Abstract

A compact and efficient algorithm for digital envelope detection in white light interferograms is derived from a well-known phase-shifting algorithm. The performance of the new algorithm is compared with that of other schemes currently used. Principal criteria considered are computational efficiency and accuracy in the presence of miscalibration. The new algorithm is shown to be near optimal in terms of computational efficiency and can be represented as a second-order nonlinear filter. In combination with a carefully designed peak detection method the algorithm exhibits exceptionally good performance on simulated interferograms.

© 1996 Optical Society of America

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  1. P. A. Flourney, R. W. McClure, G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. 11, 1907–1915 (1972).
    [CrossRef]
  2. T. Dresel, G. Häusler, H. Venzke, “Three dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
    [CrossRef] [PubMed]
  3. M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
    [CrossRef]
  4. B. S. Lee, T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
    [CrossRef] [PubMed]
  5. S. S. C. Chim, G. S. Kino, “Correlation microscope,” Opt. Lett. 15, 579–581 (1990).
    [CrossRef] [PubMed]
  6. S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).
    [CrossRef] [PubMed]
  7. G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef] [PubMed]
  8. S. S. C. Chim, G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. 31, 2550–2553 (1992).
    [CrossRef] [PubMed]
  9. M. Davidson, “Method and apparatus for using a two beam interference microscope for inspection of integrated circuits and the like,” U.S. patent4,818,110 (April4, 1989).
  10. B. L. Danielson, C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30, 2975–2979 (1991).
    [CrossRef] [PubMed]
  11. K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).
    [CrossRef] [PubMed]
  12. G. Schulz, K.-E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).
    [CrossRef] [PubMed]
  13. C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
    [CrossRef] [PubMed]
  14. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic errors sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  15. P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometer: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  16. Neat: cleverly effective in character or execution. The Macquarie Dictionary, 2nd ed. (The Macquarie Library, Macquarie University, Sydney, Australia1991).
  17. P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
    [CrossRef]
  18. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  19. S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
    [CrossRef]
  20. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  21. D. A. Zweig, R. E. Hufnagel, “A Hilbert transform algorithm for fringe-pattern analysis,” in Advanced Optical Manufacturing and Testing, L. R. Baker, P. B. Reid, G. M. Sanger, eds., Proc. SPIE1333, 295–302 (1990).
    [CrossRef]
  22. K. Freischlad, C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7, 542–551 (1990).
    [CrossRef]
  23. K. G. Larkin, B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).
    [CrossRef]
  24. R. E. Bogner, A. G. Constantinides, Introduction to Digital Filtering (Wiley, New York, 1975).
  25. O. Brigham, The Fast Fourier Transform, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  26. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  27. P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
    [CrossRef] [PubMed]
  28. P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).
    [CrossRef] [PubMed]
  29. Zygo product brochure, “New view 100:3D imaging surface structure analyzer,” Zygo Corporation, Middlefield, Conn. (1993).
  30. F. G. Stremler, Introduction to Communication Systems (Addison-Wesley, Reading, Mass., 1982).
  31. P. Carré, “Installation et utilisation du comparateur photoélectrique et interferentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  32. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  33. M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993).
  34. Analytically, Eq. (20) is an approximate solution of the linear demodulation problem g(z) = a+ c(z)cos(2πu0z+ α) using five equally spaced measurements to solve for cand having only second-order errors [related to the derivatives (∂c/∂z)2and ∂2c/∂z2]. The other algorithms have various first-order errors.
  35. B. Bushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
    [CrossRef]
  36. There are a number of review papers and book chapters that consider the ever increasing range of phase-shifting algorithms. One of the more recent is K. Creath, “Temporal phase-measurement methods,” in Interferogram Analysis: Digital Processing Techniques for Fringe Pattern Measurement, D. W. Robinson, G. T. Reid (Institute of Physics, Bristol, UK, 1993).
  37. The nonlinear filter defined by relation (20a) can be represented by a bandpass filter (equivalent to a finite-difference filter) followed by a quadratic Volterra series filter. The bandpass filter has the same form as that of F1shown in Figs. 2 and 3 when the sampling is the nominal four per period. Interestingly the Hilbert envelope detector can be expressed as a quadratic filter with many terms, but strangely truncation to the first few terms does not give relation (20a).
  38. P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
    [CrossRef]
  39. D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).
  40. Closely related to Savitsky–Golay, or digital smoothing polynomial, filters. A LSF over a symmetrical domain causes many off-diagonal elements in the matrix equation to be zero, and hence matrix inversion is trivial. The resulting kernels have even or odd symmetry. See the following reference for more information.
  41. W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  42. Typical peak detection algorithms involve a combination of samples in both numerator and denominator of a quotient resembling Eq. (23). Linear combinations are equivalent to correlation or convolution with a kernel function. In the Fourier domain the transformed signal is multiplied by the kernel transform. If the kernel is suitably chosen yet still satisfies the LSF criteria, then zeros (of the transform) can occur at certain frequencies and these frequencies are thus removed from the signal. Essentially filtering and peak detection have been combined into one. Equation (23) resembles the filtered signal quotient analyzed in detail in Refs. 22 and 23.
  43. H.-H. Liu, P.-H. Cheng, J. Wang, “Spatially coherent white-light interferometer based on a point fluorescent source,” Opt. Lett. 18, 678–680 (1993).
    [CrossRef] [PubMed]
  44. N. Bareket, “Undersampling errors in measuring the moments of images aberrated by turbulence,” Appl. Opt. 18, 3064–3069 (1979).
    [CrossRef] [PubMed]
  45. R. P. Loce, R. E. Jodoin, “Sampling theorem for geometric moment determination and its application to a laser beam position detector,” Appl. Opt. 29, 3835–3843 (1990).
    [CrossRef] [PubMed]
  46. B. F. Alexander, K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320–1331 (1991).
    [CrossRef]
  47. I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer, Boston, 1990).
  48. A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971), p. 200.
  49. B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
    [CrossRef]
  50. J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, N.M., 1990 (IEEE, New York, 1990), pp. 381–384.
    [CrossRef]
  51. P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
    [CrossRef]
  52. P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
    [CrossRef]
  53. A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).
    [CrossRef]
  54. K. G. Larkin, “Efficient demodulator for bandpass sampled AM signals,” Electron. Lett. 32(2) (1996).
    [CrossRef]

