Abstract

A new method of estimating the phase-shift between interferograms is introduced. The method is based on a recently introduced two-dimensional Fourier-Hilbert demodulation technique. Three or more interferogram frames in an arbitrary sequence are required. The first stage of the algorithm calculates frame differences to remove the fringe pattern offset; allowing increased fringe modulation. The second stage is spatial demodulation to estimate the analytic image for each frame difference. The third stage robustly estimates the inter-frame phase-shifts and then uses the generalised phase-shifting algorithm of Lai and Yatagai to extract the offset, the modulation and the phase exactly. Initial simulations of the method indicate that high accuracy phase estimates are obtainable even in the presence of closed or discontinuous fringe patterns.

© Optical Society of America

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References

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  1. P. Carre, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
    [CrossRef]
  2. W. R. C. Rowley, and J. Hamon, "Quelques mesures de dyssymetrie de profils spectraux," Revue d'Optique 9, 519-531 (1963).
  3. Y.-Y. Cheng, and J. C. Wyant, "Phase-shifter calibration in phase-shifting interferometry," App. Opt. 24, 3049-3052 (1985).
    [CrossRef]
  4. P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometer: a simple error-compensating phase calculation algorithm," App. Opt. 26, 2504-2506 (1987).
    [CrossRef]
  5. Y. Ishii, and R. Onodera, "Phase-extraction algorithm in laser diode phase-shifting interferometry," Opt. Lett. 20, 1883-1885 (1995).
    [CrossRef] [PubMed]
  6. K. G. Larkin, and B. F. Oreb, "Design and assessment of Symmetrical Phase-Shifting Algorithms," J. Opt. Soc. Am. A 9, 1740-1748 (1992).
    [CrossRef]
  7. K. G. Larkin, and B. F. Oreb, "A new seven sample phase-shifting algorithm," SPIE International Symposium on Optical Applied Science and Engineering, San Diego, Gordon M. Brown, Osuk Y. Kwon, Malgorzata Kujawinska, Graeme T. Reid, eds., SPIE Proc. 1755, California, (1992).
  8. Y. Surrel, "Phase stepping: a new self-calibrating algorithm," App. Opt. 32, 3598-3600 (1993).
    [CrossRef]
  9. P. L. Wizinowich, "Phase-shifting interferometry in the presence of vibration: a new algorithm and system," App. Opt. 29, 3271-3279 (1990).
    [CrossRef]
  10. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, et al., "New compensating four-phase algorithm for phase-shift interferometry," Opt. Eng. 32, 1883-1885 (1993).
    [CrossRef]
  11. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting interferometry for non-sinusoidal waveforms with phase-shift errors," J. Opt. Soc. Am. A 12, 761-768 (1995).
    [CrossRef]
  12. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts," J. Opt. Soc. Am. A 14, 918-930 (1997).
    [CrossRef]
  13. L. Mertz, "Real-time fringe pattern analysis," App. Opt. 22, 1535-1539 (1983).
    [CrossRef]
  14. W. W. Macy Jr, "Two-dimensional fringe-pattern analysis," App. Opt. 22, 3898-3901 (1983).
    [CrossRef]
  15. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  16. P. L. Ransom, and J. V. Kokal, "Interferogram analysis by a modified sinusoid fitting technique," App. Opt. 25, 4199-4204 (1986).
    [CrossRef]
  17. 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]
  18. M. Takeda, and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes," App. Opt. 22, 3977-3982 (1983).
    [CrossRef]
  19. K. H. Womack, "Frequency domain desciption of interferogram analysis," Opt. Eng. 23, 396-400 (1984).
  20. K. A. Nugent, "Interferogram analysis using an accurate fully automatic algorithm," App. Opt. 24, 3101-3105 (1985).
    [CrossRef]
  21. D. J. Bone, H.-A. Bachor, and R. J. Sandeman, "Fringe-pattern analysis using a 2-D Fourier transform," App. Opt. 25, 1653-1660 (1986).
    [CrossRef]
  22. C. Roddier, and F. Roddier, "Interferogram analysis using Fourier transform techniques," App. Opt. 26, 1668-1673 (1987).
    [CrossRef]
  23. M. Takeda, and M. Kitoh, "Spatiotemporal multiplex heterodyne interferometry," J. Opt. Soc. Am. A 9, 1607-1614 (1992).
    [CrossRef]
  24. G. Lai, and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822-827 (1991).
    [CrossRef]
  25. C. J. Morgan, "Least squares estimation in phase-measurement interferometry," Opt. Lett. 7, 368-370 (1982).
    [CrossRef] [PubMed]
  26. J. E. Grievenkamp, "Generalised data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).
  27. T. Kreis, Holographic interferometry. Principles and methods, 1, Akademie Verlag GmbH, Berlin, 1996.
  28. K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase in phase shifting interferometry," Opt. Comm. 84, 118-124 (1991).
    [CrossRef]
  29. G. D. Lassahn, J. K. Lassaahn, P. L. Taylor, and V. A. Deason, "Multiphase fringe analysis with unkown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
    [CrossRef]
