Abstract

A new algorithm is proposed for solving the problems associated with discontinuity sources in phase maps. It is based on the stable-marriages algorithm and is implemented as a recursive procedure.

With this technique, discontinuity sources of opposite sign are connected by a set of cut lines that fulfills a stability criterion and possesses the minimum cut length of the stable sets. The algorithm is fast and easy to implement and has proved efficient, as experimental results show.

© 1995 Optical Society of America

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References

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  1. K. A. Stetson, “Phase-step interferometry of irregular shapes by using an edge-following algorithm,” Appl. Opt. 31, 5320–5325 (1992).
    [CrossRef] [PubMed]
  2. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).
  3. H. A. Vrooman, A. M. Mass, “Image processed algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [CrossRef] [PubMed]
  4. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  5. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  6. P. Andrä, U. Mieth, W. Osten, “Strategies for unwrapping noisy interferograms in phase-sampling interferometry,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 50–60 (1991).
  7. M. Servin, R. Rodríguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
    [CrossRef]
  8. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  9. J. M. Huntley, R. Cusack, H. Saldner, “New phase unwrapping algorithms,” Fringe ’93, Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jueptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1993), pp. 148–153.
  10. D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69(1), 9–14 (1962).
    [CrossRef]
  11. N. Wirth, Algorithms + Data Structure = Programs (Prentice-Hall, Englewood Cliffs, N.J., 1976).

1994

M. Servin, R. Rodríguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

1992

1991

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]

1962

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69(1), 9–14 (1962).
[CrossRef]

Andrä, P.

P. Andrä, U. Mieth, W. Osten, “Strategies for unwrapping noisy interferograms in phase-sampling interferometry,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 50–60 (1991).

Bone, D. J.

Bryanston-Cross, P. J.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

Cusack, R.

J. M. Huntley, R. Cusack, H. Saldner, “New phase unwrapping algorithms,” Fringe ’93, Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jueptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1993), pp. 148–153.

Gale, D.

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69(1), 9–14 (1962).
[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. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

J. M. Huntley, R. Cusack, H. Saldner, “New phase unwrapping algorithms,” Fringe ’93, Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jueptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1993), pp. 148–153.

Judge, T. R.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

Mass, A. M.

Mieth, U.

P. Andrä, U. Mieth, W. Osten, “Strategies for unwrapping noisy interferograms in phase-sampling interferometry,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 50–60 (1991).

Moore, A. J.

M. Servin, R. Rodríguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

Osten, W.

P. Andrä, U. Mieth, W. Osten, “Strategies for unwrapping noisy interferograms in phase-sampling interferometry,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 50–60 (1991).

Rodríguez-Vera, R.

M. Servin, R. Rodríguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

Saldner, H.

J. M. Huntley, R. Cusack, H. Saldner, “New phase unwrapping algorithms,” Fringe ’93, Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jueptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1993), pp. 148–153.

Servin, M.

M. Servin, R. Rodríguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[CrossRef]

Shapley, L. S.

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69(1), 9–14 (1962).
[CrossRef]

Stetson, K. A.

Towers, D. P.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

Vrooman, H. A.

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]

Wirth, N.

N. Wirth, Algorithms + Data Structure = Programs (Prentice-Hall, Englewood Cliffs, N.J., 1976).

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.

D. Gale, L. S. Shapley, “College admissions and the stability of marriage,” Am. Math. Mon. 69(1), 9–14 (1962).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

M. Servin, R. Rodríguez-Vera, A. J. Moore, “A robust cellular processor for phase unwrapping,” J. Mod. Opt. 41, 119–127 (1994).
[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. M. Huntley, R. Cusack, H. Saldner, “New phase unwrapping algorithms,” Fringe ’93, Proceedings of the Second International Workshop on Automatic Processing of Fringe Patterns, W. Jueptner, W. Osten, eds. (Akademie-Verlag, Berlin, 1993), pp. 148–153.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

P. Andrä, U. Mieth, W. Osten, “Strategies for unwrapping noisy interferograms in phase-sampling interferometry,” in Industrial Applications of Holographic and Speckle Measuring Techniques, W. P. Jueptner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1508, 50–60 (1991).

N. Wirth, Algorithms + Data Structure = Programs (Prentice-Hall, Englewood Cliffs, N.J., 1976).

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Figures (9)

Fig. 1
Fig. 1

Illustration of the problems caused by discontinuity sources in phase maps. The phase difference between P 1 and P 2 is different when calculated by path A than by path B.

Fig. 2
Fig. 2

Illustration of the standard stability criterion for the stable-marriages algorithm. According to this criterion, solution (a) is the stable one. However, the total cut length of solution (b) is evidently smaller, and therefore solution (b) is better for our purposes.

Fig. 3
Fig. 3

Construction for the stability criterion adopted by us. n * is the first unmarried negative pole in the list of preferences of p different than n. Similarly, p* is the first unmarried positive pole in the list of preferences of n different than p. The marriage between p and n is stable if d 1 + d 2d 3 + d 4.

Fig. 4
Fig. 4

Problems of the stability criterion adopted by us. If p is the first positive pole to seek a partner, no marriage for it is stable. The logical marriage between p and n is unstable because d 1 + d 2 > d 3 + d 4, so the desirable solution, represented by solid arrows, is unstable. This solution is, however, stable with our criterion if p is not the first processed pole.

Fig. 5
Fig. 5

Illustration of the criterion used for generation of image poles. According to the criterion, image poles p ib , n ib , and n ic are created for n b , p b , and p c , respectively, while no image poles are generated for poles p a and n a (that is to say, p ia and n ia are not created).

Fig. 6
Fig. 6

(a) Real phase map. (b) Poles of the phase map. Positive poles are represented by black dots and negative poles by white dots. (c) Cut lines obtained by the algorithm. (d) Resulting unwrapped phase when the cut lines are used.

Fig. 7
Fig. 7

Cut lines for the phase map of Fig. 6 when a safety margin of two pixels is used.

Fig. 8
Fig. 8

(a) Real phase map. (b) Poles of the phase map. Positive poles are represented by black dots and negative poles by white dots. (c) Cut lines obtained by the algorithm when a safety margin of 0 is used. (d) Resulting unwrapped phase when the cut lines are used.

Fig. 9
Fig. 9

(a) Real phase map. (b) Poles of the phase map. Positive poles are represented by black dots and negative poles by white dots. (c) Cut lines obtained by our algorithm. (d) Cut lines obtained by a nearest-neighbor method. (e) Resulting unwrapped phase when the cut lines obtained by our algorithm are used.

Equations (7)

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n = i [ ϕ ( i ) - ϕ ( i - 1 ) 2 π ] ,
D [ p , n ] = dish ( p , n ) ,
D = [ 10 4 8 4 16 7 3 1 14 ] ;
N P = [ 2 3 1 1 3 2 2 1 3 ] ,
P N = [ 3 2 1 3 1 2 2 1 3 ] .
d 1 = D [ p , n ] , d 2 = D [ p * , n * ] , d 3 = D [ p , n * ] , d 4 = D [ p * , n ] .
P = P R P I , N = N R N I ,

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