Abstract

In most phase unwrapping algorithms, the image reconstruction results are obtained by shifting the phase jumps in the wrapped phase map by 2π. The performance of such algorithms is degraded by the presence of speckle noise, residual noise, noise at the height discontinuities and holes in the wrapped phase map. Thus, a filtering operation is performed prior to the unwrapping process in order to remove the noise. However, the filtering process smears the phase jumps in the wrapped phase map and therefore causes a phase shifting error during the reconstruction process. Moreover, the noise errors, hole errors and shifting errors are accumulated path-by-path during unwrapping. Accordingly, the present study proposes a new rotation algorithm for phase unwrapping applications which resolves the noise error, the error of hole, the shifting error. Existing phase unwrapping algorithms are designed to operate only on those pixels whose phase values have no noise or holes. Or they are designed to operate the three-dimensional unwrapping paths in the row and column directions to avoid the noise or holes. By contrast, the rotation algorithm proposed in this study operates on all the pixels in the wrapped phase map, including those affected by noise or holes. As a result, the noise errors and hole errors produced in existing 2π phase shifting unwrapping algorithms are eliminated. Furthermore, since in the proposed approach, the wrapped phase map is not filtered prior to the unwrapping process, the phase shifting errors induced in existing algorithms are also eliminated. The robustness of the proposed algorithm to various noise errors, hole errors and phase shifting errors is demonstrated both numerically and experimentally.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49(11), 2143–2150 (2010).
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    [CrossRef]
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    [CrossRef] [PubMed]
  15. S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007).
    [CrossRef] [PubMed]
  16. A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  19. S. Zhang, X. L. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46(1), 50–57 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  24. K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36(20), 4830–4838 (1997).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
    [CrossRef]
  29. J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Eng. 50(8), 1026–1029 (2011).
  30. X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Nav. 5(3), 296–304 (2011).
    [CrossRef]
  31. J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a Grid-Based Filter to Solve InSAR Phase Unwrapping,” IEEE Geosci. Remote Sens. 5(2), 147–151 (2008).
    [CrossRef]
  32. K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010).
    [CrossRef] [PubMed]
  33. K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24(18), 3053–3058 (1985).
    [CrossRef] [PubMed]
  34. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
    [CrossRef] [PubMed]

2012 (2)

2011 (4)

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[CrossRef]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Eng. 50(8), 1026–1029 (2011).

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Nav. 5(3), 296–304 (2011).
[CrossRef]

J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express 19(4), 3086–3105 (2011).
[CrossRef] [PubMed]

2010 (2)

2009 (2)

2008 (3)

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008).
[CrossRef]

A. Wada, M. Kato, and Y. Ishii, “Large step-height measurements using multiple-wavelength holographic interferometry with tunable laser diodes,” J. Opt. Soc. Am. A 25(12), 3013–3020 (2008).
[CrossRef] [PubMed]

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a Grid-Based Filter to Solve InSAR Phase Unwrapping,” IEEE Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

2007 (3)

2005 (2)

2001 (1)

1999 (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

1998 (1)

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

1997 (5)

1995 (1)

1993 (1)

1991 (1)

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

1988 (1)

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

1987 (2)

1985 (1)

1983 (1)

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

Bertani, D.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Brady, D.

Capanni, A.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Cetica, M.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Chang, H. Y.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Chen, C. W.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Cheng, J.

Cheng, X.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[CrossRef]

Creath, K.

Cui, H.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[CrossRef]

Dai, N.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[CrossRef]

Dubey, S. K.

Eiju, T.

Estrada, J. C.

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Eng. 50(8), 1026–1029 (2011).

Fetterman, M.

Flynn, T. J.

Francini, F.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Gauthier, P.

Ghiglia, D. C.

Goldstein, R.

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

Hahn, J.

Hao, Q.

Hariharan, P.

Hassebrook, L. G.

Hirose, A.

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

Hossain, M. M.

Hu, C. P.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Huang, M. J.

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008).
[CrossRef]

Huntley, J. M.

Ishii, Y.

Javidi, B.

Jiang, J.

Kato, M.

Kim, E. H.

Kim, H.

Krishnaswamy, S.

Lau, D. L.

Lee, B.

Lee, C. K.

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Li, X. L.

Liao, W.

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[CrossRef]

Liou, J. K.

