Abstract

We show that an analysis of the mean and variance of discrete wavelet coefficients of coaveraged time-domain interferograms can be used as a specification for determining when to stop coaveraging. We also show that, if a prediction model built in the wavelet domain is used to determine the composition of unknown samples, a stopping criterion for the coaveraging process can be developed with respect to the uncertainty tolerated in the prediction.

© 2002 Optical Society of America

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References

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  1. I. Daubechies, Ten Lectures on Wavelets (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
    [CrossRef]
  2. O. Rioul and M. Vetterli, IEEE Signal Process. Mag. 8, 14 (1991).
    [CrossRef]
  3. H. Martens and T. Naes, Multivariate Calibration (Wiley, London, 1993).
  4. B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
    [CrossRef]
  5. B. K. Alsberg, A. M. Woodward, and D. B. Kell, Chemom. Intell. Lab. Syst. 37, 215 (1997).
    [CrossRef]
  6. U. L. Pen, Philos. Trans. R. Soc. London Ser. A 357, 2561 (1999).
    [CrossRef]
  7. R. R. Coifman and M. V. Wickerhauser, IEEE Eng. Med. Biol. Mag. 14, 578 (1995).
    [CrossRef]
  8. J. R. Birch and T. J. Parker, Infrared and Millimeter Waves 2: Instrumentation (Academic, San Diego, 1979), p. 137.
  9. L. Duvillaret, F. Garet, and J. L. Coutaz, J. Opt. Soc. Am. B 17, 452 (2000).
    [CrossRef]
  10. Z. Jiang and X.-C. Zhang, Opt. Lett. 23, 1114 (1998).
    [CrossRef]
  11. B. B. Hu and M. C. Nuss, Opt. Lett. 20, 1716 (1995).
    [CrossRef]
  12. D. M. Mittleman, S. Hunsche, and M. C. Nuss, Opt. Lett. 22, 904 (1997).
    [CrossRef] [PubMed]
  13. P. Moulin, M. Anitescu, K. O. Kortanke, and F. A. Potra, IEEE Trans. Signal Process. 45, 2160 (1997).
    [CrossRef]

2000

1999

U. L. Pen, Philos. Trans. R. Soc. London Ser. A 357, 2561 (1999).
[CrossRef]

1998

B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
[CrossRef]

Z. Jiang and X.-C. Zhang, Opt. Lett. 23, 1114 (1998).
[CrossRef]

1997

D. M. Mittleman, S. Hunsche, and M. C. Nuss, Opt. Lett. 22, 904 (1997).
[CrossRef] [PubMed]

P. Moulin, M. Anitescu, K. O. Kortanke, and F. A. Potra, IEEE Trans. Signal Process. 45, 2160 (1997).
[CrossRef]

B. K. Alsberg, A. M. Woodward, and D. B. Kell, Chemom. Intell. Lab. Syst. 37, 215 (1997).
[CrossRef]

1995

R. R. Coifman and M. V. Wickerhauser, IEEE Eng. Med. Biol. Mag. 14, 578 (1995).
[CrossRef]

B. B. Hu and M. C. Nuss, Opt. Lett. 20, 1716 (1995).
[CrossRef]

1991

O. Rioul and M. Vetterli, IEEE Signal Process. Mag. 8, 14 (1991).
[CrossRef]

Alsberg, B. K.

B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
[CrossRef]

B. K. Alsberg, A. M. Woodward, and D. B. Kell, Chemom. Intell. Lab. Syst. 37, 215 (1997).
[CrossRef]

Anitescu, M.

P. Moulin, M. Anitescu, K. O. Kortanke, and F. A. Potra, IEEE Trans. Signal Process. 45, 2160 (1997).
[CrossRef]

Birch, J. R.

J. R. Birch and T. J. Parker, Infrared and Millimeter Waves 2: Instrumentation (Academic, San Diego, 1979), p. 137.

Coifman, R. R.

R. R. Coifman and M. V. Wickerhauser, IEEE Eng. Med. Biol. Mag. 14, 578 (1995).
[CrossRef]

Coutaz, J. L.

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

Duvillaret, L.

Garet, F.

Hu, B. B.

Hunsche, S.

Jiang, Z.

Kell, D. B.

B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
[CrossRef]

B. K. Alsberg, A. M. Woodward, and D. B. Kell, Chemom. Intell. Lab. Syst. 37, 215 (1997).
[CrossRef]

Kortanke, K. O.

P. Moulin, M. Anitescu, K. O. Kortanke, and F. A. Potra, IEEE Trans. Signal Process. 45, 2160 (1997).
[CrossRef]

Martens, H.