1996 (1)

K. G. Larkin, “Efficient demodulator for bandpass sampled AM signals,” Electron. Lett. 32(2) (1996).
[CrossRef]

1995 (1)

1994 (2)

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).
[CrossRef]

1993 (6)

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
[CrossRef]

H.-H. Liu, P.-H. Cheng, J. Wang, “Spatially coherent white-light interferometer based on a point fluorescent source,” Opt. Lett. 18, 678–680 (1993).
[CrossRef] [PubMed]

P. de Groot, L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferograms,” Opt. Lett. 18, 1462–1464 (1993).
[CrossRef] [PubMed]

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
[CrossRef] [PubMed]

1992 (3)

1991 (4)

1990 (5)

1989 (1)

1987 (1)

1985 (1)

1984 (1)

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

1983 (1)

1980 (1)

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
[CrossRef]

1979 (2)

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

N. Bareket, “Undersampling errors in measuring the moments of images aberrated by turbulence,” Appl. Opt. 18, 3064–3069 (1979).
[CrossRef] [PubMed]

1972 (1)

1966 (1)

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

Alexander, B. F.

B. F. Alexander, K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320–1331 (1991).
[CrossRef]

Bareket, N.

Bogner, R. E.

R. E. Bogner, A. G. Constantinides, Introduction to Digital Filtering (Wiley, New York, 1975).

Boisrobert, C. Y.

Bovik, A. C.

