Abstract

A windowed Fourier ridges (WFR) algorithm and a windowed Fourier filtering (WFF) algorithm have been proposed for fringe pattern analysis and have been demonstrated to be versatile and effective. Theoretical analyses of their performances are of interest. Local frequency and phase extraction errors by the WFR and WFF algorithms are analyzed in this paper. Effectiveness of the WFR and WFF algorithms will thus be theoretically proven. Consider four phase-shifted fringe patterns with local quadric phase [c20=c02=0.005  rad/(pixel)2], and assume that the noise in these fringe patterns have mean values of zero and standard deviations the same as the fringe amplitude. If the phase is directly obtained using the four-step phase-shifting algorithm, the phase error has a mean of zero and a standard deviation of 0.7rad. However, when using the WFR algorithm with a window size of σx=σy=10 pixels, the local frequency extraction error has a mean of zero and a standard deviation of less than 0.01  rad/pixel and the phase extraction error in the WFR algorithm has a mean of zero and a standard deviation of about 0.02rad. When using the WFF algorithm with the same window size, the phase extraction error has a mean of zero and a standard deviation of less than 0.04rad and the local frequency extraction error also has a mean of zero and a standard deviation of less than 0.01  rad/pixel. Thus, an unbiased estimation with very low standard deviation is achievable for local frequencies and phase distributions through windowed Fourier transforms. Algorithms applied to different fringe patterns, different noise models, and different dimensions are discussed. The theoretical analyses are verified by numerical simulations.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
    [CrossRef] [PubMed]
  2. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis: addendum,” Appl. Opt. 43, 3472-3473 (2004).
    [CrossRef] [PubMed]
  3. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
    [CrossRef]
  4. Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
    [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. M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Marcel Dekker, 2001), Chap. 12, pp. 373-426.
  7. H. Y. Yun, C. K. Hong, and S. W. Chang, “Least-square phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry,” J. Opt. Soc. Am. A 20, 240-247 (2003).
    [CrossRef]
  8. R. Kramer and O. Loffeld, “Presentation of an improved phase unwrapping algorithm based on Kalman filers combined with local slope estimation,” in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (1996).
  9. E. Trouvé, M. Caramma, and H. Maître, “Fringe detection in noisy complex interferograms,” Appl. Opt. 35, 3799-3806(1996).
    [CrossRef] [PubMed]
  10. E. Trouve, J. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
    [CrossRef]
  11. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205-210 (1999).
    [CrossRef]
  12. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742-1753 (1997).
    [CrossRef]
  13. J. L. Marroquin, M. Rivera, S. Botello, R. Rodriguez-Vera, and M. Servin, “Regularization methods for processing fringe-pattern images,” Appl. Opt. 38, 788-794 (1999).
    [CrossRef]
  14. Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
    [CrossRef]
  15. Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm: addendum,” Opt. Lasers Eng. 45, 1193-1195 (2007).
    [CrossRef]
  16. M. Takeda, H. Ina, and S. Kobayashi, “Fourier transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160(1982).
    [CrossRef]
  17. K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
    [CrossRef]
  18. U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).
  19. P. Guillemain and R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561-581 (1996).
    [CrossRef]
  20. Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098(2008).
    [CrossRef]
  21. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts & Company, 2007).
  22. K. Qian and S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
    [CrossRef]
  23. J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe pattern,” Appl. Opt. 46, 2670-2675 (2007).
    [CrossRef] [PubMed]
  24. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 1999).
  25. Y. Fu, R. M. Groves, G. Pedrini, and W. Osten, “Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence,” Appl. Opt. 46, 8645-8654(2007).
    [CrossRef] [PubMed]
  26. Q. Kemao and S. H. Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32, 127-129 (2007).
    [CrossRef]
  27. G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9-14 (1996).
    [CrossRef]
  28. 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]

2008

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098(2008).
[CrossRef]

2007

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts & Company, 2007).

