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.

© 2001 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  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, and 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]
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    [CrossRef]
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    [CrossRef]
  15. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
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    [CrossRef]
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  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).
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  22. C. Roddier and F. Roddier, “Interferogram analysis using Fourier transform techniques,” App. Opt. 26, 1668–1673 (1987).
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    [CrossRef]
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    [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 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, and Graeme T. Reid, eds., Proc. SPIE1755, (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. Bülow 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. H. Stark (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]

2001 (3)

2000 (2)

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

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

1999 (3)

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]

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]

1998 (1)

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]

1997 (2)

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]

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]

1996 (1)

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

1995 (5)

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

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

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 and J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[CrossRef]

1994 (3)

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]

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

1993 (2)

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

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

1992 (3)

1991 (2)

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

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]

1990 (3)

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

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]

1988 (1)

1987 (2)

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]

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

1986 (3)

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]

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

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

1985 (2)

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

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

1984 (3)

K. H. Womack, “Frequency domain desciption of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).

J. E. Grievenkamp, “Generalised data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

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

1983 (3)

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

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

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

1982 (2)

1966 (1)

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

1963 (1)

W. R. C. Rowley and J. Hamon, “Quelques mesures de dyssymetrie de profils spectraux,” Revue d’Optique 9, 519–531 (1963).

1947 (1)

D. Gabor, “Theory of communications,” Journal of the Institution of Electrical Engineers,  93, 429–457 (1947).

Alonso, M.

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

Bachor, H.-A.

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]

Barrett, H. H.

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]

Bokor, J.

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

Bone, D.

Bone, D. J.

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]

Bovik, A. C.

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.

Brophy, C.

Bülow, T.

T. Bülow 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]

Carre, P.

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

Chen, X.

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

Cheng, Y.-Y.

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

Deason, V. A.

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]

Dresel, T.

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]

Eiju, T.

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]

Falkenstorfer, O.

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

Farrant, D. I.

Farrell, C. T.

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

Fiddy, M. A.

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

Fisher, R. B.

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]

Fitzgibbon, A. W.

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]

Forbes, G. W.

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

Gabor, D.

D. Gabor, “Theory of communications,” Journal of the Institution of Electrical Engineers,  93, 429–457 (1947).

Goldberg, K. A.

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

Gramaglia, M.

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

Granlund, G. H.

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

Grievenkamp, J. E.

J. E. Grievenkamp, “Generalised data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Guo, L.-R.

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]

Hamon, J.

W. R. C. Rowley and J. Hamon, “Quelques mesures de dyssymetrie de profils spectraux,” Revue d’Optique 9, 519–531 (1963).

Han, G.-S.

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]

Hariharan, P.

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]

Havlicek, J. P.

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.

Havlicek, J. W.

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.

Hegedus, Z. S.

Hibino, K.

Ina, H.

Ishii, Y.

Kim, S.-W.

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]

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]

Kitoh, M.

Knutsson, H.

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

Kobayashi, S.

Kokal, J. V.

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

Kong, I.-B.

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]

Kreis, T.

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

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

Lai, G.

Larkin, K. G.

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]

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. 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]

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. 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, “Topics in Multi-dimensional Signal Demodulation”, PhD. University of Sydney, 2001.

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, and Graeme T. Reid, eds., SPIE Proc.1755, California, (1992).

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, and Graeme T. Reid, eds., Proc. SPIE1755, (1992), 219–227.
[CrossRef]

Lassaahn, J. K.

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]

Lassahn, G. D.

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]

Li, J.

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]

Macy, W. W.

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

Manzke, B.

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]

McKelvie, J.

K. E. Perry and J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[CrossRef]

Mertz, L.

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

Morgan, C. J.

Mutoh, K.

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

Nugent, K. A.

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

Okada, K.

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]

Oldfield, M. A.

Onodera, R.

Oreb, B. F.

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]

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

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, and Graeme T. Reid, eds., SPIE Proc.1755, California, (1992).

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, and Graeme T. Reid, eds., Proc. SPIE1755, (1992), 219–227.
[CrossRef]

Perry, K. E.

K. E. Perry and J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[CrossRef]

Pilu, M.

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]

Player, M. A.

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

Ransom, P. L.

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

Roddier, C.

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

Roddier, F.

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

Rogala, E. W.

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]

Rowley, W. R. C.

W. R. C. Rowley and J. Hamon, “Quelques mesures de dyssymetrie de profils spectraux,” Revue d’Optique 9, 519–531 (1963).

Sandeman, R. J.

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]

Sato, A.

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]

Schreiber, H.

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

Schwider, J.

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]

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

Sheppard, C. J. R.

Sommer, G.

T. Bülow 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]

Strobel, B.

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

Su, X.-Y.

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]

Surrel, Y.

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

Takeda, M.

Taylor, P. L.

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]

Tiziani, H. J.

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

Tsujiuchi, J.

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]

van Brug, H.

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

Windecker, R.

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

Wizinowich, P. L.

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

Womack, K. H.

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

K. H. Womack, “Frequency domain desciption of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).

Wyant, J. C.

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

Yatagai, T.

Yeazell, J. A.

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

Zhao, B.

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]

Zoller, A.

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

Am. J. Phys. (1)

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

App. Opt. (19)

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

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]

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]

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. 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]

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

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

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

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]

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]

IEEE Transactions on Pattern Analysis and Machine Intelligence (1)

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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

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]

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]

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]

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]

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]

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

C. J. R. Sheppard and Z. S. Hegedus, “Axial behavior of pupil plane filters,” J. Opt. Soc. Am. A 5, 643–664 (1988).
[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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D. Gabor, “Theory of communications,” Journal of the Institution of Electrical Engineers,  93, 429–457 (1947).

Meas. Sci. technol. (1)

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]

Meas. Sci. Techol. (2)

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]

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

Opt. Comm. (1)

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]

Opt. Eng. (7)

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]

J. E. Grievenkamp, “Generalised data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

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]

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

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

K. H. Womack, “Frequency domain desciption of interferogram analysis,” Opt. Eng. 23, 396–400 (1984).

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]

Opt. Lasers Eng. (1)

K. E. Perry and J. McKelvie, “Reference phase shift determination in phase shifting interferometry,” Opt. Lasers Eng. 22, 77–90 (1995).
[CrossRef]

Opt. Lett. (2)

Revue d’Optique (1)

W. R. C. Rowley and J. Hamon, “Quelques mesures de dyssymetrie de profils spectraux,” Revue d’Optique 9, 519–531 (1963).

Other (10)

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, and Graeme T. Reid, eds., SPIE Proc.1755, California, (1992).

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

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.

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, and Graeme T. Reid, eds., Proc. SPIE1755, (1992), 219–227.
[CrossRef]

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

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. Bülow 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. H. Stark (Orlando, Florida: Academic Press, 1987).

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.

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

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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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