A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Brigham, O.

O. Brigham, The Fast Fourier Transform, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1988).

Brophy, C. P.

D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).

Burow, R.

Bushan, B.

Caber, P. J.

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
[CrossRef] [PubMed]

D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).

Carré, P.

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

Cheng, P.-H.

Chim, S. S. C.

Cohen, D. K.

D. K. Cohen, P. J. Caber, C. P. Brophy, “Rough surface profiler and method,” U.S. patent5,133,601 (July28, 1992).

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

Constantinides, A. G.

R. E. Bogner, A. G. Constantinides, Introduction to Digital Filtering (Wiley, New York, 1975).

Creath, K.

K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).
[CrossRef] [PubMed]

There are a number of review papers and book chapters that consider the ever increasing range of phase-shifting algorithms. One of the more recent is K. Creath, “Temporal phase-measurement methods,” in Interferogram Analysis: Digital Processing Techniques for Fringe Pattern Measurement, D. W. Robinson, G. T. Reid (Institute of Physics, Bristol, UK, 1993).

Danielson, B. L.

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

M. Davidson, “Method and apparatus for using a two beam interference microscope for inspection of integrated circuits and the like,” U.S. patent4,818,110 (April4, 1989).

de Groot, P.

Deck, L.

Dresel, T.

Eiju, T.

Elssner, K.-E.

Flannery, B. P.

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Flourney, P. A.

Freischlad, K.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Grzanna, J.

Hariharan, P.

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

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

Häusler, G.

Hufnagel, R. E.

D. A. Zweig, R. E. Hufnagel, “A Hilbert transform algorithm for fringe-pattern analysis,” in Advanced Optical Manufacturing and Testing, L. R. Baker, P. B. Reid, G. M. Sanger, eds., Proc. SPIE1333, 295–302 (1990).
[CrossRef]

Jodoin, R. E.

Kaiser, J. F.

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Albuquerque, N.M., 1990 (IEEE, New York, 1990), pp. 381–384.
[CrossRef]

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

Kino, G. S.

Koliopoulos, C. L.

Kujawinska, M.

M. Kujawinska, “Spatial phase measurement methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993).

Larkin, K. G.

K. G. Larkin, “Efficient demodulator for bandpass sampled AM signals,” Electron. Lett. 32(2) (1996).
[CrossRef]

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
[CrossRef] [PubMed]

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

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

Lee, B. S.

Liu, H.-H.

Loce, R. P.

Maragos, P.

A. C. Bovik, P. Maragos, “Conditions for positivity of an energy operator,” IEEE Trans. Signal Process. 42, 469–471 (1994).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” in Integrated Circuit Metrology, Inspection, and Process Control, K. M. Monahan, ed., Proc. SPIE775, 233–247 (1987).
[CrossRef]

McClure, R. W.

Merkel, K.

Ng, K. C.

B. F. Alexander, K. C. Ng, “Elimination of systematic error in subpixel accuracy centroid estimation,” Opt. Eng. 30, 1320–1331 (1991).
[CrossRef]

Oreb, B. F.

Pitas, I.

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer, Boston, 1990).

Pohlig, S. C.

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Quatieri, T. F.

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulations with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Roy, M.

P. Hariharan, K. G. Larkin, M. Roy, “The geometric phase: interferometric observations with white light,” J. Mod. Opt. 41, 663–667 (1994).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, “Optical bilinear transformations: general properties,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

Sandoz, P.

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
[CrossRef]

Schulz, G.

Schwider, J.

Sheppard, C. J. R.

Spolaczyk, R.

Strand, T. C.

Stremler, F. G.

F. G. Stremler, Introduction to Communication Systems (Addison-Wesley, Reading, Mass., 1982).

Teulolsky, S. A.

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Tribillon, G.

P. Sandoz, G. Tribillon, “Profilometry by zero-order interference fringe identification,” J. Mod. Opt. 40, 1691–1700 (1993).
[CrossRef]

Venetsanopoulos, A. N.