  30. G.-S. Han, and S.-W. Kim, "Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting," App. Opt. 33, 7321-7325 (1994).
    [CrossRef]
  31. I.-B. Kong, and S.-W. Kim, "General algorithm of phase-shifting interferometry by iterative least-squares fitting," Opt. Eng. 34, 183-188 (1995).
    [CrossRef]
  32. K. E. Perry_Jr, and J. McKelvie, "Reference phase shift determination in phase shifting interferometry," Opt. Lasers Eng. 22, 77-90 (1995).
    [CrossRef]
  33. Tests by the author have shown that the error term to be minimised can vanish for the common four sample algorithm ( with nominal phase steps of 90 degrees), even when phase step errors are present. This can be seen as stagnation in the optimisation procedure.
  34. E. W. Rogala, and H. H. Barrett, "Phase-Shifting Interferometry and Maximum-Likelihood Estimation Theory - II - a Generalized Solution," App. Opt. 37, 7253-7258 (1998).
    [CrossRef]
  35. C. T. Farrell, and M. A. Player, "Phase step measurement and variable step algorithms in phase-shifting interferometry," Meas. Sci. Techol. 3, 953-958 (1992).
    [CrossRef]
  36. C. T. Farrell, and M. A. Player, "Phase-step insensitive algorithms for phase-shifting interferometry," Meas. Sci. Techol. 5, 648-652 (1994).
    [CrossRef]
  37. A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, "Direct least squares fitting of ellipses," IEEE Transactions on Pattern Analysis and Machine Intelligence 21, 476-480 (1999).
    [CrossRef]
  38. H. van Brug, "Phase-step calibration for phase-stepped interferometry," App. Opt. 38, 3549-3555 (1999).
    [CrossRef]
  39. X. Chen, M. Gramaglia, and J. A. Yeazell, "Phase-shifting interferometry with uncalibrated phase shifts," App. Opt. 39, 585-591 (2000).
    [CrossRef]
  40. J. Schwider, T. Dresel, and B. Manzke, "Some considerations of reduction of reference phase error in phase-stepping interferometry," App. Opt. 38, 655-659 (1999).
    [CrossRef]
  41. K. G. Larkin, and B. F. Oreb, "Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function," SPIE International Symposium on Optical Applied Science and Engineering, San Diego, California, Gordon M. Brown, Osuk Y. Kwon, Malgorzata Kujawinska, Graeme T. Reid , eds., Proc. SPIE 1755, (1992), 219-227.
    [CrossRef]
  42. J. Li, X.-Y. Su, and L.-R. Guo, "Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes," Opt. Eng. 29, 1439-1444 (1990).
    [CrossRef]
  43. R. Windecker, and H. J. Tiziani, "Semispatial, robust, and accurate phase evaluation algorithm," App. Opt. 34, 7321-7326 (1995).
    [CrossRef]
  44. K. A. Goldberg, and J. Bokor, "Fourier-transform method of phase-shift determination," App. Opt. 40, 2886-2894 (2001).
    [CrossRef]
  45. K. G. Larkin, D. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1862-1870 (2001).
    [CrossRef]
  46. G. H. Granlund, and H. Knutsson, Signal processing for computer vision, Kluwer, Dordrecht, Netherlands, 1995.
  47. K. G. Larkin, "Natural demodulation of two-dimensional fringe patterns: II. Stationary phase analysis of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1871-1881 (2001).
    [CrossRef]
  48. K. G. Larkin, "Topics in Multi-dimensional Signal Demodulation", PhD. University of Sydney, 2001.
  49. T. Kreis, "Digital holographic interference-phase measurement using the Fourier transform method," J. Opt. Soc. Am. A 3, 847-855 (1986).
    [CrossRef]
  50. Explicit orientation estimation can be eliminated from the technique at the expense of clarity. The orientation estimate is replaced by a random but consistent choice of polarity h(x,y). The two levels of differencing in the algorithm ultimately remove the function h(x,y) from the final result.
  51. D. Gabor, "Theory of communications," Journal of the Institution of Electrical Engineers, 93, 429-457 (1947).
  52. J. P. Havlicek, J. W. Havlicek, and A. C. Bovik, "The Analytic Image,," IEEE International Conference on Image Processing, Santa Barbara,California, (1997), 446-449.
  53. T. B?low, and G. Sommer, "A Novel Approach to the 2D Analytic Signal," Computer Analysis of Images and Patterns, CAIP'99, Ljubljana, Slovenia, (1999), 25-32.
    [CrossRef]
  54. M. A. Fiddy, "The role of analyticity in image recovery", Image recovery: theory and application, ed. Stark, H. (Orlando, Florida: Academic Press, 1987).
  55. M. Alonso, and G. W. Forbes, "Measures of spread for periodic distributions and the associated uncertainty relations," Am. J. Phys. 69, 340-347 (2000).
  56. C. J. R. Sheppard, and Z. S. Hegedus, "Axial behavior of pupil plane filters," J. Opt. Soc. Am. A 5, 643-664 (1988).
    [CrossRef]
  57. B. Strobel, "Processing of Interferometric Phase Maps As Complex-Valued Phasor Images," App. Opt. 35, 2192-2198 (1996).
    [CrossRef]
  58. B. Zhao, "A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm," Meas. Sci. technol. 8, 147-153 (1997).
    [CrossRef]
  59. C. Brophy, "Effect of intensity error correlation on the computed phase of phase-shifting interferometry," J. Opt. Soc. Am. A 7, 537-541 (1990).
    [CrossRef]