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008).
[CrossRef]

Liu, K.

Liu, Z.

Lo, Y. L.

Lopez-Sanchez, J. M.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a Grid-Based Filter to Solve InSAR Phase Unwrapping,” IEEE Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

Luong, B.

Macy, W. W.

Martinez-Espla, J. J.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a Grid-Based Filter to Solve InSAR Phase Unwrapping,” IEEE Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

Martinez-Marin, T.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a Grid-Based Filter to Solve InSAR Phase Unwrapping,” IEEE Geosci. Remote Sens. 5(2), 147–151 (2008).
[CrossRef]

Mastin, G.

Mehta, D. S.

Moon, I.

Navarro, M. A.

Oreb, B. F.

Pezzati, L.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Potuluri, P.

Pouet, B. F.

Quiroga, J. A.

Reichard, K.

Robinson, D. W.

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

Romero, L. A.

Saldner, H.

Saldner, H. O.

Servin, M.

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Eng. 50(8), 1026–1029 (2011).

Shakher, C.

Shi, K.

Spik, A.

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

Stetson, K. A.

Su, W. H.

Vargas, J.

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Eng. 50(8), 1026–1029 (2011).

Wada, A.

Wahid, J.

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

Wang, B.

Wang, Y. C.

Weng, J. F.

Werner, C.

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

Xianming, X.

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Nav. 5(3), 296–304 (2011).
[CrossRef]

Yamaki, R.

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

Yau, S. T.

Yiming, P.

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Nav. 5(3), 296–304 (2011).
[CrossRef]

Yin, S.

Yuqing, S.

Zebker, H.

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

Zhang, S.

Appl. Opt. (10)

H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997).
[CrossRef] [PubMed]

J. Jiang, J. Cheng, and B. Luong, “Unsupervised-clustering-driven noise-residue filter for phase images,” Appl. Opt. 49(11), 2143–2150 (2010).
[CrossRef] [PubMed]

J. M. Huntley and H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32(17), 3047–3052 (1993).
[CrossRef] [PubMed]

D. S. Mehta, S. K. Dubey, M. M. Hossain, and C. Shakher, “Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry,” Appl. Opt. 44(35), 7515–7521 (2005).
[CrossRef] [PubMed]

S. Zhang, X. L. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46(1), 50–57 (2007).
[CrossRef] [PubMed]

W. W. Macy., “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22(23), 3898–3901 (1983).
[CrossRef] [PubMed]

H. O. Saldner and J. M. Huntley, “Temporal phase unwrapping: application to surface profiling of discontinuous objects,” Appl. Opt. 36(13), 2770–2775 (1997).
[CrossRef] [PubMed]

K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36(20), 4830–4838 (1997).
[CrossRef] [PubMed]

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24(18), 3053–3058 (1985).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
[CrossRef] [PubMed]

IEEE Geosci. Remote Sens. (1)

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “Using a Grid-Based Filter to Solve InSAR Phase Unwrapping,” IEEE Geosci. Remote Sens. 5(2), 147–151 (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]

IET Radar Sonar Nav. (1)

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Nav. 5(3), 296–304 (2011).
[CrossRef]

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

Opt. Commun. (1)

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162(4–6), 205–210 (1999).
[CrossRef]

Opt. Eng. (3)

H. Cui, W. Liao, N. Dai, and X. Cheng, “Reliability-guided phase-unwrapping algorithm for the measurement of discontinuous three-dimensional objects,” Opt. Eng. 50(6), 063602 (2011).
[CrossRef]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: a review and comparison,” Opt. Eng. 50(8), 1026–1029 (2011).

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, “Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping,” Opt. Eng. 36(9), 2466–2472 (1997).
[CrossRef]

Opt. Express (8)

K. Liu, Y. C. Wang, D. L. Lau, Q. Hao, and L. G. Hassebrook, “Dual-frequency pattern scheme for high-speed 3-D shape measurement,” Opt. Express 18(5), 5229–5244 (2010).
[CrossRef] [PubMed]

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[CrossRef] [PubMed]

S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007).
[CrossRef] [PubMed]

J. F. Weng and Y. L. Lo, “Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment,” Opt. Express 19(4), 3086–3105 (2011).
[CrossRef] [PubMed]

J. F. Weng and Y. L. Lo, “Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities,” Opt. Express 20(10), 10896–10920 (2012).
[CrossRef] [PubMed]

