Abstract

In this work, a new method for surface extraction in white light scanning interferometry (WLSI) is introduced. The proposed extraction scheme is based on the Teager-Kaiser energy operator and its extended versions. This non-linear class of operators is helpful to extract the local instantaneous envelope and frequency of any narrow band AM-FM signal. Namely, the combination of the envelope and frequency information, allows effective surface extraction by an iterative re-estimation of the phase in association with a new correlation technique, based on a recent TK cross-energy operator. Through the experiments, it is shown that the proposed method produces substantially effective results in term of surface extraction compared to the peak fringe scanning technique, the five step phase shifting algorithm and the continuous wavelet transform based method. In addition, the results obtained show the robustness of the proposed method to noise and to the fluctuations of the carrier frequency.

© 2014 Optical Society of America

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  1. C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. 14, 1807 (2003).
    [CrossRef]
  2. K. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” JOSA A 13, 832–843 (1996).
    [CrossRef]
  3. P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt. 41, 4571–4578 (2002).
    [CrossRef]
  4. S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
    [CrossRef]
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    [CrossRef] [PubMed]
  6. M. Li, C. Quan, and C. Tay, “Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry,” Opt. Laser Technol. 40, 920–929 (2008).
    [CrossRef]
  7. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” App. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  8. H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
    [CrossRef]
  9. J. Kaiser, “On a simple algorithm to calculate the energy of a signal,” in Proc. ICASSP, (1990), pp. 381–384.
  10. P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Sig. Proc. Lett. 2, 152–154 (1995).
    [CrossRef]
  11. P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. 41, 3024–3051 (1993).
    [CrossRef]
  12. P. Maragos and A. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A 12, 1867–1876 (1995).
    [CrossRef]
  13. A. Boudraa, F. Salzenstein, and J. Cexus, “Two-dimensional continuous higher-order energy operators,” Opt. Eng. 44, 7001–7010 (2005).
  14. K. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express 13, 8097–8121 (2005).
    [CrossRef] [PubMed]
  15. F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc. 24, 543–572 (2013).
    [CrossRef]
  16. J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy operator,” Elect. Lett. 40, 778–780 (2004).
    [CrossRef]
  17. A. Boudraa, “Relationships between ΨB-energy operator and some time-frequency representations,” IEEE Sig. Proc. Lett. 17, 527–530 (2010).
    [CrossRef]
  18. A. Boudraa, T. Chonavel, and J. Cexus, “ΨB-energy operator and cross-power spectral density,” Sig. Proc. 94, 236–240 (2014).
    [CrossRef]
  19. A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-ΨB-energy operator-based signal detection,” JASA 123, 4283–4289 (2008).
    [CrossRef]
  20. F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. 17, 2804–2815 (2005).
    [CrossRef]
  21. S. Petitgrand, “Méthodes de microscopie interférométrique 3D statiques et dynamiques pour la caractérisation de la technologie et du comportement des microsystèmes,” Ph.D. thesis (2005).
  22. A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-ΨB energy operator,” Int. J. Electron. Comm. 63, 728–735 (2009).
    [CrossRef]
  23. A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. 135892, 1–8 (2008).
  24. W. Zhang, C. Liu, and H. Yan, “Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,” Patt. Recong. 43, 3969–3976 (2010).
    [CrossRef]
  25. P. Hariharan, B. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef] [PubMed]
  26. P. Montgomery and J. Fillard, “Peak fringe scanning microscopy (pfsm): submicron 3d measurement of semiconductor components,” Interferometry: Techniques and Analysis pp. 12–23 (1755).
  27. R. Schafer, “What is a savitzky-golay filter?[lecture notes],” IEEE Sig. Proc. Mag. 28, 111–117 (2011).
    [CrossRef]
  28. J. Estrada, M. Servin, and J. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express 18, 2632–2638 (2010).
    [CrossRef] [PubMed]
  29. E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
    [CrossRef]