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm: addendum,” Opt. Lasers Eng. 45, 1193-1195 (2007).
[CrossRef]

Q. Kemao and S. H. Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32, 127-129 (2007).
[CrossRef]

J. Zhong and H. Zeng, “Multiscale windowed Fourier transform for phase extraction of fringe pattern,” Appl. Opt. 46, 2670-2675 (2007).
[CrossRef] [PubMed]

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

Y. Fu, R. M. Groves, G. Pedrini, and W. Osten, “Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence,” Appl. Opt. 46, 8645-8654(2007).
[CrossRef] [PubMed]

2005

K. Qian and S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
[CrossRef]

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

2004

2003

2001

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Marcel Dekker, 2001), Chap. 12, pp. 373-426.

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

1998

E. Trouve, J. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

1997

1996

E. Trouvé, M. Caramma, and H. Maître, “Fringe detection in noisy complex interferograms,” Appl. Opt. 35, 3799-3806(1996).
[CrossRef] [PubMed]

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9-14 (1996).
[CrossRef]

R. Kramer and O. Loffeld, “Presentation of an improved phase unwrapping algorithm based on Kalman filers combined with local slope estimation,” in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (1996).

P. Guillemain and R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561-581 (1996).
[CrossRef]

1982

Aebischer, H. A.

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

Botello, S.

Caramma, M.

Chang, S. W.

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]

Feng, L.

Fu, Y.

Galizzi, G. E.

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9-14 (1996).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts & Company, 2007).

Groves, R. M.

Guillemain, P.

P. Guillemain and R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561-581 (1996).
[CrossRef]

Hong, C. K.

Ina, H.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

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]

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9-14 (1996).
[CrossRef]

Kemao, Q.

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098(2008).
[CrossRef]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
[CrossRef]

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm: addendum,” Opt. Lasers Eng. 45, 1193-1195 (2007).
[CrossRef]

Q. Kemao and S. H. Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32, 127-129 (2007).
[CrossRef]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
[CrossRef] [PubMed]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis: addendum,” Appl. Opt. 43, 3472-3473 (2004).
[CrossRef] [PubMed]

Kobayashi, S.

Kramer, R.

R. Kramer and O. Loffeld, “Presentation of an improved phase unwrapping algorithm based on Kalman filers combined with local slope estimation,” in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (1996).

Kronland-Martinet, R.

P. Guillemain and R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561-581 (1996).
[CrossRef]

Kujawinska, M.

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Marcel Dekker, 2001), Chap. 12, pp. 373-426.

Loffeld, O.

R. Kramer and O. Loffeld, “Presentation of an improved phase unwrapping algorithm based on Kalman filers combined with local slope estimation,” in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (1996).

Maitre, H.

E. Trouve, J. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

Maître, H.

Malacara, D.

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 1999).

Marroguin, J. L.

Marroquin, J. L.

Nam, L. T. H.

Nicolas, J.

E. Trouve, J. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

Osten, W.

Pedrini, G.

Qian, K.

K. Qian and S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
[CrossRef]

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
[CrossRef]

Rivera, M.

Rodriguez-Vera, R.

Schnars, U.

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Servin, M.

Soon, S. H.

Takeda, M.

Trouve, E.

E. Trouve, J. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

Trouvé, E.

Waldner, S.

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

Yun, H. Y.

Zeng, H.

Zhong, J.

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

E. Trouve, J. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

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

Opt. Eng.

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
[CrossRef]

G. H. Kaufmann and G. E. Galizzi, “Speckle noise reduction in television holography fringes using wavelet thresholding,” Opt. Eng. 35, 9-14 (1996).
[CrossRef]

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]

K. Qian and S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
[CrossRef]

Opt. Laser Technol.

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098(2008).
[CrossRef]

Opt. Lasers Eng.

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm: addendum,” Opt. Lasers Eng. 45, 1193-1195 (2007).
[CrossRef]

Opt. Lett.

Proc. IEEE

P. Guillemain and R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561-581 (1996).
[CrossRef]

Other

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 1999).

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Marcel Dekker, 2001), Chap. 12, pp. 373-426.

R. Kramer and O. Loffeld, “Presentation of an improved phase unwrapping algorithm based on Kalman filers combined with local slope estimation,” in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (1996).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, (Roberts & Company, 2007).

