Abstract

Phase unwrapping is a challenging task for interferometry based techniques in the presence of noise. The majority of existing phase unwrapping techniques are path-following methods, which explicitly or implicitly define an intelligent path and integrate phase difference along the path to mitigate the effect of erroneous pixels. In this paper, a path-independent unwrapping method is proposed where the unwrapped phase gradient is determined from the wrapped phase and subsequently denoised by a TV minimization based method. Unlike the wrapped phase map where 2πphase jumps are present, the gradient of the unwrapped phase map is smooth and slowly-varying at noise-free areas. On the other hand, the noise is greatly amplified by the differentiation process, which makes it easier to separate from the smooth phase gradient. Thus an approximate unwrapped phase can be obtained by integrating the denoised phase gradient. The final unwrapped phase map is subsequently determined by adding the first few modes of the unwrapped phase. The proposed method is most suitable for unwrapping phase maps without abrupt phase changes. Its capability has been demonstrated both numerically and by experimental data from shearography and electronic speckle pattern interferometry (ESPI).

© 2012 OSA

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  27. Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
    [CrossRef]
  28. Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. 36(4), 526–528 (2011).
    [CrossRef] [PubMed]
  29. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20(1/2), 89–97 (2004).
    [CrossRef]
  30. Y. H. Huang, S. Y. Hung, F. Janabi-Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Lasers Eng. 50(4), 534–539 (2012).
    [CrossRef]

2012 (3)

2011 (5)

2010 (2)

2009 (2)

M. Gdeisat, M. Arevalillo-Herráez, D. Burton, and F. Lilley, “Three-dimensional phase unwrapping using the Hungarian algorithm,” Opt. Lett. 34(19), 2994–2996 (2009).
[CrossRef] [PubMed]

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[CrossRef] [PubMed]

2008 (5)

I. Shalem and I. Yavneh, “A multilevel graph algorithm for two dimensional phase unwrapping,” Comput. Vis. Sci. 11(2), 89–100 (2008).
[CrossRef]

T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16(10), 6985–6998 (2008).
[CrossRef] [PubMed]

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Absolute phase estimation: adaptive local denoising and global unwrapping,” Appl. Opt. 47(29), 5358–5369 (2008).
[CrossRef] [PubMed]

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).
[CrossRef]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
[CrossRef]

2007 (1)

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007).
[CrossRef]

2004 (2)

C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43(5), 1152–1159 (2004).
[CrossRef]

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20(1/2), 89–97 (2004).
[CrossRef]

2003 (1)

1999 (1)

1997 (1)

1993 (1)

Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE 2003, 254–263 (1993).
[CrossRef]

1989 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Ai, C.

Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE 2003, 254–263 (1993).
[CrossRef]

Arevalillo-Herráez, M.

Astola, J.

Bioucas-Dias, J.

Burton, D.

Busca, G.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).
[CrossRef]

Cabral-Cano, E.

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20(1/2), 89–97 (2004).
[CrossRef]

Chen, Y. S.

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
[CrossRef]

Dixon, T. H.

Egiazarian, K.

Estrada, J. C.

Flynn, T. J.

Gdeisat, M.

Ghiglia, D. C.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Gorthi, S. S.

Heshmat, S.

Hirose, A.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007).
[CrossRef]

Huang, Y. H.

Y. H. Huang, S. Y. Hung, F. Janabi-Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Lasers Eng. 50(4), 534–539 (2012).
[CrossRef]

Y. H. Huang, F. Janabi-Sharifi, Y. Liu, and Y. Y. Hung, “Dynamic phase measurement in shearography by clustering method and Fourier filtering,” Opt. Express 19(2), 606–615 (2011).
[CrossRef] [PubMed]

Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. 36(4), 526–528 (2011).
[CrossRef] [PubMed]

Y. H. Huang, Y. S. Liu, S. Y. Hung, C. G. Li, and F. Janabi-Sharifi, “Dynamic phase evaluation in sparse-sampled temporal speckle pattern sequence,” Opt. Lett. 36(4), 526–528 (2011).
[CrossRef] [PubMed]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
[CrossRef]

C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43(5), 1152–1159 (2004).
[CrossRef]

Hung, M. Y. Y.

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
[CrossRef]

Hung, S. Y.

Hung, Y. Y.

Janabi-Sharifi, F.