E. H. Kim, J. Hahn, H. Kim, and B. Lee, “Profilometry without phase unwrapping using multi-frequency and four-step phase-shift sinusoidal fringe projection,” Opt. Express 17(10), 7818–7830 (2009).
[CrossRef] [PubMed]

W. H. Su, K. Shi, Z. Liu, B. Wang, K. Reichard, and S. Yin, “A large-depth-of-field projected fringe profilometry using supercontinuum light illumination,” Opt. Express 13(3), 1025–1032 (2005).
[CrossRef] [PubMed]

P. Potuluri, M. Fetterman, and D. Brady, “High depth of field microscopic imaging using an interferometric camera,” Opt. Express 8(11), 624–630 (2001).
[CrossRef] [PubMed]

Opt. Lasers Eng. (2)

A. Spik and D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Lasers Eng. 14(1), 25–37 (1991).
[CrossRef]

H. Y. Chang, C. W. Chen, C. K. Lee, and C. P. Hu, “The Tapestry Cellular Automata phase unwrapping algorithm for interferogram analysis,” Opt. Lasers Eng. 30(6), 487–502 (1998).
[CrossRef]

Opt. Lett. (2)

Radio Sci. (1)

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

Strain (1)

M. J. Huang and J. K. Liou, “Retrieving ESPI map of discontinuous objects via a novel phase unwrapping algorithm,” Strain 44(3), 239–247 (2008).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

2D wrapped phase. Note that the blue-solid lines represent wrapped phase lines. Note also that Phase 1 extends from a point to the left of D2 to D2 while Phase 2 extends fully from D3 to D4. Finally, note that the red triangles represent noise or holes.

Fig. 2
Fig. 2

Results of first relative rotation procedure. Note that the red line indicates the result obtained by rotating Phase 1 about C1 by θ 1 , while the blue line indicates the result obtained by rotating Phase 2 about C2 by θ 2 . Note also that the phase values, noise and holes are all preserved in the rotation results.

Fig. 3
Fig. 3

Results of second relative rotation procedure in which rotated Phase 1 from D1’ to D2’ is further rotated by θ' ' fix leading to new rotated Phase 1 from D1” to D2”. A stitching process is then performed in which new rotated Phase 1 is shifted in the y-direction and stitched to Phase 2 from D3′ to D4’.

Fig. 4
Fig. 4

Starting position for new first relative rotation procedure based on stitching results shown in Fig. 3. The final cycle of the relative rotation stage is terminated when eliminating all of the 2π phase jumps. Totally, two cycles of the relative rotation stage are implemented in Fig. 1.

Fig. 5
Fig. 5

(a). Noise and holes are located outside of phase jump region. (b) Noise and holes straddle phase jump region. (c) Phase jump contains shifting error.

Fig. 6
Fig. 6

(a) Detection result obtained using the detection scheme of Eq. (2) with σ Α = 2.8. (b) Detection result obtained using the additional phase jump detection scheme of Eq. (4).

Fig. 7
Fig. 7

(a) Results of first relative rotation procedure (Steps (i) to (vi)), indicated by black-dashed line and black-solid line. (b) Results of second relative rotation procedure (Steps (vii) to (viii)), indicated by green line.

Fig. 8
Fig. 8

(a) Results of stitching process (Steps (ix) to (x)), indicated by red line. (b) Starting point for new relative rotation cycle using new Phase 1 (black-dashed line) and new Phase 2 (black-solid line).

Fig. 9
Fig. 9

(a) Results of second relative rotation procedure (Steps (vii) to (viii)) in new relative rotation cycle, indicated by green line. (b) Stitching results (Steps (ix) to (x)) in new relative rotation cycle, indicated by red line and black line.

Fig. 10
Fig. 10

(a) Phase unwrapping results obtained using MACY algorithm. (b) Phase unwrapping results obtained using CA algorithm.

Fig. 11
Fig. 11

Detection result obtained using the detection scheme of Eq. (2) (a) with σ Α = 1 and (b) with σ Α = 2.8.

Fig. 12
Fig. 12

(a) Detection result obtained using the additional phase jump detection scheme of Eq. (4). (b) Unwrapped Result obtained from relative rotation process.