2014

A. Boudraa, T. Chonavel, and J. Cexus, “ΨB-energy operator and cross-power spectral density,” Sig. Proc. 94, 236–240 (2014).
[CrossRef]

2013

F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc. 24, 543–572 (2013).
[CrossRef]

2011

S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
[CrossRef]

R. Schafer, “What is a savitzky-golay filter?[lecture notes],” IEEE Sig. Proc. Mag. 28, 111–117 (2011).
[CrossRef]

2010

J. Estrada, M. Servin, and J. Quiroga, “A self-tuning phase-shifting algorithm for interferometry,” Opt. Express 18, 2632–2638 (2010).
[CrossRef] [PubMed]

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

A. Boudraa, “Relationships between ΨB-energy operator and some time-frequency representations,” IEEE Sig. Proc. Lett. 17, 527–530 (2010).
[CrossRef]

W. Zhang, C. Liu, and H. Yan, “Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,” Patt. Recong. 43, 3969–3976 (2010).
[CrossRef]

2009

A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-ΨB energy operator,” Int. J. Electron. Comm. 63, 728–735 (2009).
[CrossRef]

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

2008

M. Li, C. Quan, and C. Tay, “Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry,” Opt. Laser Technol. 40, 920–929 (2008).
[CrossRef]

A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-ΨB-energy operator-based signal detection,” JASA 123, 4283–4289 (2008).
[CrossRef]

A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. 135892, 1–8 (2008).

2005

F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. 17, 2804–2815 (2005).
[CrossRef]

A. Boudraa, F. Salzenstein, and J. Cexus, “Two-dimensional continuous higher-order energy operators,” Opt. Eng. 44, 7001–7010 (2005).

K. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express 13, 8097–8121 (2005).
[CrossRef] [PubMed]

2004

J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy operator,” Elect. Lett. 40, 778–780 (2004).
[CrossRef]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” App. Opt. 43, 2695–2702 (2004).
[CrossRef]

2003

C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. 14, 1807 (2003).
[CrossRef]

2002

P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt. 41, 4571–4578 (2002).
[CrossRef]

1997

1996

K. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” JOSA A 13, 832–843 (1996).
[CrossRef]

1995

P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Sig. Proc. Lett. 2, 152–154 (1995).
[CrossRef]

P. Maragos and A. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A 12, 1867–1876 (1995).
[CrossRef]

1993

P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. 41, 3024–3051 (1993).
[CrossRef]

1987

Abed-Meraim, K.

A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-ΨB-energy operator-based signal detection,” JASA 123, 4283–4289 (2008).
[CrossRef]

Barillon, R.

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

Benramdane, S.

A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-ΨB energy operator,” Int. J. Electron. Comm. 63, 728–735 (2009).
[CrossRef]

Boudraa, A.

A. Boudraa, T. Chonavel, and J. Cexus, “ΨB-energy operator and cross-power spectral density,” Sig. Proc. 94, 236–240 (2014).
[CrossRef]

F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc. 24, 543–572 (2013).
[CrossRef]

A. Boudraa, “Relationships between ΨB-energy operator and some time-frequency representations,” IEEE Sig. Proc. Lett. 17, 527–530 (2010).
[CrossRef]

A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-ΨB energy operator,” Int. J. Electron. Comm. 63, 728–735 (2009).
[CrossRef]

A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-ΨB-energy operator-based signal detection,” JASA 123, 4283–4289 (2008).
[CrossRef]

A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. 135892, 1–8 (2008).

F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. 17, 2804–2815 (2005).
[CrossRef]

A. Boudraa, F. Salzenstein, and J. Cexus, “Two-dimensional continuous higher-order energy operators,” Opt. Eng. 44, 7001–7010 (2005).

J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy operator,” Elect. Lett. 40, 778–780 (2004).
[CrossRef]

Bovik, A.

Brunagel, P.

A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. 135892, 1–8 (2008).

Brunet, M.

C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. 14, 1807 (2003).
[CrossRef]

Cexus, J.

