the matrix in Fig. 4(b); independent zeros are represented by solid lines, other zeros by dashed lines. (b) Graph representing the zeros in the matrix in Fig. 4(c); solid lines represent the optimal matching.

Fig. 7
Fig. 7

Phase maps from an unfiltered speckle interferogram with 7 fringes (left column) and 30 fringes (right column) across the field of view: (a), (e) wrapped maps; (b), (f) unwrapped without branch cuts; (c), (g) unwrapped with branch cuts placed by modified nearest-neighbor method; (d), (h) unwrapped with minimum-cost-matching branch cuts. The linear ramp is subtracted from (c), (d), (g), and (h) to show the absolute value of the phase error. White and black represent zero and 4π phase errors, respectively.

Fig. 8
Fig. 8

Cut distributions for a phase map consisting of the top one ninth of Fig. 7(e) placed by (a) the nearest-neighbor method, (b) the modified nearest-neighbor method, and (c) the minimum-cost-matching algorithm.

Fig. 9
Fig. 9

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

Fig. 10
Fig. 10

(a) Wrapped phase map for a disk undergoing compression between two anvils; phase maps from (a), unwrapped by (b) the modified nearest-neighbor method and (c) the minimum-cost-matching method. The magnitude of the phase is shown (negative phases displayed as positive) to enhance contrast.

Equations (4)

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d ( i ) = [ Φ ( i ) - Φ ( i - 1 ) 2 π ] ,
ν = i = 1 N d ( i ) .
C = i j c i j m i j ,
u x = Ω y + u 0 ,

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