Abstract

The paper proposes a novel approach for estimating multiple phases in holographic moiré. The need to design such an algorithm is necessitated by the development of optical configurations containing two phase stepping devices, e.g. PZTs, with a view to measure simultaneously two phase distributions. The approach consists of first applying minimum-norm algorithm to extract phase steps imparted to the PZTs. Salient feature of the algorithm lies in its ability to handle nonsinusoidal waveforms and noise. This approach also provides the flexibility of using arbitrary phase steps, a feature most commonly attributed to generalized phase shifting interferometry. Once the phase steps are estimated for each PZT, the Vandermonde system of equations is designed to estimate the phase distributions.

© 2005 Optical Society of America

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  1. K. Creath, “Phase-shifting holographic interferometry,” Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, Berlin1994),  Vol.68, pp. 109–150.
  2. T. Kreis, Holographic interferometry Principles and Methods (Akademie Verlag, 1996) pp. 101–170.
  3. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  4. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses, App. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  5. J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley) 501–598 (1992).
  6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  7. Y. Zhu and T. Gemma, “Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry,” App. Opt. 40, 4540–4546 (2001).
    [CrossRef]
  8. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” App. Opt. 26, 2504–2506 (1987).
    [CrossRef]
  9. J. Schwider, O. Falkenstorfer, H. Schreiber, and 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, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
  13. 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]
  14. K. Hibino, B. F. Oreb, D. I. Farrant, and 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 and J. C. Wyant, “Phase-shifter calibration in phase-shifting interferometry,” App. 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 and 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, and M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing, Opt. Engg. 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 and L. L. deck, “Numerical simulations of vibration in phase-shifting interferometry,” App. Opt. 35, 2172–2178 (1996).
    [CrossRef]
  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 and 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, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. 27, 5082–5089 (1988).
    [CrossRef]
  25. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” App. Opt. 31, 1680–1681 (1992).
    [CrossRef]
  26. 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,” App. Opt. 32, 3669–3675 (1993).
    [CrossRef]
  27. P. K. Rastogi and E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” App. Opt. 31, 2402–2404 (1992).
    [CrossRef]
  28. P. K. Rastogi, M. Barillot, and G. Kaufmann“Comparative phase shifting holographic interferometry,” Appl. Opt.30, 722–728 (1991) C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett.7, 368–370 (1982).
    [CrossRef] [PubMed]
  29. P. K. Rastogi, M. Spajer, and J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
    [CrossRef]
  30. A. Patil, R. Langoju, and P Rastogi, “An integral approach to phase shifting interferometry using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681.
    [CrossRef] [PubMed]
  31. R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems AES-19, 134–139 (1983).
    [CrossRef]
  32. M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34, 331–341 (1986).
    [CrossRef]
  33. J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  34. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982).
    [CrossRef] [PubMed]
  35. R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoustics, Speech, and Signal Processing 37, 984–995 (1989).
    [CrossRef]
  36. R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” in Proceedings RADC, Spectral Estimation Workshop, Rome, NY, (243–258) 1979.
  37. G. Bienvenu, “Influence of the spatial coherence of the background noise on high resolution passive methods,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Washington, DC, 306–309 (1979).
  38. T. Söderström and P. Stoica, “Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F 140, 71–80 (1993).
  39. A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Boston, MA, 336–339 (1983).
  40. J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846–1853 (1988).
    [CrossRef]
  41. B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison”, IEEE Trans. Signal Processing 41, 788–803 (1993).
    [CrossRef]

2004 (1)

2001 (1)

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

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

1996 (2)

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

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

1995 (3)

1994 (1)

K. Creath, “Phase-shifting holographic interferometry,” Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, Berlin1994),  Vol.68, pp. 109–150.

1993 (5)

J. Schwider, O. Falkenstorfer, H. Schreiber, and 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]

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,” App. Opt. 32, 3669–3675 (1993).
[CrossRef]

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

B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison”, IEEE Trans. Signal Processing 41, 788–803 (1993).
[CrossRef]

1992 (5)

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

P. K. Rastogi and E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” App. Opt. 31, 2402–2404 (1992).
[CrossRef]

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

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

1991 (1)

1990 (1)

1989 (1)

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

1988 (2)

J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing 36, 1846–1853 (1988).
[CrossRef]

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

1987 (1)

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

1986 (1)

M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34, 331–341 (1986).
[CrossRef]

1985 (1)

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

1984 (1)

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

1983 (2)

R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems AES-19, 134–139 (1983).
[CrossRef]

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

1982 (1)

1981 (1)

P. K. Rastogi, M. Spajer, and J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

1974 (1)

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

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]

Barabell, A. J

M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34, 331–341 (1986).
[CrossRef]

Barabell, A. J.

A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Boston, MA, 336–339 (1983).

Barillot, M.

P. K. Rastogi, M. Barillot, and G. Kaufmann“Comparative phase shifting holographic interferometry,” Appl. Opt.30, 722–728 (1991) C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett.7, 368–370 (1982).
[CrossRef] [PubMed]

Bienvenu, G.