H. Martens and T. Naes, Multivariate Calibration (Wiley, London, 1993).

Mittleman, D. M.

Moulin, P.

P. Moulin, M. Anitescu, K. O. Kortanke, and F. A. Potra, IEEE Trans. Signal Process. 45, 2160 (1997).
[CrossRef]

Naes, T.

H. Martens and T. Naes, Multivariate Calibration (Wiley, London, 1993).

Nuss, M. C.

Parker, T. J.

J. R. Birch and T. J. Parker, Infrared and Millimeter Waves 2: Instrumentation (Academic, San Diego, 1979), p. 137.

Pen, U. L.

U. L. Pen, Philos. Trans. R. Soc. London Ser. A 357, 2561 (1999).
[CrossRef]

Potra, F. A.

P. Moulin, M. Anitescu, K. O. Kortanke, and F. A. Potra, IEEE Trans. Signal Process. 45, 2160 (1997).
[CrossRef]

Rioul, O.

O. Rioul and M. Vetterli, IEEE Signal Process. Mag. 8, 14 (1991).
[CrossRef]

Rowland, J. J.

B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
[CrossRef]

Vetterli, M.

O. Rioul and M. Vetterli, IEEE Signal Process. Mag. 8, 14 (1991).
[CrossRef]

Wickerhauser, M. V.

R. R. Coifman and M. V. Wickerhauser, IEEE Eng. Med. Biol. Mag. 14, 578 (1995).
[CrossRef]

Winson, M. K.

B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
[CrossRef]

Woodward, A. M.

B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
[CrossRef]

B. K. Alsberg, A. M. Woodward, and D. B. Kell, Chemom. Intell. Lab. Syst. 37, 215 (1997).
[CrossRef]

Zhang, X.-C.

Anal. Chim. Acta

B. K. Alsberg, A. M. Woodward, M. K. Winson, J. J. Rowland, and D. B. Kell, Anal. Chim. Acta 368, 29 (1998).
[CrossRef]

Chemom. Intell. Lab. Syst.

B. K. Alsberg, A. M. Woodward, and D. B. Kell, Chemom. Intell. Lab. Syst. 37, 215 (1997).
[CrossRef]

IEEE Eng. Med. Biol. Mag.

R. R. Coifman and M. V. Wickerhauser, IEEE Eng. Med. Biol. Mag. 14, 578 (1995).
[CrossRef]

IEEE Signal Process. Mag.

O. Rioul and M. Vetterli, IEEE Signal Process. Mag. 8, 14 (1991).
[CrossRef]

IEEE Trans. Signal Process.

P. Moulin, M. Anitescu, K. O. Kortanke, and F. A. Potra, IEEE Trans. Signal Process. 45, 2160 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Philos. Trans. R. Soc. London Ser. A

U. L. Pen, Philos. Trans. R. Soc. London Ser. A 357, 2561 (1999).
[CrossRef]

Other

H. Martens and T. Naes, Multivariate Calibration (Wiley, London, 1993).

I. Daubechies, Ten Lectures on Wavelets (Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

J. R. Birch and T. J. Parker, Infrared and Millimeter Waves 2: Instrumentation (Academic, San Diego, 1979), p. 137.

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Figures (4)

Fig. 1
Fig. 1

SNR of the coaveraged interferogram (a) without and (b) with wavelet filtering. The thick curves are estimates, SNRt, and the thin lines are the true values, SNRt. The inset shows (a) the sample time-domain signal and (b) an interferogram degraded by reduction of the SNR by a factor of 50.

Fig. 2
Fig. 2

Sum-squared errors of the coaveraged (thin curve) and the filtered (thick curve) interferograms with respect to the original sample interferogram. The inset shows the gradual change of the crossing point with increasing rms noise level [with respect to the noise level in the interferogram shown in inset (a) of Fig. 1] added to the individual interferograms during the coaveraging process.

Fig. 3
Fig. 3

Absolute value of the differences in complex insertion loss of the polyvinyl chloride sample calculated from the coaveraged (thin curve) and wavelet-filtered (thick curve) time-domain signatures, respectively.

Fig. 4
Fig. 4

Standard deviation (SD) estimate for the prediction of y, obtained by a coaveraging process after the SNR of the original interferogram was degraded by a factor of 20 and all model weights wj were set to 1. The tolerance level, shown in the inset, was obtained as 1% of the model prediction at each coaveraging step.

Equations (1)

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SNRtk=1Ni=1Nμti2kHμtik-2.326 σtik1/21Ni=1Nσti2kHμtik-2.326 σtik1/2,

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