Other

P. Carre, "Installation et utilisation du comparateur photoelectrique et interferentiel du Bureau International des Poids et Mesures," Metrologia 2, 13-23 (1966).
[CrossRef]

W. R. C. Rowley, and J. Hamon, "Quelques mesures de dyssymetrie de profils spectraux," Revue d'Optique 9, 519-531 (1963).

Y.-Y. Cheng, and J. C. Wyant, "Phase-shifter calibration in phase-shifting interferometry," App. Opt. 24, 3049-3052 (1985).
[CrossRef]

P. Hariharan, B. F. Oreb, and T. Eiju, "Digital phase-shifting interferometer: a simple error-compensating phase calculation algorithm," App. Opt. 26, 2504-2506 (1987).
[CrossRef]

Y. Ishii, and R. Onodera, "Phase-extraction algorithm in laser diode phase-shifting interferometry," Opt. Lett. 20, 1883-1885 (1995).
[CrossRef] [PubMed]

K. G. Larkin, and B. F. Oreb, "Design and assessment of Symmetrical Phase-Shifting Algorithms," J. Opt. Soc. Am. A 9, 1740-1748 (1992).
[CrossRef]

K. G. Larkin, and B. F. Oreb, "A new seven sample phase-shifting algorithm," SPIE International Symposium on Optical Applied Science and Engineering, San Diego, Gordon M. Brown, Osuk Y. Kwon, Malgorzata Kujawinska, Graeme T. Reid, eds., SPIE Proc. 1755, California, (1992).

Y. Surrel, "Phase stepping: a new self-calibrating algorithm," App. Opt. 32, 3598-3600 (1993).
[CrossRef]

P. L. Wizinowich, "Phase-shifting interferometry in the presence of vibration: a new algorithm and system," App. Opt. 29, 3271-3279 (1990).
[CrossRef]

J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, et al., "New compensating four-phase algorithm for phase-shift interferometry," Opt. Eng. 32, 1883-1885 (1993).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting interferometry for non-sinusoidal waveforms with phase-shift errors," J. Opt. Soc. Am. A 12, 761-768 (1995).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, "Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts," J. Opt. Soc. Am. A 14, 918-930 (1997).
[CrossRef]

