Abstract

A windowed Fourier filtered and quality guided (WFF-QG) phase unwrapping algorithm was proposed recently [Appl. Opt. 47, 5420–5428 (2008)], based on the windowed Fourier transform [Appl. Opt. 47, 5408–5419 (2008)] where the phase is assumed to be locally quadric. We consider a locally higher order polynomial phase. After the phase is filtered and unwrapped by the WFF-QG, it is postprocessed by a congruence operation (CO), so that the unwrapped phase is congruent to the original wrapped phase. The unwrapped phase can now be assumed to be a locally high-order polynomial, and, consequently, least squares fitting (LSF) is proposed to suppress the noise. This postprocessing algorithm is abbreviated as CO-LSF. The CO-LSF is theoretically a reasonable choice to improve the WFF-QG results, especially when the noise is severe. This is because for severe noise, a large window is necessary for reliable phase extraction in the WFF-QG. However, this large window makes the quadric phase assumption less reasonable and leads to a large phase error. The CO-LSF thus helps to reduce the phase error by more reasonably assuming that the phase is a high-order polynomial. The polynomial order of 4 is suggested for the CO-LSF, as higher order polynomials do not give significant improvement to the WFF-QG. One disadvantage of the CO-LSF is that it is more sensitive to phase discontinuities than the WFF-QG.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm and Software (Wiley, 1998).
  2. Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41, 2650-2654 (2002).
    [CrossRef] [PubMed]
  3. Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
    [CrossRef]
  4. M. Servin, J. L. Marroguin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540-4548(1997).
    [CrossRef] [PubMed]
  5. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934-1941 (1999).
    [CrossRef]
  6. Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408-5419 (2008).
    [CrossRef] [PubMed]
  7. Q. Kemao, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5408 (2008).
    [CrossRef] [PubMed]
  8. B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
    [CrossRef]
  9. S. S. Gorthi and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomials phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
    [CrossRef]
  10. S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949-954 (2009).
    [CrossRef]
  11. S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11, 065405 (2009).
    [CrossRef]
  12. S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Instrum. 80, 073109 (2009).
    [CrossRef] [PubMed]
  13. S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575-2577 (2009).
    [CrossRef] [PubMed]
  14. S. Peleg and B. Friedlander, “The discrete polynomial-phase transform,” IEEE Trans. Signal Process. 43, 1901-1914 (1995).
    [CrossRef]
  15. S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560-565 (2010).
    [CrossRef] [PubMed]
  16. P. O'Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385-393 (2004).
    [CrossRef]
  17. A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
    [CrossRef]
  18. W. Gao, N. T. H. Huyen, H. S. Loi, and Q. Kemao, “Real-time 2D parallel windowed Fourier transform for fringe pattern analysis using Graphics Processing Unit,” Opt. Express 17, 23147-23152 (2009).
    [CrossRef]

2010 (1)

2009 (6)

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575-2577 (2009).
[CrossRef] [PubMed]

W. Gao, N. T. H. Huyen, H. S. Loi, and Q. Kemao, “Real-time 2D parallel windowed Fourier transform for fringe pattern analysis using Graphics Processing Unit,” Opt. Express 17, 23147-23152 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomials phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949-954 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11, 065405 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
[CrossRef]

2004 (1)

P. O'Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385-393 (2004).
[CrossRef]

2002 (1)

2001 (1)

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

1999 (1)

1997 (1)

1996 (1)

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

1995 (1)

S. Peleg and B. Friedlander, “The discrete polynomial-phase transform,” IEEE Trans. Signal Process. 43, 1901-1914 (1995).
[CrossRef]

Cuevas, F. J.

Federico, A.

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

Francos, J. M.

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

Friedlander, B.

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

S. Peleg and B. Friedlander, “The discrete polynomial-phase transform,” IEEE Trans. Signal Process. 43, 1901-1914 (1995).
[CrossRef]

Fu, S.

Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
[CrossRef]

Gao, W.

Ghiglia, D. C.

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

Gorthi, S. S.

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560-565 (2010).
[CrossRef] [PubMed]

S. S. Gorthi and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomials phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949-954 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11, 065405 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef] [PubMed]

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575-2577 (2009).
[CrossRef] [PubMed]

Huyen, N. T. H.

Kaufmann, G. H.

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

Kemao, Q.

Liu, X.

Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
[CrossRef]

Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41, 2650-2654 (2002).
[CrossRef] [PubMed]

Loi, H. S.

Malacara, D.

Marroguin, J. L.

O'Shea, P.

P. O'Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385-393 (2004).
[CrossRef]

Peleg, S.

S. Peleg and B. Friedlander, “The discrete polynomial-phase transform,” IEEE Trans. Signal Process. 43, 1901-1914 (1995).
[CrossRef]

Pritt, M. D.

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

Qiu, Z.