Katkovnik, V.

Langley, J.

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[CrossRef] [PubMed]

Li, C. G.

Lilley, F.

Liu, L.

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
[CrossRef]

Liu, Y.

Liu, Y. S.

Lo, Y. L.

Miyamoto, N.

Moore, C.

Navarro, M. A.

Ng, S. P.

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
[CrossRef]

Nishiyama, S.

Osmanoglu, B.

Parkhurst, J.

Price, G.

Quan, C.

C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43(5), 1152–1159 (2004).
[CrossRef]

Quiroga, J. A.

Rajshekhar, G.

Rastogi, P.

Romero, L. A.

Schmidt, J. D.

Schofield, M. A.

Servin, M.

Shalem, I.

I. Shalem and I. Yavneh, “A multilevel graph algorithm for two dimensional phase unwrapping,” Comput. Vis. Sci. 11(2), 89–100 (2008).
[CrossRef]

Sharrock, P.

Strand, J.

Taxt, T.

Tay, C. J.

C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43(5), 1152–1159 (2004).
[CrossRef]

Tomioka, S.

Vargas, J.

Venema, T. M.

Wang, W.

Y. H. Huang, S. Y. Hung, F. Janabi-Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Lasers Eng. 50(4), 534–539 (2012).
[CrossRef]

Wdowinski, S.

Weng, J. F.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Wu, T.

C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43(5), 1152–1159 (2004).
[CrossRef]

Xu, Y.

Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE 2003, 254–263 (1993).
[CrossRef]

Yamaki, R.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007).
[CrossRef]

Yavneh, I.

I. Shalem and I. Yavneh, “A multilevel graph algorithm for two dimensional phase unwrapping,” Comput. Vis. Sci. 11(2), 89–100 (2008).
[CrossRef]

Zappa, E.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Zhao, Q.

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[CrossRef] [PubMed]

Zhu, Y. M.

Appl. Opt. (5)

Comput. Vis. Sci. (1)

I. Shalem and I. Yavneh, “A multilevel graph algorithm for two dimensional phase unwrapping,” Comput. Vis. Sci. 11(2), 89–100 (2008).
[CrossRef]

IEEE Trans. Geosci. Rem. Sens. (1)

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Rem. Sens. 45(10), 3240–3251 (2007).
[CrossRef]

J. Math. Imaging Vis. (1)

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vis. 20(1/2), 89–97 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

Magn. Reson. Imaging (1)

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[CrossRef] [PubMed]

Opt. Eng. (2)

C. J. Tay, C. Quan, T. Wu, and Y. H. Huang, “Integrated method for 3-D rigid-body displacement measurement using fringe projection,” Opt. Eng. 43(5), 1152–1159 (2004).
[CrossRef]

Y. H. Huang, S. P. Ng, L. Liu, Y. S. Chen, and M. Y. Y. Hung, “Shearographic phase retrieval using one single specklegram: a clustering approach,” Opt. Eng. 47(5), 054301 (2008).
[CrossRef]

Opt. Express (5)

Opt. Lasers Eng. (2)

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46(2), 106–116 (2008).
[CrossRef]

Y. H. Huang, S. Y. Hung, F. Janabi-Sharifi, W. Wang, and Y. S. Liu, “Quantitative phase retrieval in dynamic laser speckle interferometry,” Opt. Lasers Eng. 50(4), 534–539 (2012).
[CrossRef]

Opt. Lett. (5)

Proc. SPIE (1)

Y. Xu and C. Ai, ““Simple and effective phase unwrapping technique,” Interferometry IV: Techniques and Analysis,” Proc. SPIE 2003, 254–263 (1993).
[CrossRef]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[CrossRef]

Other (4)

C. Prati, M. Giani, and N. Leuratti, “SAR interferometry: a 2-D phase unwrapping technique based on phase and absolute values information,” in Proceedings of the 1990 International Geoscience and Remote Sensing Symposium (IEEE, 1990), pp. 2043–2046.

E. Volkl, L. F. Allard, and D. C. Joy, eds., Introduction to Electron holography (Plenum, 1999).

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

M. D. Pritt, “Congruence in least-squares phase unwrapping,” in Proceedings Vol. II: Remote Sensing - a Scientific Vision for Sustainable Development, IGARSS '97 - 1997 International Geoscience and Remote Sensing Symposium, 1997), pp.875–877.

