Abstract

We provide a unified framework for a range of linear transforms that can be used for the analysis of terahertz spectroscopic data, with particular emphasis on their application to the measurement of leaf water content. The use of linear transforms for filtering, regression, and classification is discussed. For illustration, a classification problem involving leaves at three stages of drought and a prediction problem involving simulated spectra are presented. Issues resulting from scaling the data set are discussed. Using Lagrange multipliers, we arrive at the transform that yields the maximum separation between the spectra and show that this optimal transform is equivalent to computing the Euclidean distance between the samples. The optimal linear transform is compared with the average for all the spectra as well as with the Karhunen–Loève transform to discriminate a wet leaf from a dry leaf. We show that taking several principal components into account is equivalent to defining new axes in which data are to be analyzed. The procedure shows that the coefficients of the Karhunen–Loève transform are well suited to the process of classification of spectra. This is in line with expectations, as these coefficients are built from the statistical properties of the data set analyzed.

© 2002 Optical Society of America

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  9. R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
    [CrossRef]
  10. H. C. Goicoechea, A. C. Olivieri, “Wavelength selection by net analyte signals calculated with multivariate factor-based hybrid linear analysis (HLA). A theoretical and experimental comparison with partial least-squares (PLS),” Analyst (London) 124, 725–731 (1999).
  11. J. Sustek, “Method for choice of optimal analytical positions in spectrophotometric analysis of multicomponent systems,” Ann. Geophys. 46, 1676–1679 (1974).
  12. S. Qian, D. Chen, Joint Time–Frequency Analysis—Methods and Applications (Prentice-Hall, Upper Saddle River, N.J., 1996).
  13. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
  14. M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, Wavelet Toolbox User’s Guide (Mathworks, Natick, Mass., 1996).
  15. H. Martens, T. Naes, Multivariate Calibration (Wiley, London, 1993).
  16. B. G. Tabachnick, L. S. Fidell, Using Multivariate Statistics, 4th ed. (Allyn and Bacon, Boston, Mass., 2001).
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    [CrossRef]
  21. R. K. H. Galvão, S. Hadjiloucas, J. W. Bowen, “Use of the statistical properties of the wavelet transform coefficients for the optimization of integration time in Fourier transform spectrometry,” Opt. Lett. 27, 643–645 (2002).
    [CrossRef]
  22. D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
    [CrossRef]
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    [CrossRef]
  24. B. Ferguson, D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32, 943–953 (2001).
    [CrossRef]
  25. T. Naes, B. H. Mevik, “Understanding the collinearity problem in regression and discriminant analysis,” J. Chemom. 15, 413–426 (2001).
    [CrossRef]
  26. T. D. Dorney, R. G. Baraniuk, D. M. Mittleman, “Material parameter estimation using terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18, 1562–1571 (2001).
    [CrossRef]
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  30. R. H. Jacobsen, D. M. Mittleman, M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
    [CrossRef] [PubMed]
  31. D. S. Venables, C. A. Schmuttenmaer, “Far-infrared spectra and associated dynamics in acetonitrile–water mixtures measured with femtosecond THz pulse spectroscopy,” J. Chem. Phys. 108, 4935–4944 (1998).
    [CrossRef]
  32. I. Facchin, C. Mello, M. I. M. S. Bueno, R. J. Poppi, “Simultaneous determination of lead and sulfur by energy-dispersive x-ray spectrometry. Comparison between artificial neural networks and other multivariate calibration methods,” X-Ray Spectrom. 28, 173–177 (1999).
    [CrossRef]
  33. P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
    [CrossRef]
  34. Z. Jiang, X.-C. Zhang, “Single-shot spatial-temporal THz field imaging,” Opt. Lett. 23, 1114–1116 (1998).
    [CrossRef]
  35. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

2002 (2)

2001 (6)

R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
[CrossRef]

B. Ferguson, D. Abbott, “Wavelet de-noising of optical terahertz imaging data,” Fluctuat. Noise Lett. 1, L65–L70 (2001).
[CrossRef]

B. Ferguson, D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32, 943–953 (2001).
[CrossRef]

T. Naes, B. H. Mevik, “Understanding the collinearity problem in regression and discriminant analysis,” J. Chemom. 15, 413–426 (2001).
[CrossRef]