I. Pitas, A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications (Kluwer, Boston, 1990).

Venzke, H.

Vetterling, W. T.

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Wang, J.

Whalen, A. D.

A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971), p. 200.

Womack, K. H.

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

Wyant, J. C.

Wyntjes, G.

Zweig, D. A.

D. A. Zweig, R. E. Hufnagel, “A Hilbert transform algorithm for fringe-pattern analysis,” in Advanced Optical Manufacturing and Testing, L. R. Baker, P. B. Reid, G. M. Sanger, eds., Proc. SPIE1333, 295–302 (1990).
[CrossRef]

Appl. Opt. (16)

P. A. Flourney, R. W. McClure, G. Wyntjes, “White-light interferometric thickness gauge,” Appl. Opt. 11, 1907–1915 (1972).
[CrossRef]

N. Bareket, “Undersampling errors in measuring the moments of images aberrated by turbulence,” Appl. Opt. 18, 3064–3069 (1979).
[CrossRef] [PubMed]

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

B. Bushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of surface topography of magnetic tapes by Mirau interferometry,” Appl. Opt. 24, 1489–1497 (1985).
[CrossRef]

K. Creath, “Calibration of numerical aperture effects in interferometric microscope objectives,” Appl. Opt. 28, 3333–3338 (1989).
[CrossRef] [PubMed]

G. S. Kino, S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
[CrossRef] [PubMed]

B. S. Lee, T. C. Strand, “Profilometry with a coherence scanning microscope,” Appl. Opt. 29, 3784–3788 (1990).
[CrossRef] [PubMed]

R. P. Loce, R. E. Jodoin, “Sampling theorem for geometric moment determination and its application to a laser beam position detector,” Appl. Opt. 29, 3835–3843 (1990).
[CrossRef] [PubMed]

B. L. Danielson, C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30, 2975–2979 (1991).
[CrossRef] [PubMed]

S. S. C. Chim, G. S. Kino, “Phase measurements using the Mirau correlation microscope,” Appl. Opt. 30, 2197–2201 (1991).
[CrossRef] [PubMed]

S. S. C. Chim, G. S. Kino, “Three-dimensional image realization in interference microscopy,” Appl. Opt. 31, 2550–2553 (1992).
[CrossRef] [PubMed]

P. J. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32, 3438–3441 (1993).
[CrossRef] [PubMed]

C. J. R. Sheppard, K. G. Larkin, “Effect of numerical aperture on interference fringe spacing,” Appl. Opt. 34, 4731–4734 (1995).
[CrossRef] [PubMed]

G. Schulz, K.-E. Elssner, “Errors in phase-measurement interferometry with high numerical apertures,” Appl. Opt. 30, 4500–4506 (1991).
[CrossRef] [PubMed]

T. Dresel, G. Häusler, H. Venzke, “Three dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31, 919–925 (1992).
[CrossRef] [PubMed]

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Analytically, Eq. (20) is an approximate solution of the linear demodulation problem g(z) = a+ c(z)cos(2πu0z+ α) using five equally spaced measurements to solve for cand having only second-order errors [related to the derivatives (∂c/∂z)2and ∂2c/∂z2]. The other algorithms have various first-order errors.

There are a number of review papers and book chapters that consider the ever increasing range of phase-shifting algorithms. One of the more recent is K. Creath, “Temporal phase-measurement methods,” in Interferogram Analysis: Digital Processing Techniques for Fringe Pattern Measurement, D. W. Robinson, G. T. Reid (Institute of Physics, Bristol, UK, 1993).

The nonlinear filter defined by relation (20a) can be represented by a bandpass filter (equivalent to a finite-difference filter) followed by a quadratic Volterra series filter. The bandpass filter has the same form as that of F1shown in Figs. 2 and 3 when the sampling is the nominal four per period. Interestingly the Hilbert envelope detector can be expressed as a quadratic filter with many terms, but strangely truncation to the first few terms does not give relation (20a).

