Abstract

A novel generalized approach to phase-shifting interferometry in which phase distribution in an interferogram is evaluated in the presence of nonsinusoidal waveforms and piezoactuator device miscalibration is proposed. The approach is based on the underlying rotational invariance of signal subspaces spanned by two temporally displaced data sets. The advantage of the proposed method lies in its ability to identify arbitrary phase-step values pixelwise from an interference signal buried in noise. The robustness of the proposed method is investigated by addition of white Gaussian noise during the simulations.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  3. K. Creath, “Phase-shifting holographic interferometry,” in Holographic Interferometry, P. K. Rastogi, ed., Vol. 68 of Springer Series in Optical Sciences (Springer-Verlag, 1994), pp. 109–150.
    [CrossRef]
  4. T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Verlag, 1996), pp. 101–170.
  5. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501–598.
  6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  7. Y. Zhu, T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40, 4540–4546 (2001).
    [CrossRef]
  8. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  9. J. Schwider, O. Falkenstorfer, H. Schreiber, A. Zoller, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32, 1883–1885 (1993).
    [CrossRef]
  10. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32, 3598–3600 (1993).
    [CrossRef] [PubMed]
  11. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
    [CrossRef] [PubMed]
  12. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  13. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  14. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  15. Y.-Y. Cheng, J. C. Wyant, “Phase-shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
    [CrossRef]
  16. 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]
  17. B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
    [CrossRef] [PubMed]
  18. R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
    [CrossRef]
  19. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
    [CrossRef]
  20. P. de Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
    [CrossRef] [PubMed]
  21. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [CrossRef]
  22. P. L. Wizinowich, “Phase-shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29, 3271–3279 (1990).
    [CrossRef] [PubMed]
  23. C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
    [CrossRef]
  24. K. Kinnnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef]
  25. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  26. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  27. R. Roy, T. Kailath, “ESPRIT—Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust. Speech Signal Process. 37, 984–995 (1989).
    [CrossRef]
  28. J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust. Speech Signal Process. 36, 1846–1853 (1988).
    [CrossRef]
  29. T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F 140, 71–80 (1993).
  30. R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. 19, 134–139 (1983).
    [CrossRef]
  31. M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities (Dover, 1992), pp. 15–16.
  32. K. M. Hoffman, R. Kunze, Linear Algebra, 2nd ed. (Prentice-Hall, 1971).
  33. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
    [CrossRef] [PubMed]
  34. P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
    [CrossRef] [PubMed]

2001 (1)

1997 (2)

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]

B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (3)

1993 (4)

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F 140, 71–80 (1993).

P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
[CrossRef] [PubMed]

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

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

1992 (4)

K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
[CrossRef]

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
[CrossRef] [PubMed]

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

1991 (1)

1990 (1)

1989 (1)

R. Roy, T. Kailath, “ESPRIT—Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust. Speech Signal Process. 37, 984–995 (1989).
[CrossRef]

1988 (2)

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust. Speech Signal Process. 36, 1846–1853 (1988).
[CrossRef]

K. Kinnnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

1987 (1)

1985 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

1983 (2)

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. 19, 134–139 (1983).
[CrossRef]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

1982 (1)

1974 (1)

1966 (1)

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

Brangaccio, D. J.

Bruning, J. H.

Burow, R.

Carré, P.

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

Cheng, Y.-Y.

Creath, K.

K. Creath, “Phase-shifting holographic interferometry,” in Holographic Interferometry, P. K. Rastogi, ed., Vol. 68 of Springer Series in Optical Sciences (Springer-Verlag, 1994), pp. 109–150.
[CrossRef]

de Groot, P.

Deck, L. L.

Eiju, T.

Elssner, K. E.

Falkenstorfer, O.

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

Farrant, D. I.

Frankena, H. J.

Fuchs, J. J.

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust. Speech Signal Process. 36, 1846–1853 (1988).
[CrossRef]

Gallagher, J. E.

