Abstract

Phase unwrapping with the branch-cut method has been successfully used in many different applications in recent years. Most methods to set the branch cuts minimize the overall cut length. However, this technique fails in different cases, since this criterion is based mainly on statistical examinations. We show how the orientation and direction of the phase map help to create additional physical criteria that can be used to optimize the setting of the branch cuts. We show how these new criteria can be implemented into an energy function that will be minimized by a simulated annealing algorithm in order of a correct setting of the branch cuts. Finally, we present experimental results from electronic speckle pattern interferometry and digital holography phase maps.

© 2000 Optical Society of America

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  1. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  2. H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
    [CrossRef]
  3. J. M. Huntley, H. O. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
    [CrossRef] [PubMed]
  4. M. Takeda, “Recent progress in phase unwrapping techniques,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 334–343 (1996).
    [CrossRef]
  5. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).
  6. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  7. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  8. 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 Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
    [CrossRef]
  9. R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase unwrapping algorithm,” Appl. Opt. 35, 781–789 (1995).
    [CrossRef]
  10. M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
    [CrossRef]
  11. D. Winter, D. Bergmann, B.-W. Lührig, R. Ritter, “Evaluation of ESPI phase images with regional discontinuities: area based unwrapping,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, pp. 301–311 (1993).
  12. J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [CrossRef] [PubMed]
  13. J. A. Quiroga, A. González-Cano, E. Bernabeu, “Stable-marriages algorithm for preprocessing phase maps with discontinuity sources,” Appl. Opt. 34, 5029–5038 (1995).
    [CrossRef] [PubMed]
  14. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  15. R. V. V. Vidal ed., Applied Simulated Annealing (Springer-Verlag, Berlin, 1993).
    [CrossRef]
  16. B. Gutmann, H. Weber, “Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing,” Appl. Opt. 38, 5577–5593 (1999).
    [CrossRef]
  17. B. Jähne, Image Processing for Scientific Applications (CRC Press, Boca Raton, Fla., 1997).
  18. J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A 12, 1990–1996 (1995).
    [CrossRef]
  19. E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, Chichester, N.Y., 1989).
  20. Images used with the kind permission of Giancarlo Pedrini, Institute for Technical Optics, University of Stuttgart, Stuttgart, Germany.
  21. Image used with the kind permission of the Departement of Physics, Applied Optics, Carl von Ossietzky University, Oldenburg, Germany.

1999 (1)

1997 (1)

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

1996 (1)

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

1995 (4)

1993 (1)

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 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]

1987 (1)

1983 (1)

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

Aarts, E.

E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, Chichester, N.Y., 1989).

Abe, T.

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[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 Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Bergmann, D.

D. Winter, D. Bergmann, B.-W. Lührig, R. Ritter, “Evaluation of ESPI phase images with regional discontinuities: area based unwrapping,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, pp. 301–311 (1993).

Bernabeu, E.

Buckland, J. R.

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 Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Cusack, R.

R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase unwrapping algorithm,” Appl. Opt. 35, 781–789 (1995).
[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 Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Gelatt, C. D.

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

Ghiglia, D. C.

Goldrein, H. T.

R. Cusack, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase unwrapping algorithm,” Appl. Opt. 35, 781–789 (1995).
[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]

González-Cano, A.

Gutmann, B.

Huntley, J. M.

Jähne, B.

B. Jähne, Image Processing for Scientific Applications (CRC Press, Boca Raton, Fla., 1997).

Kirkpatrick, S.

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

Korst, J.

E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, Chichester, N.Y., 1989).

Lührig, B.-W.

D. Winter, D. Bergmann, B.-W. Lührig, R. Ritter, “Evaluation of ESPI phase images with regional discontinuities: area based unwrapping,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, pp. 301–311 (1993).

Mastin, G. A.

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 Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

Quiroga, J. A.

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 Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Ritter, R.

D. Winter, D. Bergmann, B.-W. Lührig, R. Ritter, “Evaluation of ESPI phase images with regional discontinuities: area based unwrapping,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, pp. 301–311 (1993).

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 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 Astronomy Observatories,” Opt. Eng. 30, 1405–1414 (1991).
[CrossRef]

Romero, L. A.

Saldner, H. O.

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

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

Takeda, M.

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

M. Takeda, “Recent progress in phase unwrapping techniques,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 334–343 (1996).
[CrossRef]

Turner, S. R. E.

Vecchi, M. P.

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

Weber, H.

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]

Winter, D.

