Abstract

In the process of phase unwrapping for an image obtained by an interferometer or in-line holography, noisy image data may pose difficulties. Traditional phase unwrapping algorithms used to estimate a two-dimensional phase distribution include much estimation error, due to the effect of singular points. This paper introduces an accurate phase-unwrapping algorithm based on three techniques: a rotational compensator, unconstrained singular point positioning, and virtual singular points. The new algorithm can confine the effect of singularities to the local region around each singular point. The phase- unwrapped result demonstrates that accuracy is improved, compared with past methods based on the least-squares approach.

© 2010 Optical Society of America

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    [CrossRef]
  3. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
    [CrossRef]
  4. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing,” Appl. Opt. 38, 5577–5593 (1999).
    [CrossRef]
  14. S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730–743 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sensing 45, 3240–3251 (2007).
    [CrossRef]
  24. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  25. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.
  26. S. Tomioka, S. Nisiyama, and T. Enoto, “Nonlinear least square regression by adaptive domain method with multiple genetic algorithms,” IEEE Trans. Evol. Comput. 11, 1–16(2007).
    [CrossRef]
  27. J. A. Quiroga, A. González-Cano, and E. Bernabeu, “Stable-marriages algorithm for preprocessing phase maps with discontinuity sources,” Appl. Opt. 34, 5029–5038(1995).
    [CrossRef] [PubMed]

2007 (3)

S. Tomioka, S. Nisiyama, and T. Enoto, “Nonlinear least square regression by adaptive domain method with multiple genetic algorithms,” IEEE Trans. Evol. Comput. 11, 1–16(2007).
[CrossRef]

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sensing 45, 3240–3251 (2007).
[CrossRef]

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730–743 (2007).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

1998 (2)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sensing 36, 813–821 (1998).
[CrossRef]

1995 (4)

1994 (1)

1993 (1)

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

1991 (1)

1989 (2)

1988 (3)

1986 (1)

1985 (1)

1982 (1)

1979 (1)

1977 (2)

1974 (1)

1953 (1)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.

Bachor, H.-A.

Bernabeu, E.

Bone, D. J.

Brangaccio, D. J.

Breuckmann, B.

Bruning, J. H.

Buckland, J. R.

Burton, D. R.

Costantini, M.

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sensing 36, 813–821 (1998).
[CrossRef]

Cuche, E.

Cusack, R.

Depeursinge, C.

Enoto, T.

S. Tomioka, S. Nisiyama, and T. Enoto, “Nonlinear least square regression by adaptive domain method with multiple genetic algorithms,” IEEE Trans. Evol. Comput. 11, 1–16(2007).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.

Fried, D. L.

Gallagher, J. E.

Gdeisat, M. A.

Ghiglia, D. C.

Goldrein, H. T.

Goldstein, R. M.

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

González-Cano, A.

Grebe, R.

Gutmann, B.

Herriott, D. R.

Hirose, A.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sensing 45, 3240–3251 (2007).
[CrossRef]

Hudgin, R. H.

Hunt, B. R.

Huntley, J. M.

Ina, H.

Karout, S. A.

Kobayashi, S.

Lalor, M. J.

Marquet, P.

McKelvie, J.

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.

Nisiyama, S.

S. Tomioka, S. Nisiyama, and T. Enoto, “Nonlinear least square regression by adaptive domain method with multiple genetic algorithms,” IEEE Trans. Evol. Comput. 11, 1–16(2007).
[CrossRef]

Perry, K. E.

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Quiroga, J. A.

Romero, L. A.

Rosenfeld, D. P.

Sandeman, R. J.

Takahashi, T.

Takajo, H.

Takeda, M.

Thieme, W.

Tomioka, S.

S. Tomioka, S. Nisiyama, and T. Enoto, “Nonlinear least square regression by adaptive domain method with multiple genetic algorithms,” IEEE Trans. Evol. Comput. 11, 1–16(2007).
[CrossRef]

Turner, S. R. E.

Vandenhouten, R.

Weber, H.

Werner, C. L.

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

White, A. D.

Yamaki, R.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sensing 45, 3240–3251 (2007).
[CrossRef]

Zebker, H. A.

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

Appl. Opt. (12)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

B. Breuckmann and W. Thieme, “Computer-aided analysis of holographic interferograms using the phase-shift method,” Appl. Opt. 24, 2145–2149 (1985).
[CrossRef] [PubMed]

D. J. Bone, H.-A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
[CrossRef] [PubMed]

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

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

B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing,” Appl. Opt. 38, 5577–5593 (1999).
[CrossRef]

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

R. Vandenhouten and R. Grebe, “Phase reconstruction and unwrapping from holographic interferograms of partially absorbent phase objects,” Appl. Opt. 34, 1401–1406 (1995).
[CrossRef] [PubMed]

J. A. Quiroga, A. González-Cano, and E. Bernabeu, “Stable-marriages algorithm for preprocessing phase maps with discontinuity sources,” Appl. Opt. 34, 5029–5038(1995).
[CrossRef] [PubMed]

J. R. Buckland, J. M. Huntley, and S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
[CrossRef] [PubMed]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730–743 (2007).
[CrossRef] [PubMed]

IEEE Trans. Evol. Comput. (1)

S. Tomioka, S. Nisiyama, and T. Enoto, “Nonlinear least square regression by adaptive domain method with multiple genetic algorithms,” IEEE Trans. Evol. Comput. 11, 1–16(2007).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (2)

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sensing 36, 813–821 (1998).
[CrossRef]

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sensing 45, 3240–3251 (2007).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Opt. Lasers Eng. (1)

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

Opt. Lett. (1)

Radio Sci. (1)

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

Other (2)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.