Gemma, T.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501–598.

Grzanna, J.

Hariharan, P.

Herriott, D. R.

Hibino, K.

Hoffman, K. M.

K. M. Hoffman, R. Kunze, Linear Algebra, 2nd ed. (Prentice-Hall, 1971).

Joenathan, C.

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

Józwicki, R.

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Kailath, T.

R. Roy, T. Kailath, “ESPRIT—Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust. Speech Signal Process. 37, 984–995 (1989).
[CrossRef]

Khorana, B. M.

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

Kinnnstaetter, K.

Kreis, T.

T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Verlag, 1996), pp. 101–170.

Kujawinska, M.

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Kumaresan, R.

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. 19, 134–139 (1983).
[CrossRef]

Kunze, R.

K. M. Hoffman, R. Kunze, Linear Algebra, 2nd ed. (Prentice-Hall, 1971).

Larkin, K. G.

Lohmann, A. W.

Marcus, M.

M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities (Dover, 1992), pp. 15–16.

Merkel, K.

Minc, H.

M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities (Dover, 1992), pp. 15–16.

Morgan, C. J.

Oreb, B. F.

Rastogi, P. K.

Rathjen, C.

Rosenfeld, D. P.

Roy, R.

R. Roy, T. Kailath, “ESPRIT—Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust. Speech Signal Process. 37, 984–995 (1989).
[CrossRef]

Salbut, M.

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

Schreiber, H.

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

Schwider, J.

Smorenburg, C.

Söderström, T.

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F 140, 71–80 (1993).

Spolaczyk, R.

Stoica, P.

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F 140, 71–80 (1993).

Streibl, N.

Surrel, Y.

Tufts, D. W.

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. 19, 134–139 (1983).
[CrossRef]

van Wingerden, J.

White, A. D.

Wizinowich, P. L.

Wyant, J. C.

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]

B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

Zhu, Y.

Zoller, A.

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

Appl. Opt. (14)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
[CrossRef] [PubMed]

Y. Zhu, T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” Appl. Opt. 40, 4540–4546 (2001).
[CrossRef]

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

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

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
[CrossRef] [PubMed]

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

B. Zhao, Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997).
[CrossRef] [PubMed]

P. de Groot, L. L. Deck, “Numerical simulations of vibration in phase-shifting interferometry,” Appl. Opt. 35, 2172–2178 (1996).
[CrossRef] [PubMed]

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

K. Kinnnstaetter, A. W. Lohmann, J. Schwider, N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
[CrossRef]

P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992).
[CrossRef] [PubMed]

P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt. 32, 3669–3675 (1993).
[CrossRef] [PubMed]

IEE Proc. F (1)

T. Söderström, P. Stoica, “Accuracy of high-order Yule–Walker methods for frequency estimation of complex sine waves,” IEE Proc. F 140, 71–80 (1993).

IEEE Trans. Acoust. Speech Signal Process. (2)

R. Roy, T. Kailath, “ESPRIT—Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust. Speech Signal Process. 37, 984–995 (1989).
[CrossRef]

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Trans. Acoust. Speech Signal Process. 36, 1846–1853 (1988).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

R. Kumaresan, D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Trans. Aerosp. Electron. Syst. 19, 134–139 (1983).
[CrossRef]

J. Mod. Opt. (1)

C. Joenathan, B. M. Khorana, “Phase measurement by differentiating interferometric fringes,” J. Mod. Opt. 39, 2075–2087 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

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]

Metrologia (1)

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

Opt. Eng. (3)

R. Józwicki, M. Kujawinska, M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31, 422–433 (1992).
[CrossRef]

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

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
[CrossRef]

Opt. Lett. (1)

Other (5)

M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities (Dover, 1992), pp. 15–16.