L. Mertz, "Real-time fringe pattern analysis," App. Opt. 22, 1535-1539 (1983).
[CrossRef]

W. W. Macy Jr, "Two-dimensional fringe-pattern analysis," App. Opt. 22, 3898-3901 (1983).
[CrossRef]

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

P. L. Ransom, and J. V. Kokal, "Interferogram analysis by a modified sinusoid fitting technique," App. Opt. 25, 4199-4204 (1986).
[CrossRef]

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]

M. Takeda, and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes," App. Opt. 22, 3977-3982 (1983).
[CrossRef]

K. H. Womack, "Frequency domain desciption of interferogram analysis," Opt. Eng. 23, 396-400 (1984).

K. A. Nugent, "Interferogram analysis using an accurate fully automatic algorithm," App. Opt. 24, 3101-3105 (1985).
[CrossRef]

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

C. Roddier, and F. Roddier, "Interferogram analysis using Fourier transform techniques," App. Opt. 26, 1668-1673 (1987).
[CrossRef]

M. Takeda, and M. Kitoh, "Spatiotemporal multiplex heterodyne interferometry," J. Opt. Soc. Am. A 9, 1607-1614 (1992).
[CrossRef]

G. Lai, and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822-827 (1991).
[CrossRef]

C. J. Morgan, "Least squares estimation in phase-measurement interferometry," Opt. Lett. 7, 368-370 (1982).
[CrossRef] [PubMed]

J. E. Grievenkamp, "Generalised data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

T. Kreis, Holographic interferometry. Principles and methods, 1, Akademie Verlag GmbH, Berlin, 1996.

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase in phase shifting interferometry," Opt. Comm. 84, 118-124 (1991).
[CrossRef]

G. D. Lassahn, J. K. Lassaahn, P. L. Taylor, and V. A. Deason, "Multiphase fringe analysis with unkown phase shifts," Opt. Eng. 33, 2039-2044 (1994).
[CrossRef]

G.-S. Han, and S.-W. Kim, "Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting," App. Opt. 33, 7321-7325 (1994).
[CrossRef]

I.-B. Kong, and S.-W. Kim, "General algorithm of phase-shifting interferometry by iterative least-squares fitting," Opt. Eng. 34, 183-188 (1995).
[CrossRef]

K. E. Perry_Jr, and J. McKelvie, "Reference phase shift determination in phase shifting interferometry," Opt. Lasers Eng. 22, 77-90 (1995).
[CrossRef]

Tests by the author have shown that the error term to be minimised can vanish for the common four sample algorithm ( with nominal phase steps of 90 degrees), even when phase step errors are present. This can be seen as stagnation in the optimisation procedure.

E. W. Rogala, and H. H. Barrett, "Phase-Shifting Interferometry and Maximum-Likelihood Estimation Theory - II - a Generalized Solution," App. Opt. 37, 7253-7258 (1998).
[CrossRef]

C. T. Farrell, and M. A. Player, "Phase step measurement and variable step algorithms in phase-shifting interferometry," Meas. Sci. Techol. 3, 953-958 (1992).
[CrossRef]

C. T. Farrell, and M. A. Player, "Phase-step insensitive algorithms for phase-shifting interferometry," Meas. Sci. Techol. 5, 648-652 (1994).
[CrossRef]

A. W. Fitzgibbon, M. Pilu, and R. B. Fisher, "Direct least squares fitting of ellipses," IEEE Transactions on Pattern Analysis and Machine Intelligence 21, 476-480 (1999).
[CrossRef]

H. van Brug, "Phase-step calibration for phase-stepped interferometry," App. Opt. 38, 3549-3555 (1999).
[CrossRef]

X. Chen, M. Gramaglia, and J. A. Yeazell, "Phase-shifting interferometry with uncalibrated phase shifts," App. Opt. 39, 585-591 (2000).
[CrossRef]

J. Schwider, T. Dresel, and B. Manzke, "Some considerations of reduction of reference phase error in phase-stepping interferometry," App. Opt. 38, 655-659 (1999).
[CrossRef]

K. G. Larkin, and B. F. Oreb, "Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function," SPIE International Symposium on Optical Applied Science and Engineering, San Diego, California, Gordon M. Brown, Osuk Y. Kwon, Malgorzata Kujawinska, Graeme T. Reid , eds., Proc. SPIE 1755, (1992), 219-227.
[CrossRef]

J. Li, X.-Y. Su, and L.-R. Guo, "Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes," Opt. Eng. 29, 1439-1444 (1990).
[CrossRef]