A. Boudraa, T. Chonavel, and J. Cexus, “ΨB-energy operator and cross-power spectral density,” Sig. Proc. 94, 236–240 (2014).
[CrossRef]

A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-ΨB energy operator,” Int. J. Electron. Comm. 63, 728–735 (2009).
[CrossRef]

A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. 135892, 1–8 (2008).

A. Boudraa, J. Cexus, and K. Abed-Meraim, “Cross-ΨB-energy operator-based signal detection,” JASA 123, 4283–4289 (2008).
[CrossRef]

A. Boudraa, F. Salzenstein, and J. Cexus, “Two-dimensional continuous higher-order energy operators,” Opt. Eng. 44, 7001–7010 (2005).

J. Cexus and A. Boudraa, “Link between cross-Wigner distribution and cross-Teager energy operator,” Elect. Lett. 40, 778–780 (2004).
[CrossRef]

Chen, L.

S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
[CrossRef]

Chonavel, T.

A. Boudraa, T. Chonavel, and J. Cexus, “ΨB-energy operator and cross-power spectral density,” Sig. Proc. 94, 236–240 (2014).
[CrossRef]

F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc. 24, 543–572 (2013).
[CrossRef]

A. Boudraa, S. Benramdane, J. Cexus, and T. Chonavel, “Some useful properties of cross-ΨB energy operator,” Int. J. Electron. Comm. 63, 728–735 (2009).
[CrossRef]

de Groot, P.

P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt. 41, 4571–4578 (2002).
[CrossRef]

de Lega, X. C.

P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt. 41, 4571–4578 (2002).
[CrossRef]

Duane, R.

C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. 14, 1807 (2003).
[CrossRef]

Eiju, T.

Estrada, J.

Fillard, J.

P. Montgomery and J. Fillard, “Peak fringe scanning microscopy (pfsm): submicron 3d measurement of semiconductor components,” Interferometry: Techniques and Analysis pp. 12–23 (1755).

Galindo, C.

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

Gao, Z.

S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
[CrossRef]

Georg, S.

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

Groussat, M.

A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. 135892, 1–8 (2008).

Halter, E.

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

Hariharan, P.

Hill, M.

C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. 14, 1807 (2003).
[CrossRef]

Kaiser, J.

P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. 41, 3024–3051 (1993).
[CrossRef]

J. Kaiser, “On a simple algorithm to calculate the energy of a signal,” in Proc. ICASSP, (1990), pp. 381–384.

Kemao, Q.

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” App. Opt. 43, 2695–2702 (2004).
[CrossRef]

Kramer, J.

P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt. 41, 4571–4578 (2002).
[CrossRef]

Larkin, K.

K. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Opt. Express 13, 8097–8121 (2005).
[CrossRef] [PubMed]

K. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” JOSA A 13, 832–843 (1996).
[CrossRef]

Li, M.

M. Li, C. Quan, and C. Tay, “Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry,” Opt. Laser Technol. 40, 920–929 (2008).
[CrossRef]

Liu, C.

W. Zhang, C. Liu, and H. Yan, “Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,” Patt. Recong. 43, 3969–3976 (2010).
[CrossRef]

Ma, S.

S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
[CrossRef]

Maragos, P.

P. Maragos and A. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A 12, 1867–1876 (1995).
[CrossRef]

P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Sig. Proc. Lett. 2, 152–154 (1995).
[CrossRef]

P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. 41, 3024–3051 (1993).
[CrossRef]

Mathewson, A.

C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. 14, 1807 (2003).
[CrossRef]

Montaner, D.

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. 17, 2804–2815 (2005).
[CrossRef]

Montgomery, P.

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. 17, 2804–2815 (2005).
[CrossRef]

P. Montgomery and J. Fillard, “Peak fringe scanning microscopy (pfsm): submicron 3d measurement of semiconductor components,” Interferometry: Techniques and Analysis pp. 12–23 (1755).

Nero, M. D.