G. Bienvenu, “Influence of the spatial coherence of the background noise on high resolution passive methods,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, Washington, DC, 306–309 (1979).

Brangaccio, D. J.

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

Bruning, J. H.

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

J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley) 501–598 (1992).

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.

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

Creath, K.

K. Creath, “Phase-shifting holographic interferometry,” Holographic Interferometry, P. K. Rastogi, ed. (Springer Series in Optical Sciences, Berlin1994),  Vol.68, pp. 109–150.

de Groot, P.

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

P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12, 354–365 (1995).
[CrossRef]

deck, L. L.

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

Denarié, E.

P. K. Rastogi and E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” App. Opt. 31, 2402–2404 (1992).
[CrossRef]

Eiju, T.

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

Elssner, K. E.

Falkenstorfer, O.

J. Schwider, O. Falkenstorfer, H. Schreiber, and 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 Transactions on Acoustics, Speech, and Signal Processing 36, 1846–1853 (1988).
[CrossRef]

Gallagher, J. E.

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

Gemma, T.

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

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, Phase shifting interferometry Optical Shop Testing ed D. Malacara (New York: Wiley) 501–598 (1992).

Grievenkamp, J. E.

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

Grzanna, J.

Hari, K. V. S.

B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison”, IEEE Trans. Signal Processing 41, 788–803 (1993).
[CrossRef]

Hariharan, P.

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

Herriott, D. R.

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

Hibino, K.

Joenathan, C.

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

Józwicki, R.

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

Kailath, T.

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

Kaufmann, G.

P. K. Rastogi, M. Barillot, and G. Kaufmann“Comparative phase shifting holographic interferometry,” Appl. Opt.30, 722–728 (1991) C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett.7, 368–370 (1982).
[CrossRef] [PubMed]

Kaveh, M.

M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34, 331–341 (1986).
[CrossRef]

Khorana, B. M.

C. Joenathan and 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, and M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing, Opt. Engg. 31, 422–433 (1992).
[CrossRef]

Kumaresan, R.

R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems AES-19, 134–139 (1983).
[CrossRef]

Langoju, R.

Larkin, K. G.

Lohmann, A. W.

Merkel, K.

Monneret, J.

P. K. Rastogi, M. Spajer, and J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Morgan, C. J.

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

P. K. Rastogi, M. Barillot, and G. Kaufmann“Comparative phase shifting holographic interferometry,” Appl. Opt.30, 722–728 (1991) C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett.7, 368–370 (1982).
[CrossRef] [PubMed]

Oreb, B. F.

Patil, A.

Rao, B. D.

B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison”, IEEE Trans. Signal Processing 41, 788–803 (1993).
[CrossRef]

Rastogi, P

Rastogi, P. K.

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,” App. Opt. 32, 3669–3675 (1993).
[CrossRef]

P. K. Rastogi and E. Denarié, “Visualization of in-plane displacement fields using phase shifting holographic moiré: application to crack detection and propagation,” App. Opt. 31, 2402–2404 (1992).
[CrossRef]

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

P. K. Rastogi, M. Spajer, and J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

P. K. Rastogi, M. Barillot, and G. Kaufmann“Comparative phase shifting holographic interferometry,” Appl. Opt.30, 722–728 (1991) C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett.7, 368–370 (1982).
[CrossRef] [PubMed]

Rathjen, C.

Rosenfeld, D. P.

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

Roy, R.

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

Salbut, M.

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

Schmidt, R. O.

R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” in Proceedings RADC, Spectral Estimation Workshop, Rome, NY, (243–258) 1979.

Schreiber, H.

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

Schwider, J

Schwider, J.

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

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

Smorenburg, C.

Söderström, T.

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

Spajer, M.

P. K. Rastogi, M. Spajer, and J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. 2, 79–103 (1981).
[CrossRef]

Spolaczyk, R.

Stoica, P.

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

Streibl, N.

Surrel, Y.

Tufts, D. W.

R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems AES-19, 134–139 (1983).
[CrossRef]

van Wingerden, J.

White, A. D.

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

Wizinowich, P. L.

Wyant, J. C.

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

Zhao, B.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the optical setup in holographic moiré.

Fig. 2.
Fig. 2.

Fringe map corresponding to a) κ=1 and pure signal, a) κ=1 and 10 SNR, a) κ=2 and pure signal, a) κ=2 and 10 SNR.

Fig. 3.
Fig. 3.

Plot for phase step values α and β (in degrees) obtained at an arbitrary pixel location on a data frame for different values of N and m using the forward approach. During the simulation the phase steps are assumed to be α=45° and β=70°.

Fig. 4.
Fig. 4.

Plot for phase step values α and β (in degrees) obtained at an arbitrary pixel location on a data frame for different values of N and m using the forward-backward approach. During the simulation the phase steps are assumed to be α=45° and β=70°.

Fig. 5.
Fig. 5.

Plots show typical error in computation of phase distribution a) φ 1 (in radians), and b) φ 2 (in radians), for phase step obtained from Fig. 4(c) for 30dB noise.

Fig. 6.
Fig. 6.

Plot shows wrapped phase for a) φ 1 and b) φ 2 for phase step obtained from Fig. 4(c) for 30dB noise.