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Closely related to Savitsky–Golay, or digital smoothing polynomial, filters. A LSF over a symmetrical domain causes many off-diagonal elements in the matrix equation to be zero, and hence matrix inversion is trivial. The resulting kernels have even or odd symmetry. See the following reference for more information.

W. H. Press, S. A. Teulolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Typical peak detection algorithms involve a combination of samples in both numerator and denominator of a quotient resembling Eq. (23). Linear combinations are equivalent to correlation or convolution with a kernel function. In the Fourier domain the transformed signal is multiplied by the kernel transform. If the kernel is suitably chosen yet still satisfies the LSF criteria, then zeros (of the transform) can occur at certain frequencies and these frequencies are thus removed from the signal. Essentially filtering and peak detection have been combined into one. Equation (23) resembles the filtered signal quotient analyzed in detail in Refs. 22 and 23.

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Figures (7)

Fig. 1
Fig. 1

Typical white light correlogram, g(z). Note the fringe phase offset α = π/4.

Fig. 2
Fig. 2

Modulus of the Fourier-transformed correlogram, |G(w)|. The normally large dc term has been reduced in magnitude to fit within the chosen scale. Also shown for comparison is the FT F1(w) of the finite-difference filter f1(z) with separation Δ = 1/4w0.

Fig. 3
Fig. 3

FT’s F1(w) and F2(w) of the five-step phase-shifting filters f1(z) and f2(z). Note particularly the stationary points at w = ±1/4Δ.

Fig. 4
Fig. 4

Envelope detection for all three algorithms with use of a 90° step size. In this particular case the curves produced by algorithms (2) and (3) are the same. Algorithm (1) has noticeable fringe structure.

Fig. 5
Fig. 5

Envelope detection for all three algorithms with use of a 45° step size. Both algorithms (1) and (2) have significant fringe structure visible. Algorithm (3) performs exceptionally well and has a 50% reduction as predicted.

Fig. 6
Fig. 6

Envelope detection for all three algorithms with use of a 135° step size. Again both algorithms (1) and (2) have significant fringe structure visible, whereas algorithm (3) shows only a trace of the second-harmonic fringe structure.

Fig. 7
Fig. 7

Error in the predicted peak position with the weighted LSF defined by Eq. (23). Note that, for an initial estimate of peak position within two samples of the actual value (delimited by the vertical dotted lines), the peak prediction has only 1/20 sample error. The small bias in the predicted peak is related to the nonzero phase change on reflection. The range of z is just half that shown in Figs. 1 and 46. Clearly the error is related to the second harmonic of the original fringe. A second application of the peak detection process ensures that the estimate is within one-half sample, and errors decrease proportionately.

Tables (2)

Tables Icon

Table 1 rms Peak Location Error for Various Algorithms with Four Samples per Period

Tables Icon

Table 2 rms Peak Location Error for Various Algorithms with 4/3 Samples per Period (3 Times Undersampling)

Equations (31)