K. M. Hoffman, R. Kunze, Linear Algebra, 2nd ed. (Prentice-Hall, 1971).

K. Creath, “Phase-shifting holographic interferometry,” in Holographic Interferometry, P. K. Rastogi, ed., Vol. 68 of Springer Series in Optical Sciences (Springer-Verlag, 1994), pp. 109–150.
[CrossRef]

T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Verlag, 1996), pp. 101–170.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501–598.

Cited By

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

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Fringe maps corresponding to Eq. (18) for κ = 2 and SNRs of (a) 10 and (b) 60 dB.

Fig. 2
Fig. 2

Estimation of the number of harmonics present in the signal. Diagonal entries are shown for matrix S obtained from singular-value decomposition of RI in Eq. (3). The curve for a noiseless signal shows five significant diagonal values of matrix S. Similarly, the curve for a noisy signal with a SNR of 10 dB shows five significant values in the diagonal of matrix S. The number of harmonics can therefore be reliably estimated from n = 2κ + 1, where n is the number of frequencies, to be κ = 2.

Fig. 3
Fig. 3

Phase step α (in degrees) versus SNR at an arbitrary pixel location for several values of N and m.

Fig. 4
Fig. 4

Typical absolute errors in computation of phase φ (in radians) at a SNR of 30 dB. (a) Phase obtained with our proposed method with the phase-step value obtained from Fig. 3(e). (b) Phase obtained with the algorithm proposed by Surrel.11

Fig. 5
Fig. 5

Typical wrapped phase φ (in radians) obtained for the phase-step value determined from Fig. 3(e) for a SNR of 30 dB.

Equations (31)