Equations (39)

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I ( t ) = I dc + k = 1 κ a k exp [ ik ( φ 1 + t α ) ] + k = 1 κ a k exp [ ik ( φ 1 + t α ) ] +
k = 1 κ b k exp [ ik ( φ 2 + t β ) ] + k = 1 κ b k exp [ ik ( φ 2 + t β ) ] ;
for t = 0 , 1 , 2 , ... , m , ... , N 1
I ( t ) = I dc + k = 1 κ k u k t + k = 1 κ k * ( u k * ) t + k = 1 κ k v k t + k = 1 κ k * ( v k * ) t + η ( t ) ;
for t = 0 , 1 , ... , m , ... , N 1
r ( p ) = E [ I ( t ) I * ( t p ) ]
I ( t ) = I dc + a 1 e i φ 1 e i α t + a 1 e i φ 1 e e α t + b 1 e i φ 2 e i β t + b 1 e i φ 2 e i β t + η ( t )
I * ( t p ) = I dc + a 1 e i φ 1 e i α ( t p ) + a 1 e φ 1 e i α ( t p ) +
b 1 e i φ 2 e i β ( t p ) + b 1 e i φ 2 e i β ( t p ) + η * ( t p )
r ( p ) = E { I dc 2 + c 1 + e i α p ( a 1 2 + c 2 ) + e i α p ( a 1 2 + c 3 ) + e i β p ( b 1 2 + c 4 ) + e i β p ( b 1 2 + c 5 ) + η ( t ) η * ( t p ) }
r ( p ) = A 0 2 + A 1 2 e i α p + A 2 2 e i α p + A 3 2 e i β p + A 4 2 e i β p + σ 2 δ p , 0
E [ η ( k ) η * ( j ) ] = σ 2 δ k , j
r ( p ) = E [ I ( t ) I * ( t p ) ] = n = 0 4 κ A n 2 e i ω n p + σ 2 δ p , 0
R I = E [ I * ( t ) I ( t ) ] = [ r ( 0 ) r * ( 1 ) r * ( 2 ) . r * ( m 1 ) r ( 1 ) r ( 0 ) . . . r ( 2 ) . . . . . . . . r * ( 1 ) r ( m 1 ) . . . r ( 0 ) ]
R I = A P A c R s + σ 2 I R ε
P = [ A 0 2 0 . 0 0 A 1 2 . . . . . . 0 . . A m 2 ]
R I G = G [ A 4 κ + 1 2 0 . . 0 0 A 4 κ + 2 2 . . . . . . . . . . . . 0 0 0 . . A m 2 ] = σ 2 G = AP A c G + σ 2 G
A c G = 0
R ( G ) = N ( A c )
S c G = 0
R ( S ) = R ( A )
a T ( z 1 ) G ̂ G ̂ c a ( z ) = 0
a T ( z 1 ) [ 1 g ̂ ] = 0
S ̂ = [ χ c S ¯ ] } m 1 } 1
S ̂ c [ 1 g ̂ ] = 0
S ¯ c g ̂ = χ
g ̂ = S ¯ ( S ¯ c S ¯ ) 1 χ
I = S ̂ c S ̂ = χ χ c + S ¯ c S ¯
χ 2 1
rank ( S ¯ ) = n
I ( x , y ; t ) = I dc + a 1 exp [ i ( φ 1 + t α ) ] + a 1 exp [ i ( φ 1 + t α ) ] +
a 2 exp [ 2 i ( φ 1 + t α ) ] + a 2 exp [ 2 i ( φ 1 + t α ) ] +
b 1 exp [ i ( φ 2 + t β ) ] + b 1 exp [ i ( φ 2 + t β ) ] +
b 2 exp [ 2 i ( φ 2 + t β ) ] + b 2 exp [ 2 i ( φ 2 + t β ) ] + η ( t )
φ 1 ( x , y ) = 2 π λ ( p x ) 2 + ( p y ) 2 + φ R 1
φ 2 ( x , y ) = 2 π λ ( p x ) 2 + ( p y ) 2 + φ R 2 .
R ̂ I = 1 N t = m N [ I * ( t 1 ) I * ( t 2 ) . . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ]
R ̂ I = 1 2 N t = m N { [ I * ( t 1 ) I * ( t 2 ) . I * ( t m ) ] [ I ( t 1 ) I ( t 2 ) . . I ( t m ) ] + [ I * ( t m ) . I * ( t 2 ) I * ( t 1 ) ] [ I ( t m ) . . I ( t 2 ) I ( t 1 ) ] }
[ e i κ α 0 e i κ α 0 e i κ β 0 . e i ( κ 1 ) α 0 . . 1 e i κ α 1 e i κ α 1 e i κ β 1 . e i ( κ 1 ) α 1 . . 1 . . . . . . . . . . . . . . . . e i κ α ( N 1 ) e i κ α ( N 1 ) e i κ β ( N 1 ) . e i ( κ 1 ) α ( N 1 ) . . 1 ] [ κ κ * κ . I dc ] = [ I 0 I 1 I 2 . I N 1 ]

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