D. Winter, D. Bergmann, B.-W. Lührig, R. Ritter, “Evaluation of ESPI phase images with regional discontinuities: area based unwrapping,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, pp. 301–311 (1993).

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. (6)

J. Opt. Soc. Am. A (2)

Opt. Eng. (3)

H. O. Saldner, J. M. Huntley, “Profilometry using temporal phase unwrapping and a spatial light modulator-based fringe projector,” Opt. Eng. 36, 610–615 (1997).
[CrossRef]

M. Takeda, T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345–2351 (1996).
[CrossRef]

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 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]

Science (1)

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

Other (8)

R. V. V. Vidal ed., Applied Simulated Annealing (Springer-Verlag, Berlin, 1993).
[CrossRef]

E. Aarts, J. Korst, Simulated Annealing and Boltzmann Machines, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley, Chichester, N.Y., 1989).

Images used with the kind permission of Giancarlo Pedrini, Institute for Technical Optics, University of Stuttgart, Stuttgart, Germany.

Image used with the kind permission of the Departement of Physics, Applied Optics, Carl von Ossietzky University, Oldenburg, Germany.

D. Winter, D. Bergmann, B.-W. Lührig, R. Ritter, “Evaluation of ESPI phase images with regional discontinuities: area based unwrapping,” in Interferometry: Techniques and Analysis II, O. Y. Kwon, G. M. Brown, M. Kujawinska, eds., Proc. SPIE2003, pp. 301–311 (1993).

M. Takeda, “Recent progress in phase unwrapping techniques,” in Optical Inspection and Micromeasurements, C. Gorecki, ed., Proc. SPIE2782, 334–343 (1996).
[CrossRef]

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping (Wiley, New York, 1998).

B. Jähne, Image Processing for Scientific Applications (CRC Press, Boca Raton, Fla., 1997).

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Figures (23)

Fig. 1
Fig. 1

Integration of phase differences in a phase map between two pixels P1 and P2 along two different paths {rn(A)} and {rn(B)}. The closed loop {rn(C)} encircles a discontinuity source. The integration of phase differences along this path does not vanish. This phase map is gained from an ESPI measurement21 after separate low-pass filtering of the sine part and the cosine part. The gaps in the 2π-discontinuity lines that create pairs of sources are due to the filtering process and the low SNR at this place.

Fig. 2
Fig. 2

Phase map of Fig. 1 with discontinuity sources and branch cuts. The branch cuts continue the 2π-discontinuity lines in regions where the filtering process has flattened these lines.

Fig. 3
Fig. 3

Four states of a cooling process for 530 discontinuity sources with the sum of cut lengths as the energy function. Energy and temperature decrease from the initial state (a), which is a random configuration to the final solution (d), which represents a set of cuts with minimal overall cut length. The temperature decreases from (a) to (d). The running time of the algorithm was 447 s.

Fig. 4
Fig. 4

Neighborhood of gray values that have an orientation in direction of vector k. Perpendicular to this vector the gray values do not change.

Fig. 5
Fig. 5

Phase map of a digital holography measurement20 and its first Gaussian pyramid plane. The original phase map was filtered with a binomial mask, which was applied separately to the sine part and the cosine part of the phase to preserve the 2π-discontinuity lines. However, owing to a low SNR, the fringes vanish in the upper right-hand area.

Fig. 6
Fig. 6

Orientation field O(r) of Fig. 5 calculated by means of its first Gaussian pyramid plane. The orientation field represents a global view on the course of the fringes.

Fig. 7
Fig. 7

Direction field P(r) of Fig. 6.

Fig. 8
Fig. 8

(a) Different sets of branch cuts for the phase map of a low-pass-filtered digital holography measurement (separate filtering of sine and cosine parts). These sets represent local minima of the linear energy function as given by Eq. (8). (b) Set of branch cuts that is almost correct except for the border connection in the upper right-hand corner. The set has an energy of E = 627. (c) Badly placed branch cuts with an energy of E = 634.

Fig. 9
Fig. 9

Example for a phase map (a) that cannot be unwrapped by means of setting the branch cuts through minimizing the overall cut length. The correct setting of cuts in (b) contains an energy of E = 187 (linear term) and is not a local minimum, because in a next step the cuts can be set as shown in (c) with an energy of E = 183. (d) Setting that minimizes the overall cut length (E = 146).

Fig. 10
Fig. 10

Relation between density of discontinuity sources and average length of branch cuts during filtering with an ESPI measurement with binomial masks of order r.