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Figures (11)

Fig. 1
Fig. 1

An interferometric fringe from shearography (a), the initial noisy wrapped phase map obtained (b), denoised wrapped phase by applying the wrapped phase filtering method 5 times (c) and 50 times (d), and the unwrapping results from raster scanning (e), Goldstein’s branch cut algorithm [result is displayed as logarithm because of the dominance of noise amplified by the algorithm at point indicated as “HN” in the picture] (f), Quality guided algorithm with maximum gradient (g), Mask cut algorithm with maximum gradient (h) and Flynn’s minimum discontinuity method (i).

Fig. 2
Fig. 2

The partial derivatives of Fig. 1(d) in x- (a) and y- directions (b), the TV denoised partial derivatives with regard to x- (c) and y- directions (d), the unwrapped phase map by integration the denoised phase derivatives (e) and its rewrapped phase map (f), the first residual wrapped phase (g), the second residual wrapped phase (h) and the final unwrapped phase map from the proposed algorithm (i).

Fig. 3
Fig. 3

Flowchart of the TV unwrapping algorithm.

Fig. 4
Fig. 4

Evaluation of the TV unwrapping algorithm using experiment data from ESPI.

Fig. 5
Fig. 5

Evaluation of the TV unwrapping algorithm using simulated data with height discontinuity.

Fig. 6
Fig. 6

Typical fringe maps (top row) with Gaussian noise, wrapped phase map (middle row) and unwrapped phase map (bottom row) obtained by the proposed TV unwrapping method. The fringe density controlling parameter φ 0 in Eq. (12) with is fixed at 20 for these simulations.

Fig. 7
Fig. 7

Iteration times required to achieve a satisfying unwrapped phase Vs. Gaussian noise level using simulated fringe maps according to Eq. (12) with φ 0 fixed at 20. The mean and standard deviation values are calculated from 100 random realizations.

Fig. 8
Fig. 8

Normalized mean squared error of the reconstructed phase Vs. Gaussian noise level using simulated fringe maps according to Eq. (12) with φ 0 fixed at 20. The mean and standard deviation values are calculated from 100 random realizations.

Fig. 9
Fig. 9

Typical fringe maps (top row) with Gaussian noise with SNR of 0.8, wrapped phase map (middle row) and unwrapped phase map (bottom row) obtained by the proposed TV unwrapping method. The fringe density controlling parameter φ 0 in Eq. (12) is changed from 10 to 40 at step 10.

Fig. 11
Fig. 11

Reconstruction error of unwrapped phase Vs. phase magnitude using simulated fringe maps according to Eq. (12) with fixed Gaussian noise and φ 0 ranging from 10 to 40. The mean and standard deviation values are calculated from 100 random realizations.

Fig. 10
Fig. 10

Iteration times required to achieve a satisfying unwrapped phase Vs. phase magnitude using simulated fringe maps according to Eq. (12) with fixed Gaussian noise and φ 0 ranging from 10 to 40. The mean and standard deviation values are calculated from 100 random realizations.

Equations (13)

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

ϕ w =ϕ2nπ         nZ,  ϕ w (π,π].
W{ψ}={ 2arctan[ sin(ψ) 1+cos(ψ) ]          cos(ψ)1              π                          cos(ψ)=1  
ϕ x = ϕ x =W{ ϕ w x }
ϕ y = ϕ y =W{ ϕ w y }.
ϕ ^ x = arg min       ϕ ^ x  {|| ϕ ^ x ϕ x | | 2 /(2μ)+TV( ϕ ^ x )}
ϕ ^ x = ϕ x π μκ ( ϕ x )
arg min       p  {||μ p ϕ x | | 2 :  | p | 2 1}
π μκ ( ϕ x )=μ  p
p n+1 = p n +τ[( p n ϕ x /μ)] 1+τ||( p n ϕ x /μ)||
ϕ r =W{ϕ ϕ ^ }=W{ ϕ w ϕ ^ }.
ϕ= k=1,2 ϕ ^ k + ϕ rf
φ( x,y )= 2π λ φ 0 xexp( ( x 2 + y 2 ) 2 a 2 )
I i (x,y)=1+cos[φ(x,y)+ δ i ]+noise(x,y)

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