U. Schnars and W. Jueptner, Digital Holography: Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

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 (1)

Fig. 1
Fig. 1

Typical example for investigation. (a) A simulated noisy wrapped phase map, (b) phase map extracted by the WFR algorithm, (c) phase map extracted by the WFF algorithm, (d) speckled phase error in the WFF algorithm.

Tables (1)

Tables Icon

Table 1 Comparison of Theoretical and Numerical Results with Regard to the EPF with Phase of Eq. (46) a

Equations (82)

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

S f ( u , v ; ξ , η ) = f ( x , y ) g u , v ; ξ , η * ( x , y ) d x d y ,
f ( x , y ) = 1 4 π 2 S f ( u , v ; ξ , η ) g u , v ; ξ , η ( x , y ) d ξ d η d u d v ,
g u , v ; ξ , η ( x , y ) = g ( x u , y v ) exp ( j ξ x + j η y ) ,
g ( x , y ) = 1 π σ x σ y exp ( x 2 2 σ x 2 y 2 2 σ y 2 ) .
[ ω x ( u , v ) , ω y ( u , v ) ] = arg max ξ , η | S f ( u , v ; ξ , η ) | ,
r ( u , v ) = | S f [ u , v ; ω x ( u , v ) , ω y ( u , v ) ] | .
φ ( u , v ) = angle { S f [ u , v ; ω x ( u , v ) , ω y ( u , v ) ] } + ω x ( u , v ) u + ω y ( u , v ) v .
f ¯ ( x , y ) = 1 4 π 2 η l η h ξ l ξ h S f ¯ ( u , v ; ξ , η ) g u , v ; ξ , η ( x , y ) d ξ d η d u d v ,
S f ¯ ( u , v ; ξ , η ) = { S f ( u , v ; ξ , η ) if     | S f ( u , v ; ξ , η ) | thr 0 if     | S f ( u , v ; ξ , η ) | < thr ,
f i ( x , y ) = a ( x , y ) + b ( x , y ) cos [ j φ ( x , y ) + ( i 1 ) π / 2 ] + n i ( x , y ) , i = 1 , 2 , 3 , 4 ,
f ( x , y ) = f 1 ( x , y ) f 3 ( x , y ) + j f 4 ( x , y ) j f 2 ( x , y ) .
f ( x , y ) = 2 b ( x , y ) exp [ j φ ( x , y ) ] + n ( x , y ) ,
n ( x , y ) = n 1 ( x , y ) n 3 ( x , y ) + j n 4 ( x , y ) j n 2 ( x , y ) .
f ( x , y ) = f 0 ( x , y ) + n ( x , y ) ,
f 0 ( x , y ) = exp [ j φ ( x , y ) ] ,
n ( x , y ) = n ( x , y ) 2 b ( x , y ) = n 1 ( x , y ) n 3 ( x , y ) + j n 4 ( x , y ) j n 2 ( x , y ) 2 b ( x , y ) .
φ ( x , y ) = c 00 + c 10 x + c 01 y + 0.5 c 20 x 2 + 0.5 c 02 y 2 ,
ω x ( x , y ) φ ( x , y ) x = c 10 + c 20 x ,
ω y ( x , y ) φ ( x , y ) y = c 01 + c 02 y .
2 φ ( x , y ) x 2 = c 20 ,