E. Halter, P. Montgomery, D. Montaner, R. Barillon, M. D. Nero, C. Galindo, and S. Georg, “Characterization of inhomogeneous colloidal layers using adapted coherence probe microscopy,” Appl. Surf. Sci. 256, 6144–6152 (2010).
[CrossRef]

Niu, H.

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

O’Mahony, C.

C. O’Mahony, M. Hill, M. Brunet, R. Duane, and A. Mathewson, “Characterization of micromechanical structures using white-light interferometry,” Measurement Sci. Technol. 14, 1807 (2003).
[CrossRef]

Oreb, B.

Petitgrand, S.

S. Petitgrand, “Méthodes de microscopie interférométrique 3D statiques et dynamiques pour la caractérisation de la technologie et du comportement des microsystèmes,” Ph.D. thesis (2005).

Potamianos, A.

P. Maragos and A. Potamianos, “Higher order differential energy operators,” IEEE Sig. Proc. Lett. 2, 152–154 (1995).
[CrossRef]

Quan, C.

S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
[CrossRef]

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

M. Li, C. Quan, and C. Tay, “Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry,” Opt. Laser Technol. 40, 920–929 (2008).
[CrossRef]

Quatieri, T.

P. Maragos, T. Quatieri, and J. Kaiser, “Energy separation in signal modulations with applications to speech analysis,” IEEE Trans. Sig. Proc. 41, 3024–3051 (1993).
[CrossRef]

Quiroga, J.

Salzenstein, F.

F. Salzenstein, A. Boudraa, and T. Chonavel, “A new class of multi-dimensional Teager-Kaiser and higher order operators based on directional derivatives,” Multidimensional Sys. Sig. Proc. 24, 543–572 (2013).
[CrossRef]

A. Boudraa, F. Salzenstein, and J. Cexus, “Two-dimensional continuous higher-order energy operators,” Opt. Eng. 44, 7001–7010 (2005).

F. Salzenstein, P. Montgomery, D. Montaner, and A. Boudraa, “Teager-Kaiser energy and higher order operators in white light interference microscopy for surface shape measurement,” J. Appl. Sig. Proc. 17, 2804–2815 (2005).
[CrossRef]

Sandoz, P.

Schafer, R.

R. Schafer, “What is a savitzky-golay filter?[lecture notes],” IEEE Sig. Proc. Mag. 28, 111–117 (2011).
[CrossRef]

Servin, M.

Tay, C.

S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
[CrossRef]

H. Niu, C. Quan, and C. Tay, “Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform,” Opt. Lasers Eng. 47, 1334–1339 (2009).
[CrossRef]

M. Li, C. Quan, and C. Tay, “Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry,” Opt. Laser Technol. 40, 920–929 (2008).
[CrossRef]

Turzhitsky, M.

P. de Groot, X. C. de Lega, J. Kramer, and M. Turzhitsky, “Determination of fringe order in white-light interference microscopy,” App. Opt. 41, 4571–4578 (2002).
[CrossRef]

Yan, H.

W. Zhang, C. Liu, and H. Yan, “Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,” Patt. Recong. 43, 3969–3976 (2010).
[CrossRef]

Zhang, W.

W. Zhang, C. Liu, and H. Yan, “Clustering of temporal gene expression data by regularized spline regression an energy based similarity measure,” Patt. Recong. 43, 3969–3976 (2010).
[CrossRef]

Zhu, R.

S. Ma, C. Quan, R. Zhu, C. Tay, L. Chen, and Z. Gao, “Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry,” Opt. Comm. 284, 2488–2493 (2011).
[CrossRef]

Adv. Sig. Proc.

A. Boudraa, J. Cexus, M. Groussat, and P. Brunagel, “An energy-based similarity measure for time series,” Adv. Sig. Proc. 135892, 1–8 (2008).

App. Opt.