R. Windecker, and H. J. Tiziani, "Semispatial, robust, and accurate phase evaluation algorithm," App. Opt. 34, 7321-7326 (1995).
[CrossRef]

K. A. Goldberg, and J. Bokor, "Fourier-transform method of phase-shift determination," App. Opt. 40, 2886-2894 (2001).
[CrossRef]

K. G. Larkin, D. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1862-1870 (2001).
[CrossRef]

G. H. Granlund, and H. Knutsson, Signal processing for computer vision, Kluwer, Dordrecht, Netherlands, 1995.

K. G. Larkin, "Natural demodulation of two-dimensional fringe patterns: II. Stationary phase analysis of the spiral phase quadrature transform," J. Opt. Soc. Am. A 18, 1871-1881 (2001).
[CrossRef]

K. G. Larkin, "Topics in Multi-dimensional Signal Demodulation", PhD. University of Sydney, 2001.

T. Kreis, "Digital holographic interference-phase measurement using the Fourier transform method," J. Opt. Soc. Am. A 3, 847-855 (1986).
[CrossRef]

Explicit orientation estimation can be eliminated from the technique at the expense of clarity. The orientation estimate is replaced by a random but consistent choice of polarity h(x,y). The two levels of differencing in the algorithm ultimately remove the function h(x,y) from the final result.

D. Gabor, "Theory of communications," Journal of the Institution of Electrical Engineers, 93, 429-457 (1947).

J. P. Havlicek, J. W. Havlicek, and A. C. Bovik, "The Analytic Image,," IEEE International Conference on Image Processing, Santa Barbara,California, (1997), 446-449.

T. B?low, and G. Sommer, "A Novel Approach to the 2D Analytic Signal," Computer Analysis of Images and Patterns, CAIP'99, Ljubljana, Slovenia, (1999), 25-32.
[CrossRef]

M. A. Fiddy, "The role of analyticity in image recovery", Image recovery: theory and application, ed. Stark, H. (Orlando, Florida: Academic Press, 1987).

M. Alonso, and G. W. Forbes, "Measures of spread for periodic distributions and the associated uncertainty relations," Am. J. Phys. 69, 340-347 (2000).

C. J. R. Sheppard, and Z. S. Hegedus, "Axial behavior of pupil plane filters," J. Opt. Soc. Am. A 5, 643-664 (1988).
[CrossRef]

B. Strobel, "Processing of Interferometric Phase Maps As Complex-Valued Phasor Images," App. Opt. 35, 2192-2198 (1996).
[CrossRef]

B. Zhao, "A statistical method for fringe intensity-correlated error in phase-shifting measurement: the effect of quantization error on the N-bucket algorithm," Meas. Sci. technol. 8, 147-153 (1997).
[CrossRef]

C. Brophy, "Effect of intensity error correlation on the computed phase of phase-shifting interferometry," J. Opt. Soc. Am. A 7, 537-541 (1990).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

The left plot shows sequential inter-frame differences (dotted connecting lines) on a unit circle representing phase angles from 0 to 2π. The right plot shows all possible interframe differences as dotted lines

Fig. 2.
Fig. 2.

(a). Flowchart for automatic phase-step calibration method.

Fig. 2.
Fig. 2.

b). Continuation of flowchart for automatic calibration method.

Fig. 3.
Fig. 3.

Simple weight function calculated from the estimated errors in the contingent analytic function (relative to the output from the PSA). Black regions indicate zero weight, white regions have a weight of 1. Note how the weight removes the contribution from regions with large fringe curvature, from regions with discontinuous phase, and from regions with stationary phase.

Fig. 4.
Fig. 4.

Fringe pattern used for testing the phase-shift calibration algorithm. Note the closed fringes, the rapid fringe spacing variation and the fringe discontinuity – factors which defeat many other techniques. The sampled intensity varies between 0 and 255 grey levels.

Fig. 5.
Fig. 5.

Phase error showing the classic second harmonic structure, and the disappearance of the vertical half-period discontinuity. Note that the phase error is just ±0.0068 radian, but is shown here normalised.