2 φ ( x , y ) y 2 = c 02 .
S f 0 ( u , v ; ξ , η ) = A 0 ( u , v ; ξ , η ) exp [ j Φ 0 ( u , v ; ξ , η ) ] ,
A 0 ( u , v ; ξ , η ) = [ 16 π 2 σ x 2 σ y 2 ( 1 + σ x 4 c 20 2 ) ( 1 + σ y 4 c 02 2 ) ] 1 / 4 exp { σ x 2 [ ξ ω x ( u , v ) ] 2 2 ( 1 + σ x 4 c 20 2 ) σ y 2 [ η ω y ( u , v ) ] 2 2 ( 1 + σ y 4 c 02 2 ) } ,
Φ 0 ( u , v ; ξ , η ) = φ ( u , v ) ξ u η v σ x 4 c 20 [ ξ ω x ( u , v ) ] 2 2 ( 1 + σ x 4 c 20 2 ) σ y 4 c 02 [ η ω y ( u , v ) ] 2 2 ( 1 + σ y 4 c 02 2 ) + 1 2 arctan ( σ x 2 c 20 ) + 1 2 arctan ( σ y 2 c 02 ) .
n ( x , y ) = n r ( x , y ) + j n i ( x , y ) ,
S n ( u , v ; ξ , η ) = R n ( u , v ; ξ , η ) + j I n ( u , v ; ξ , η ) = A n ( u , v ; ξ , η ) exp [ j Φ n ( u , v ; ξ , η ) ] ,
p R n ( R n ) = 1 2 π σ n exp ( R n 2 2 σ n 2 ) .
p I n ( I n ) = 1 2 π σ n exp ( I n 2 2 σ n 2 ) .
p A n ( A n ) = A n σ n 2 exp ( A n 2 2 σ n 2 ) .
p Φ n ( Φ n ) = { 1 2 π π Φ n < π 0 otherwise .
p A n 1 | A n 2 ( A n 1 ) = A n 1 ( 1 α 2 ) σ n 2 exp [ A n 1 2 + α 2 A n 2 2 2 ( 1 α 2 ) σ n 2 ] I 0 [ α A n 1 A n 2 ( 1 α 2 ) σ n 2 ] ,
α | μ S n 1 S n 2 * | μ A n 1 2 μ A n 2 2 .
μ A n 1 2 μ A n 2 2 = 2 σ n 2 .
| μ S n 1 S n 2 * | = 2 σ n 2 exp [ σ x 2 ( Δ ξ ) 2 + σ y 2 ( Δ η ) 2 4 ] .
α = exp [ σ x 2 ( Δ ξ ) 2 + σ y 2 ( Δ η ) 2 4 ] ,
μ A n 1 | A n 2 = 1 2 π 2 ( 1 α 2 ) σ n 2 exp ( α 2 A n 2 2 4 ( 1 α 2 ) σ n 2 ) × { [ α 2 A n 2 2 + 2 ( 1 α 2 ) σ n 2 ] I 0 [ α 2 A n 2 2 4 ( 1 α 2 ) σ n 2 ] + α 2 A n 2 2 I 1 [ α 2 A n 2 2 4 ( 1 α 2 ) σ n 2 ] } ,
μ A n 1 2 | A n 2 = α 2 A n 2 2 + 2 ( 1 α 2 ) σ n 2 .
S f ( u , v ; ξ , η ) = A ( u , v ; ξ , η ) exp [ j Φ ( u , v ; ξ , η ) ] = S f 0 ( u , v ; ξ , η ) + S n ( u , v ; ξ , η ) .
A 01 A 02 = [ 16 π 2 σ x 2 σ y 2 ( 1 + σ x 4 c 20 2 ) ( 1 + σ y 4 c 02 2 ) ] 1 / 4 { 1 exp [ σ x 2 ξ 0 2 2 ( 1 + σ x 4 c 20 2 ) ] } .
μ ( A n 1 A n 2 ) | A n 2 = μ A n 1 | A n 2 A n 2 ,
σ ( A n 1 A n 2 ) | A n 2 = σ A n 1 | A n 2 = μ A n 1 2 | A n 2 ( μ A n 1 | A n 2 ) 2 .
c 20 = c 02 = 0.005   rad / ( pixel ) 2 ,
σ n i ( x , y ) = b ( x , y ) , i = 1 , 2 , 3 , 4 ,
σ x = σ y = 10   pixels ,
ξ 0 = η 0 = 0 . 025   rad / pixel .