Rajshekhar, G.

Rastogi, P.

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560-565 (2010).
[CrossRef] [PubMed]

S. S. Gorthi and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomials phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575-2577 (2009).
[CrossRef] [PubMed]

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef] [PubMed]

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11, 065405 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949-954 (2009).
[CrossRef]

Rodriguez-Vera, R.

Servin, M.

Sun, X.

Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
[CrossRef]

Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41, 2650-2654 (2002).
[CrossRef] [PubMed]

Wang, H.

Yang, X.

Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
[CrossRef]

Yu, Q.

Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
[CrossRef]

Q. Yu, X. Sun, X. Liu, and Z. Qiu, “Spin filtering with curve windows for interferometric fringe patterns,” Appl. Opt. 41, 2650-2654 (2002).
[CrossRef] [PubMed]

Appl. Opt. (5)

IEEE Geosci. Remote Sensing Lett. (1)

Q. Yu, X. Yang, S. Fu, X. Liu, and X. Sun, “An adaptive contoured window filter for interferometric synthetic aperture radar,” IEEE Geosci. Remote Sensing Lett. 4, 23-26 (2007).
[CrossRef]

IEEE Trans. Signal Process. (3)

B. Friedlander and J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999-3007 (1996).
[CrossRef]

S. Peleg and B. Friedlander, “The discrete polynomial-phase transform,” IEEE Trans. Signal Process. 43, 1901-1914 (1995).
[CrossRef]

P. O'Shea, “A fast algorithm for estimating the parameters of a quadratic FM signal,” IEEE Trans. Signal Process. 52, 385-393 (2004).
[CrossRef]

J. Mod. Opt. (1)

S. S. Gorthi and P. Rastogi, “Numerical analysis of fringe patterns recorded in holographic interferometry using high-order ambiguity function,” J. Mod. Opt. 56, 949-954 (2009).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

S. S. Gorthi and P. Rastogi, “Piecewise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A: Pure Appl. Opt. 11, 065405 (2009).
[CrossRef]

Meas. Sci. Technol. (1)

S. S. Gorthi and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomials phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
[CrossRef]

Opt. Eng. (1)

A. Federico and G. H. Kaufmann, “Comparative study of wavelet thresholding methods for denoising electronic speckle pattern interferometry fringes,” Opt. Eng. 40, 2598-2604 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

S. S. Gorthi and P. Rastogi, “Windowed high-order ambiguity function method for fringe analysis,” Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef] [PubMed]

Other (1)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Simulation example of a peak function processed by the WFF-QG: (a) ideal phase map (rewrapped), (b) phase map with speckle noise (originally wrapped from simulation), (c) phase map by the WFF-QG ( σ = 10 ) (rewrapped), and (d) phase error by the WFF-QG ( σ = 10 ).

Fig. 2
Fig. 2

Simulation example postprocessed by the CO-LSF: (a) phase map by the WFF-QG ( σ = 10 ) (continuous), (b) congruent phase map (continuous), (c) phase map by the CO-LSF (order 4) (rewrapped), and (d) phase error by the CO-LSF (order 4).

Fig. 3
Fig. 3

Simulation example: comparison of averaged phase errors among the WFF-QG ( σ = 10 ), the CO-LSF (order 4), and the CO-LSF (order 6).

Fig. 4
Fig. 4

Simulation example of a noisy shear: (a) noisy phase map, (b) phase map by the WFF-QG ( σ = 10 ) (rewrapped), (c) quality map by the WFF-QG ( σ = 10 ), and (d) phase map by the CO-LSF (order 2) (rewrapped).

Fig. 5
Fig. 5

Experimental example from interferometric synthetic aperture radar: (a) original phase map, (b) phase map by the WFF-QG ( σ = 10 ) (rewrapped), (c) quality map by the WFF-QG ( σ = 10 ), and (d) phase map by the CO-LSF (order 4) (rewrapped).

Equations (9)

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

φ c ( x , y ) = φ ( x , y ) + W [ φ w ( x , y ) φ ( x , y ) ] ,
W [ φ c ( x , y ) ] = φ w ( x , y ) .
φ c ( x , y ) = φ ( x , y ) + h + W [ φ w ( x , y ) φ ( x , y ) h ] ,
φ p ( x , y ) = t = 0 K s s = 0 K c s , t x s y t .
E r ( c s , t , K ) = v = N 1 N 2 u = M 1 M 2 w ( u , v ) [ φ p ( u , v ) φ c ( u , v ) ] 2 min ,
E r ( c s , t , K ) c s , t = 0 ;
A C = D ,
φ ( x 0 , y 0 ) = c 0 , 0 = B 1 D .
f ( x , y ) = exp [ j φ 0 ( x , y ) ] + L × [ n r ( x , y ) + j n i ( x , y ) ] ,

Metrics