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Figures (3)

Fig. 1
Fig. 1

(a) Interferometric signal; (b) the reference surface shape; (c) a fringe intensity profile along the optical axis (40 nm step, 20% noise level); (d) sθ(z) [Eq. (8)] (i.e, θ = 0) superimposed on the true signal (offset removed) along the optical axis; (e) the final estimated signal sθ̂ (z) provided by the phase fitting, superimposed on the true signal.

Fig. 2
Fig. 2

Surface profile (80 nm step (10% noise level, constant carrier frequency) extracted by the methods: (a) PETKB; (b) FSPS; (c) 1D TK; (d) 2D TK.

Fig. 3
Fig. 3

(a) Real image. (b) Output of the 2D gradient. Surface profile (80 nm step) obtained by PETKB (c). PFSM (d). 1D TK (e). 2D TK (f).

Tables (4)

Tables Icon

Table 1 Error rate (nm) corresponding to synthetic image 1(a) containing two surfaces (surface1 and surface2) for Te = 80 nm and with a constant carrier frequency.

Tables Icon

Table 2 Error rate (nm) corresponding to synthetic image 1(a) containing two surfaces (surface1 and surface2) for Te = 80 nm with randomized carrier frequencies around the mean value with a 5% margin interval.

Tables Icon

Table 3 Error rate (nm) corresponding to synthetic image 1(a) containing two surfaces (surface1 and surface2) for Te = 40 nm and with a constant carrier frequency.

Tables Icon

Table 4 Error rate (nm) corresponding to synthetic image 1(a) containing two surfaces (surface1 and surface2) for Te = 40 nm with randomized carrier frequencies around the mean value with a 5% margin interval.

Equations (10)

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s ( x , y , z ) = a ( x , y , z ) + b ( x , y ) exp [ ( z h ( x , y ) l c ) 2 ] C ( x , y , z ) cos [ 4 π λ 0 ( z h ( x , y ) ) + α ( x , y ) ]
Ψ [ x ( t ) ] = [ x ˙ ( t ) ] 2 x ( t ) x ¨ ( t )
| ϕ ˙ ( t ) | Ψ [ s ˙ ( t ) ] Ψ [ s ( t ) ] ; | a ( t ) | Ψ [ s ( t ) ] Ψ [ s ˙ ( t ) ]
Φ [ I ( x 1 , x 2 ) ] = | | I ( x 1 , x 2 ) | | 2 I ( x 1 , x 2 ) 2 I ( x 1 , x 2 )
| ω x ( x 1 , x 2 ) | Φ [ I ( x 1 , x 2 ) ] Φ [ I ( x 1 , x 2 ) x 1 ] ; | ω y ( x 1 , x 2 ) | Φ [ I ( x 1 , x 2 ) ] Φ [ I ( x 1 , x 2 ) x 2 ] | a ( x 1 , x 2 ) | Φ [ I ( x 1 , x 2 ) ] Φ [ I ( x 1 , x 2 ) x 1 ] + Φ [ I ( x 1 , x 2 ) x 2 ]
Ψ B ( s 1 ( t ) , s 2 ( t ) ) = 1 2 [ Ψ C ( s 1 ( t ) , s 2 ( t ) ) + Ψ C ( s 2 ( t ) , s 1 ( t ) ) ]
Ψ C ( s 1 ( t ) , s 2 ( t ) ) = 1 2 [ s ˙ 1 * ( t ) s ˙ 2 ( t ) + s ˙ 2 * ( t ) s ˙ 1 ( t ) ] 1 2 [ s 1 ( t ) s ˙ 2 * ( t ) + s 1 * ( t ) s ¨ 2 ( t ) ]
s θ ( z ) = C ^ ( z θ / ( 2 π ν ^ z ) ) cos ( 2 π ν ^ z ( z z max ) + θ )
θ ^ = arg max θ [ z max T / 2 z max + T / 2 Ψ B ( s ( z ) , s θ ( z ) ) d z ]
θ ^ = arctan ( 2 I 1 2 I 1 ( I 2 I 2 ) cos ω 0 ( 2 I 0 I 2 I 2 ) sin ω 0 )

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