Tables (1)

Tables Icon

Table 1. Performance of phase-shift calibrating algorithm applied to fringe pattern of Fig. 4

Equations (32)

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

f n ( x , y ) = a ( x , y ) + b ( x , y ) cos [ ψ ( x , y ) + δ n ] .
N M 3 M + N 1 or N 3 M 1 M 1 or M N 1 N 3 .
b ( x , y ) exp [ i ψ ( x , y ) + i δ n ] ,
g n m = f n f m = g m = b { cos [ ψ + δ n ] cos [ ψ + δ m ] }
$ { f ( x , y ) } = F 1 { exp [ i ϕ ] F { f ( x , y ) } } .
F { f ( x , y ) } = F ( u , v ) = + + f ( x , y ) exp [ 2 π i ( u x + v y ) ] d x d y F 1 { F ( u , v ) } = f ( x , y ) = F ( u , v ) [ + 2 π i ( u x + v y ) ] d u d v } .
u = q cos ϕ v = q sin ϕ } .
$ { g n m } i exp [ i β ] b [ sin ( ψ + δ n ) sin ( ψ + δ m ) ]
ψ ( x , y ) = ψ 00 + 2 π u 0 ( x x 0 ) + 2 π v 0 ( y y 0 ) +
ψ ( x , y ) x x = x 0 y = y 0 2 π u 0 ψ ( x , y ) y x = x 0 y = y 0 2 π v 0 } .
g n m x b ( sin ( ψ + δ n ) + sin ( ψ + δ m ) ) ψ x g n m y b ( sin ( ψ + δ n ) + sin ( ψ + δ m ) ) ψ y }
g n m y g n m x v 0 u 0 = tan β n m
exp [ i β e ] = cos β e + i sin β e = ± u 0 + i v 0 u 0 2 + v 0 2 .
g n m = 2 b sin [ ψ + ( δ n + δ m ) 2 ] sin [ ( δ n δ m ) 2 ]
V { g n m } = i exp [ i β e ] $ { g n m } 2 b exp [ i ( β β e ) ] cos [ ψ + ( δ n + δ m ) 2 ] sin [ ( δ n δ m ) 2 ]
h ( x , y ) = exp [ i ( β β e ) ] = { + exp [ i ε ] , β e = β ε exp [ i ε ] , β e = β ε + π
g ˜ n m = V { g n m } + i g n m
g ˜ n m = 2 b sin [ ( δ n δ m ) 2 ] ( h cos [ ψ + ( δ n + δ m ) 2 ] + i sin [ ψ + ( δ n + δ m ) 2 ] )
= 2 b sin [ ( δ n δ m ) 2 ] exp { i h cos [ ψ + ( δ n + δ m ) 2 ] + i π 2 } .
α n m = Arg ( g ˜ n m ) [ ψ + ( δ n + δ m ) 2 ] h + π 2
α n m α m k = [ ψ + ( δ n + δ m ) 2 ] h [ ψ + ( δ m + δ k ) 2 ] h = [ ( δ n δ k ) 2 ] h .
δ n δ k mean = 2 S α n m α m k d x d y S d x d y
α n m α m k α n m α m k = h ( x , y ) δ n δ k δ n δ k = h ( x , y ) sgn ( δ n δ k ) .
h g ( x , y ) = h n ̂ k ̂ ( x , y ) = sgn ( α n ̂ m ̂ α m ̂ k ̂ ) ,
p n k ( x , y ) = exp [ 2 i h { Arg ( g ˜ n m ) Arg ( g ˜ m k ) } ] exp [ i ( δ n δ k ) ] .
( δ n δ k ) ¯ phase mean = Arg [ p n k ¯ ] = Arg [ S p n k ( x , y ) d x d y S d x d y ] = Arg [ S p n k ( x , y ) d x d y ]
γ n k = ( δ n δ k ) ¯ phase weighted mean = Arg [ w p n k ¯ ] = Arg [ S p n k ( x , y ) w 2 ( x , y ) d x d y S w 2 ( x , y ) d x d y ]
N ! 2 ( N 2 ) !
{ γ 12 , γ 23 , γ 34 , γ 45 , γ 13 , γ 24 , γ 35 , γ 14 , γ 25 , γ 15 ,
δ 1 = 0 , δ 2 = δ 1 + γ 21 , δ 3 = δ 2 + γ 32 , δ 4 = δ 3 + γ 43 , δ 5 = δ 4 + γ 54
σ n k 2 = S p n k w p ¯ n k 2 w 2 ( x , y ) d x d y S w 2 ( x , y ) d x d y .
σ n k 2 = 1 w p ¯ n k 2 .

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