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Figures (10)

Fig. 1
Fig. 1

Integration path to evaluate the RC.

Fig. 2
Fig. 2

Elementary path including an USP: Δ θ l means the facing angle of the side with the ends r l and r l + 1 and Δ Φ l means the difference of the wrapped phases.

Fig. 3
Fig. 3

Determination of dipole pairs. (a) Definition of the virtual SP candidates: Square frame shows the measurement area. Open and solid circles show positive and negative SPs, respectively. Each dashed arrow shows the correspondence between original and virtual SP. (b) Nearest SP search and pairing in the first iteration: Each arrow shows the nearest opposite-signed SP. The SPs encircled are paired as dipoles. The ticked virtual candidate is removed from the list of candidates in subsequent steps. (c) Second iteration: SPs enclosed with dashed ellipses have been already paired. (d) Elimination of removable virtual SP pairs.

Fig. 4
Fig. 4

Unwrapped and rewrapped phase map: In each figure, the left-hand side figure shows the unwrapped or original phase map, where the phase increases with the increase of brightness, and the right-hand side figure shows a rewrapped or wrapped phase map. (a) Original phase map. (b) Goldstein path-following method. (c) LS-DCT. (d) Proposed method using all the RC, the USP, and the VSP.

Fig. 5
Fig. 5

Distribution of SPs: All of the SPs are coupled with another SP. In the legend, both “Nearest dipole” and “Not nearest dipole” mean that both SPs of a pair are located inside the measurement area; “Nearest” shows that the SPs are nearest each other; Alternately, pairs are formed from “Dipole with virtual SP” located inside the area and “Virtual SP” located outside the area.

Fig. 6
Fig. 6

SP around the bottom border: Pixels where the wrapped data are defined are located at intersections on the grid. The points denoted by solid symbols are constrained SPs located at the center of the smallest grid; i.e., the elementary loop. The points denoted by open symbols are unconstrained SPs. The regular and the inverted triangle symbols express positive and negative SPs, respectively. The thick line at the bottom is the border of the measurement area. The SPs outside the area (under the border) correspond to VSPs.

Fig. 7
Fig. 7

Frequency histogram for distance from a positive SP to the nearest negative SP.

Fig. 8
Fig. 8

Computational time: The horizontal axis N denotes the one-dimensional area size in pixels. “ RC + USP + VSP ” shows the result by the proposed method. The computational time is measured with a PC including an Intel Core 2 DUO CPU with 2.13 GHz clock in a single CPU operation mode.

Fig. 9
Fig. 9

Unwrapped phase of fringe by Mach–Zehnder interferometer for candle flame: The I, Φ, M, ϕ, and W { } show observed fringe patterns with enhancement of contrast, wrapped phase maps by Fourier domain method, distributions of SPs (positive and negative SPs are represented by white and black dots, respectively), unwrapped phase maps by the proposed method (RC, USP, and VSP), and wrapping operators, respectively. The subscripts “obj” and “bg” mean object and background, respectively. The unwrapped phase difference, Δ ϕ = ϕ obj ϕ bg , shows the actual phase shift by the flame.

Fig. 10
Fig. 10

Comparison of unwrapped phase differences: “ RC + USP + VSP ” shows the unwrapped phase difference, Δ ϕ , along y = 85 in Fig. 9, and the “LS-DCT” shows it by the LS-DCT method.

Tables (1)

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Table 1 Accuracy Comparison among Algorithms by Planar Function Fitting

Equations (23)

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Φ = W { ϕ } ϕ Int [ ϕ 2 π ] 2 π ,
g ( r , r ) W { Φ ( r ) Φ ( r ) } s ^ ( r r ) ,
c g · s ^ d l = 0 .
c g · s ^ d l = 2 π k m k = 2 π M , m k { 1 , 0 , 1 } ,
g = ϕ + × A .
c g · s ^ d l = c × A · s ^ d l .
c × A · s ^ d l = 2 π k m k .
A = k A k .
ϕ ( r ) ϕ ( r ) = r r ϕ · s ^ d l = r r ( g k × A k ) · s ^ d l .
c × A k · s ^ d l = 2 π m k .
× A k · s ^ = a z R e R × e z · s ^ = a z R e R · n ^ ,
a z R = m k R .
c × A k · s ^ d l = c m k R e R · n ^ d l .
C k ( r 1 , r 2 ) r 1 r 2 × A k · s ^ d l = r 1 r 2 m k R e R · n ^ d l = r ϵ 1 r ϵ 2 m k R e R · n ^ d l = θ 1 , s k θ 2 , s k m k ϵ e R · ( e R ) ϵ d θ = m k ( θ 2 , s k θ 1 , s k ) ,
ϕ ( r ) = ϕ ( r ) + g ( r , r ) · s ^ ( r r ) + k C k ( r , r ) .
E monopole = 1 R e R .
E dipole = 1 R 2 ( 2 ( d · e R ) e R d ) ,
Φ ( r ) = W { m θ ( r , r s ) + ϕ ¯ + δ ϕ ( r ) } , ( m { 1 , + 1 } ) ,
Δ W { W { Φ Φ } m ( θ θ ) } ,
Δ = W { δ ϕ δ ϕ } .
minimize l = 0 3 Δ 2 = l = 0 3 ( W { W { Δ Φ l } m Δ θ l ( r s ) } ) 2 ,
Δ Φ l Φ l + 1 Φ l , Φ 4 = Φ 0 ,
Δ θ l θ l + 1 ( r s ) θ l ( r s ) > 0 , θ 4 = θ 0 + 2 π ,

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