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

I ( t ) = I dc + k = 1 κ l k u k t + k = 1 κ l k * ( u k * ) t + η ( t ) , t = 0 , 1 , 2 , , m , , N - 1 ,
r ( p ) = E [ I ( t ) I * ( t - p ) ] = n = 0 2 κ A n 2 exp ( i ω n p ) + σ 2 δ p , 0 ,
R I = E { [ I * ( t - 1 ) I * ( t - 2 ) · · I * ( t - m ) ] [ I ( t - 1 ) I ( t - 2 ) I ( t - m ) ] } = [ r ( 0 ) r * ( 1 ) · · r * ( m - 1 ) r ( 1 ) · · · · r ( 2 ) · · · · · · · · r * ( 1 ) r ( m - 1 ) r ( m - 2 ) · · r ( 0 ) ] ,
R I = APA c R s + σ 2 I R ɛ ,
A m × n = [ a ( ω 0 )             a ( ω 1 ) a ( ω 2 κ ) ] ,
P = [ A 0 2 0 · · 0 0 A 1 2 · · · · · · · · · · · · 0 0 0 · · A 2 κ 2 ] .
λ 1 λ 2 λ n σ 2 , λ n + 1 = λ n + 2 = λ m = σ 2 .
Λ = [ λ 1 - σ 2 0 · · 0 0 λ 2 - σ 2 · · · · · · · · · · · · 0 0 0 · · λ n - σ 2 ] .
R I S = S [ λ 1 0 · · 0 0 λ 2 · · · · · · · · · · · · 0 0 0 · · λ n ] = APA c S + σ 2 S .
S = A ( P A c S Λ - 1 ) Γ
A 1 ( m - 1 ) × n = [ I ( m - 1 ) × ( m - 1 ) 0 ( m - 1 ) × 1 ] A , A 2 ( m - 1 ) × n = [ 0 ( m - 1 ) × 1             I ( m - 1 ) × ( m - 1 ) ] A ,
A 2 ( m - 1 ) × n = A 1 ( m - 1 ) × n × [ exp ( i ω 1 ) exp ( i ω 2 ) · · · exp ( i ω n ) ] D n × n
S 1 ( m - 1 ) × n = [ I ( m - 1 ) × ( m - 1 )             0 ( m - 1 ) × 1 ] S , S 2 ( m - 1 ) × n = [ 0 ( m - 1 ) × 1             I ( m - 1 ) × ( m - 1 ) ] S .
S 2 = A 2 Γ = A 1 D Γ = S 1 Γ - 1 D Γ = S 1 ϒ ,
ϒ = ( S 1 c S 1 ) - 1 S 1 c S 2 .
ϒ ^ = ( S ^ 1 c S ^ 1 ) - 1 S ^ 1 c S ^ 2 .
I ( x , y ; t ) = I dc + a - 1 exp [ - i ( φ + t α ) ] + a 1 exp [ i ( φ + t α ) ] + a - 2 exp [ - 2 i ( φ + t α ) ] + a 2 exp [ 2 i ( φ + t α ) ] , for t = 0 , 1 , , N - 1.
φ ( x , y ) = Ω ( x - x ) 2 + ( x - y ) 2 ,
R ^ I = 1 N t = m N I ( t ) I c ( t ) ,
[ 1 1 1 · · · · 1 x 1 x 2 x 3 · · · · x n x 1 2 x 2 2 x 3 2 · · · · x n 2 · · · · · · · · x n n - 1 x 2 n - 1 x 3 n - 1 · · · · x n n - 1 ] .
[ exp ( i κ α 0 ) exp ( - i κ α 0 ) exp [ i ( κ - 1 ) α 0 ] · · · · 1 exp ( i κ α 1 ) exp ( - i κ α 1 ) exp [ i ( κ - 1 ) α 1 ] · · · · 1 · · · · · · · · · · · · · · · · exp [ i κ α ( N - 1 ) ] exp [ - i κ α ( N - 1 ) ] exp [ i ( κ - 1 ) α ( N - 1 ) ] · · · · 1 ] [ l κ l k * l κ - 1 · I dc ] = [ I 0 I 1 I 2 · I N - 1 ] ,
I ( t ) = I dc + k = 1 κ a k exp ( i k φ ) exp ( i α k t ) + k = 1 κ a k exp ( - i k φ ) exp ( - i α κ t ) + η ( t ) , t = 0 , 1 , 2 , m , , N - 1.
r ( p ) = E [ I ( t ) I * ( t - p ) ] ,
I ( t ) = I dc + a 1 exp ( i φ ) exp ( i α t ) + a 1 exp ( - i φ ) exp ( - i α t ) + η ( t ) .
I * ( t - p ) = I dc + a 1 exp ( i φ ) exp [ i α ( t - p ) ] + a 1 × exp ( - i φ ) exp [ - i α ( t - p ) ] + η * ( t - p ) .
r ( p ) = E [ I ( t ) I * ( t - p ) ] = E { I dc 2 + I dc a 1 exp ( - i φ ) exp ( - i α t ) + I dc a 1 exp ( i φ ) exp ( i α t ) + exp ( i α p ) [ a 1 2 + I dc a 1 exp ( - i φ ) exp ( - i α t ) + a 1 2 exp ( - 2 i φ ) × exp ( - 2 i α t ) ] + exp ( - i α p ) [ a 1 2 + I dc a 1 exp ( i φ ) exp ( i α t ) + a 1 2 exp ( 2 i α t ) ] + η ( t ) η * ( t - p ) } .
r ( p ) = E [ I dc 2 + c 1 + exp ( i α p ) ( a 1 2 + c 2 ) + exp ( - i α p ) × ( a 1 2 + c 3 ) + η ( t ) η * ( t - p ) ] ,
r ( p ) = A 0 2 + A 1 2 exp ( i α p ) + A 2 2 exp ( - i α p ) + σ 2 δ p , 0 .
E [ η ( k ) η * ( j ) ] = σ 2 δ k , j .
0 2 π exp ( i ψ ) d ψ = 0.
r ( p ) = E [ I ( t ) I * ( t - p ) ] = n = 0 2 κ A n 2 exp ( i ω n p ) + σ 2 δ p , 0 .

Metrics