Fig. 11
Fig. 11

Phase map with direction field P(r) that indicates the direction of the global phase gradient. The 2π-discontinuity lines are always perpendicular to the phase gradient, because they are equipotential lines. Thus the branch cuts that substitute the discontinuity line also have to be perpendicular to the phase gradient.

Fig. 12
Fig. 12

Phase map (a) that cannot be solved by the simple condition that branch cuts have to be perpendicular to the phase gradient. (b) Incorrect set of cuts that is perpendicular to the phase gradient; however, the setting is completely wrong. The branch cuts in (c) also fulfill this condition; furthermore, the angle enclosed between branch cut a and direction field P(r) is positive.

Fig. 13
Fig. 13

Illustration of the new energy term.

Fig. 14
Fig. 14

Histogram of the direction energy J of correct branch cuts set in phase maps of ESPI measurements.

Fig. 15
Fig. 15

Histogram of the direction energy J of correct branch cuts set in phase maps of shadow moiré measurements with aliasing effects.

Fig. 16
Fig. 16

Discontinuity sources that are separated by the border of the measurement range. As the phase increases from right to left the upper border separates sources of negative sign, whereas the lower separates sources of positive sign. At the left- and the right-hand borders, both kinds of separated source are possible.

Fig. 17
Fig. 17

Energy of the settings of Fig. 8 for increasing values of the coupling parameter χ in Eq. (26). The energy values become more and more separated, thus stabilizing the correct solution.

Fig. 18
Fig. 18

Energy for the settings of Fig. 9. For increasing coupling parameter the energy of the correct setting becomes less than the energy of the next possible configuration with smaller cut length, thus creating a minimum for the correct solution. Furthermore for χ ≥ 17 the energy also separates from the setting with minimal cut length.

Fig. 19
Fig. 19

(a) Low-pass-filtered phase map of an digital holography measurement. (b) Superimposed direction field.

Fig. 20
Fig. 20

(a) Branch cuts set by minimization of the overall cut-length. (b) Fringe order after phase unwrapping with this set of branch cuts.

Fig. 21
Fig. 21

(a) Branch cuts set by minimization of Eq. (26) with χ = 40. (b) Fringe order after phase unwrapping with this set of branch cuts.

Fig. 22
Fig. 22

(a) Phase map of an out-of-plane ESPI measurement with a horizontal fringe displacement that is due to a crack in the object. (b) Phase map after separate filtering of the sine part and the cosine part. The 2π-discontinuity lines have been preserved except in some regions of low SNR. The number of sources was reduced from 32887 to 700.

Fig. 23
Fig. 23

(a) Unwrapped phase of the ESPI measurement of Fig. 22 after the branch-cuts were set with length minimization. The crack now appears in the continuous region. (b) Unwrapped phase after the branch cuts were set with length and direction optimization (χ = 60). The crack now appears at the correct place.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

ϕwpr=ϕrmod 2π.
ϕr=ϕwpr+2πΛr,  Λ.
Wˆϕri=ϕri+2πkri,  kri,
ˆϕwpri=Wˆϕwpri-ϕwpri-1.
ϕrN=ϕwpr0+i=1N ˆϕwpri.
i ˆϕwpCi=0.
i ˆϕwpCi=2πn,  n.
E=i=1Nbc |ai|,
Jpq=Dp  Dq
Jxxϕwpr=Dxϕwpr  Dxϕwpr,
Jyyϕwpr=Dyϕwpr  Dyϕwpr,
Jxyϕwpr=Jyxϕwpr=Dxϕwpr  Dyϕwpr.
αr=12atan2JxyϕwprJyy-Jxxϕwpr
Or=cos αrsin αr.
Cr=Jyy-Jxxϕwpr  Jyy-Jxxϕwpr+4Jxyϕwpr  JxyϕwprJxx+Jyyϕwpr  Jxx+Jyyϕwpr.
Dx=-303-10010-303,  Dy=-3-10-30003103,
r=λ2/π2.
Δϕwpr=i=-kk Wˆϕwpr+Or · i+1-ϕwpr+Or · i,k=λr/2.
Dr=0if Δϕwpr01if Δϕwpr<0.
Pr=Orif Dr=0-Orif Dr=1.
a·grad ϕ=0
νi·grad ϕ=1.
J=-i=1Nbc1|ai|t=t1t2Prt·νˆidt.
JC=-i=1Nbc1|ai|t=t1t2Prt·νˆiCrtdt.
E=L1+μJC,
EJ=L1+χJc=i=1Nbc |ai|-χ i=1Nbc1|ai|t=t1t2Prt·νˆiCrtdt,

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