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g ( x , y , z ) = a ( x , y ) + b ( x , y ) c [ z - 2 h ( x , y ) ] × cos [ 2 π w 0 z - α ( x , y ) ] .
G ( x , y , w ) = - g ( x , y , z ) exp ( - 2 π i w z ) d z .
G ( x , y , w ) = a ( x , y ) δ ( w ) + b ( x , y ) 2 C ( w ) exp [ - 4 π i w h ( x , y ) ] * [ exp ( i α ) δ ( w - w 0 ) + exp ( - i α ) δ ( w + w 0 ) ] .
G ( w ) = a δ ( w ) + b 2 exp [ + i ( α + 4 π w 0 h ) ] × C ( w - w 0 ) exp ( - 4 π i w h ) + b 2 exp [ - i ( α + 4 π w 0 h ) ] × C ( w + w 0 ) exp ( - 4 π i w h ) .
G ( w ) = G ( w ) exp [ i ϕ ( w ) ] ,
G ( w ) a δ ( w ) + b 2 C ( w - w 0 ) + b 2 C ( w + w 0 ) ,
ϕ ( w ) { - α - 4 π ( w - w 0 ) h w > 0 + α - 4 π ( w + w 0 ) h w < 0 .
s ( z ) = b c ( z - h ) exp [ i ( α - 4 π w o h ) ] .
f 1 ( t ) = 2 [ δ ( t - Δ ) - δ ( t + Δ ) ] ,
f 2 ( t ) = - δ ( t - 2 Δ ) + 2 δ ( t ) - δ ( t + 2 Δ ) .
f 1 ( z ) = f ( z ) F ( w ) = F ( u ) exp [ i χ ( w ) ] .
f 2 ( z ) = f ( - z ) F * ( w ) = F ( u ) exp [ - i χ ( w ) ] .
f 0 ( z ) = f ( z ) + f ( - z ) 2 F cos χ .
f 90 ( z ) = f ( z ) - f ( - z ) 2 i F sin χ .
g 1 ( z ) = f 1 ( z ) g ( z ) ,             g 2 ( z ) = f 2 ( z ) g ( z ) .
ϕ ( x , y ) = tan - 1 { 2 [ I 2 ( x , y ) - I 4 ( x , y ) ] - I 1 ( x , y ) + 2 I 3 ( x , y ) - I 5 ( x , y ) } ,
M ( x , y ) = 1 / 4 [ 4 ( I 2 - I 4 ) 2 + ( - I 1 + 2 I 3 - I 5 ) 2 ] 1 / 2 .
ϕ = tan - 1 { [ 4 ( I 2 - I 4 ) 2 - ( I 1 - I 5 ) 2 ] 1 / 2 - I 1 + 2 I 3 - I 5 } ,
M = ( I 2 - I 4 ) 2 { [ 4 ( I 2 - I 4 ) 2 - ( I 1 - I 5 ) 2 ] + ( - I 1 + 2 I 3 - I 5 ) 2 } 1 / 2 4 ( I 2 - I 4 ) 2 - ( I 1 - I 5 ) 2 .
4 M 2 sin 4 ψ = ( I 2 - I 4 ) 2 - ( I 1 - I 3 ) ( I 3 - I 5 ) ,
M 2 ( I 2 - I 4 ) 2 - ( I 1 - I 3 ) ( I 3 - I 5 ) .
M ( x , y ) = 1 / 2 [ ( I 3 - I 2 ) 2 + ( I 2 - I 1 ) 2 ] 1 / 2
z p ( x , y ) = h ( x , y ) .
z p = 0.4 Δ ( L 1 + 3 L 2 + 0 L 3 - 3 L 4 - L 5 L 1 + 0 L 2 - 2 L 3 + 0 L 4 + L 5 ) .
ϕ 1 = mod 2 π × ( tan - 1 { [ 4 ( I 2 - I 4 ) 2 - ( I 1 - I 5 ) 2 ] 1 / 2 - I 1 + 2 I 3 - I 5 } - 4 π w 0 z ) .
α p = ϕ 1 + 4 π w 0 z .
f a ( z ) = g ( z + Δ ) - g ( z - Δ )             finite difference ,
f b ( z ) = f a 2 ( z ) - f a ( z - Δ ) f a ( z + Δ )             nonlinear difference .
F a ( w ) = 2 i sin ( 2 π w Δ ) G ( w )             bandpass filter ,
F b ( w ) = F a ( w ) * F a ( w ) - [ exp ( - 2 π i w Δ ) F a ( w ) ] * [ exp ( 2 π i w Δ ) F a ( w ) ] .
g ( x , z ) = INT [ 128 + 100 exp ( - z s 2 / σ 2 ) cos ( 4 π z s / λ m ) + n ( x , z ) ] .

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