φ ( x , y ) = 0.5 c 20 ( x 128 ) 2 + 0.5 c 02 ( y 128 ) 2 , 1 x , y 256 ,
p Δ ω x ( Δ ω x ) = { 1 / ξ 0 , ξ 0 / 2 Δ ω x ξ 0 / 2 0 otherwise ,
p Δ ω y ( Δ ω y ) = { 1 / η 0 , η 0 / 2 Δ ω y η 0 / 2 0 otherwise .
μ Δ ω x = μ Δ ω y = 0 ,
σ Δ ω x = ξ 0 / 12 ,
σ Δ ω y = η 0 / 12 .
( Δ φ ) 1 = σ x 4 c 20 [ Δ ω x ( u , v ) ] 2 2 ( 1 + σ x 4 c 20 2 ) ,
( Δ φ ) 2 = σ y 4 c 02 [ Δ ω y ( u , v ) ] 2 2 ( 1 + σ y 4 c 02 2 )
μ ( Δ φ ) 1 = ξ 0 2 / 12 ,
μ ( Δ φ ) 2 = η 0 2 / 12 ,
σ ( Δ φ ) 1 = ξ 0 2 / 180 ,
σ ( Δ φ ) 2 = η 0 2 / 180 .
p ( Φ ) = 1 2 π σ n / A 0 exp [ ( Φ Φ 0 ) 2 2 ( σ n / A 0 ) 2 ] .
( Δ Φ ) n = Φ Φ 0 ,
p [ ( Δ Φ ) n ] = 1 2 π σ n / A 0 exp [ ( Δ Φ ) n 2 2 ( σ n / A 0 ) 2 ] .
μ ( Δ φ ) n = 0 ,
σ ( Δ φ ) n = σ n / A 0 .
[ 16 π 2 σ x 2 σ y 2 ( 1 + σ x 4 c 20 2 ) ( 1 + σ y 4 c 02 2 ) ] 1 / 4 exp { σ x 2 [ ξ ω x ( u , v ) ] 2 2 ( 1 + σ x 4 c 20 2 ) σ y 2 [ η ω y ( u , v ) ] 2 2 ( 1 + σ y 4 c 02 2 ) } thr .
L ξ = 2 ( 1 + σ x 4 c 20 2 ) σ x ,
L η = 2 ( 1 + σ y 4 c 02 2 ) σ y ,
R = ln ( 8 π / L ξ L η / thr ) ,
[ ξ ω x ( u , v ) ] 2 ( L ξ R ) 2 + [ η ω y ( u , v ) ] 2 ( L η R ) 2 1 ,
area = π R 2 L ξ L η .
f ¯ 0 ( x , y ) = [ 1 exp ( 2 R 2 ) ] f 0 ( x , y ) ,
μ [ n ¯ ( x , y ) ] = 0.
| n ( x , y ) | 2 d x d y = 1 4 π 2 | S n ( u , v ; ξ , η ) | 2 d ξ d η d u d v ,
| n ¯ ( x , y ) | 2 d x d y = 1 4 π 2 | S n ¯ ( u , v ; ξ , η ) | 2 d ξ d η d u d v ,
σ n ¯ = area 4 π 2 σ n = R L ξ L η 2 π σ n .
f ¯ ( x , y ) = f 0 ¯ ( x , y ) + n ¯ ( x , y ) .
μ ( Δ φ ) n = 0 ,
σ ( Δ φ ) n = σ n ¯ | f ¯ 0 ( x , y ) | = R L ξ L η σ n 2 π [ 1 exp ( 2 R 2 ) ] .
2 π [ 1 exp ( 2 R 2 ) ] R L ξ L η
( thr ) opt = 2.64 L ξ L η .
e 2 ( x ) = d e 1 ( x ) d x = 4 π σ ( Δ φ ) n T x sin ( 2 π x T x ) ,
f I I ( x , y ) = exp [ j φ W ( x , y ) ] = f I ( x , y ) / | f I ( x , y ) | .
f I I I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ω c x x + ω c y y + φ ( x , y ) ] + n I I I ( x , y ) = a ( x , y ) + b ( x , y ) 2 exp [ j ω c x x j ω c y y j φ ( x , y ) ] + b ( x , y ) 2 exp [ j ω c x x + j ω c y y + j φ ( x , y ) ] + n I I I ( x , y ) ,
f I V ( x , y ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) ] + n I V ( x , y ) ,

Metrics