Fig. 13
Fig. 13

(a) Phase unwrapping results obtained using MACY algorithm. (b) Phase unwrapping results obtained using CA algorithm.

Fig. 14
Fig. 14

Filtered wrapped phase following 10 times by Filter A [11].

Fig. 15
Fig. 15

(a) Detection result obtained using the detection scheme of Eq. (2) with σ Α = 2.8. (b) Detection result obtained using the additional phase jump detection scheme of Eq. (4). (c) Unwrapped result obtained from relative rotation process.

Fig. 16
Fig. 16

(a) Raw wrapped phase of Si sample using white-light source. (b) Unwrapped result obtained from relative rotation process.

Fig. 17
Fig. 17

(a) Raw wrapped phase of Si sample using laser source. (b) Unwrapped result obtained from relative rotation process.

Fig. 18
Fig. 18

(a) Raw wrapped phase map of step height sample with rough surface. (b) Noise map given σ Α = 2.75.

Fig. 19
Fig. 19

Results of absolute rotation procedure. (a) Without Filter B. (b) With Filter B.

Tables (2)

Tables Icon

Table 1 Phase difference values between pixels (80, 6.168) and (81, 0.1543) (row, phase) in three different unwrapping schemes.

Tables Icon

Table 2 The corresponding angles in Cycle 2 in the relative rotation procedure

Equations (7)

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S1(i,j)=[ ϕ(i+1,j)ϕ(i,j) σ Α 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j)+ σ Α 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1) σ Α 2π ]+[ ϕ(i,j)ϕ(i,j+1)+ σ Α 2π ] S2(i,j)=[ ϕ(i+1,j)ϕ(i,j)+ σ Α 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j) σ Α 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1)+ σ Α 2π ]+[ ϕ(i,j)ϕ(i,j+1) σ Α 2π ] S3(i,j)=[ ϕ(i+1,j)ϕ(i,j)+ σ Α 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j)+ σ Α 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1) σ Α 2π ]+[ ϕ(i,j)ϕ(i,j+1) σ Α 2π ] S4(i,j)=[ ϕ(i+1,j)ϕ(i,j) σ Α 2π ]+[ ϕ(i+1,j+1)ϕ(i+1,j) σ Α 2π ] +[ ϕ(i,j+1)ϕ(i+1,j+1)+ σ Α 2π ]+[ ϕ(i,j)ϕ(i,j+1)+ σ Α 2π ]
S1(i,1)=[ ϕ(i+1,1)ϕ(i,1) σ Α 2π ]+ ϕ(i,1)ϕ(i+1,1) σ Α 2π ] S2(i,1)=[ ϕ(i+1,1)ϕ(i,1)+ σ Α 2π ]+[ ϕ(i,1)ϕ(i+1,1)+ σ Α 2π ] S3(i,1)=[ ϕ(i+1,1)ϕ(i,1)+ σ Α 2π ]+[ ϕ(i,1)ϕ(i+1,1) σ Α 2π ] S4(i,1)=[ ϕ(i+1,1)ϕ(i,1) σ Α 2π ]+[ ϕ(i,1)ϕ(i+1,1)+ σ Α 2π ]
{ Pixel (i+1,j) is a 0-low phase jump position : if [ ϕ(i+1,j)ϕ(i,j) σ Α 2π ]=1 Pixel (i+1,j) is a 2π-high phase jump position: if [ ϕ(i+1,j)ϕ(i,j) σ Α 2π ]=1
{ Pixel (i+1,1) is a 0-low phase jump position : if  [ ϕ(i+1,1)ϕ(i,1) σ Α 2π ]=1 Pixel (i+1,1) is a 2π-high phase jump position: if [ ϕ(i+1,1)ϕ(i,1) σ Α 2π ]=1
row_center1= i=D1 D2 phase(i,1)*i i=D1 D2 phase(i,1) or row_center2= i=D3 D4 phase(i,1)*i i=D3 D4 phase(i,1)
phase_center1=interpolation with row_center1                                 and two neighboring integral row pixels or phase_center2=interpolation with row_center2                                  and two neighboring integral row pixels
S1(80,1)=[-1.4351]+[0.5438]=1+1=0 S2(80,1)=[-0.5438]+[1.4351]=1+1=0 S3(80,1)=[-0.5438]+[0.5438]=1+1=0 S4(80,1)=[-1.4351]+[1.4351]=1+1=0

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