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

T. D. Dorney, R. G. Baraniuk, D. M. Mittleman, “Material parameter estimation using terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18, 1562–1571 (2001).
[CrossRef]

2000 (1)

D. W. van der Weide, J. Murakowski, F. Keilmann, “Gas-absorption spectroscopy with electronic terahertz techniques,” IEEE Trans. Microwave Theory Tech. 48, 740–743 (2000).
[CrossRef]

1999 (5)

I. Facchin, C. Mello, M. I. M. S. Bueno, R. J. Poppi, “Simultaneous determination of lead and sulfur by energy-dispersive x-ray spectrometry. Comparison between artificial neural networks and other multivariate calibration methods,” X-Ray Spectrom. 28, 173–177 (1999).
[CrossRef]

L. Smrcok, M. Durik, V. Jorik, “Wavelet denoising of powder diffraction patterns,” Powder Diffr. 14, 300–304 (1999).
[CrossRef]

U. L. Pen, “Application of wavelets to filtering of noisy data,” Philos. Trans. R. Soc. London Ser. A 357, 2561–2571 (1999).
[CrossRef]

H. C. Goicoechea, A. C. Olivieri, “Wavelength selection by net analyte signals calculated with multivariate factor-based hybrid linear analysis (HLA). A theoretical and experimental comparison with partial least-squares (PLS),” Analyst (London) 124, 725–731 (1999).

S. Hadjiloucas, L. S. Karatzas, J. W. Bowen, “Measurements of leaf water content using terahertz radiation,” IEEE Trans. Microwave Theory Tech. 47, 142–149 (1999).
[CrossRef]

1998 (5)

J. R. Birch, “Pseudocoherence in dispersive Fourier transform spectroscopy,” Infrared Phys. 28, 345–352 (1998).
[CrossRef]

D. S. Venables, C. A. Schmuttenmaer, “Far-infrared spectra and associated dynamics in acetonitrile–water mixtures measured with femtosecond THz pulse spectroscopy,” J. Chem. Phys. 108, 4935–4944 (1998).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

Z. Jiang, X.-C. Zhang, “Single-shot spatial-temporal THz field imaging,” Opt. Lett. 23, 1114–1116 (1998).
[CrossRef]

M. Soriano, C. Saloma, “Improved classification robustness for noisy cell images represented as principal-component projections in a hybrid recognition system,” Appl. Opt. 37, 3628–3638 (1998).
[CrossRef]

1996 (2)

R. H. Jacobsen, D. M. Mittleman, M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[CrossRef] [PubMed]

D. M. Mittleman, R. H. Jacobsen, M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

1974 (1)

J. Sustek, “Method for choice of optimal analytical positions in spectrophotometric analysis of multicomponent systems,” Ann. Geophys. 46, 1676–1679 (1974).

Abbott, D.

B. Ferguson, D. Abbott, “Wavelet de-noising of optical terahertz imaging data,” Fluctuat. Noise Lett. 1, L65–L70 (2001).
[CrossRef]

B. Ferguson, D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32, 943–953 (2001).
[CrossRef]

Alda, J.

Araujo, M. C. U.

R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
[CrossRef]

Baraniuk, R. G.

T. D. Dorney, R. G. Baraniuk, D. M. Mittleman, “Material parameter estimation using terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18, 1562–1571 (2001).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

Bernabeu, E.

Bertie, J. E.

J. E. Bertie, “Optical constants,” in Handbook of Vibrational Spectroscopy, J. Chalmers, P. R. Griffiths, eds. (Wiley, New York, 2001).

Birch, J. R.

J. R. Birch, “Pseudocoherence in dispersive Fourier transform spectroscopy,” Infrared Phys. 28, 345–352 (1998).
[CrossRef]

J. R. Birch, T. J. Parker, “Dispersive Fourier transform spectrometry,” in Infrared and Millimeter Waves, K. Button, ed. (Academic, New York, 1979), Vol. 2, Chap. 3.

Bowen, J. W.

R. K. H. Galvão, S. Hadjiloucas, J. W. Bowen, “Use of the statistical properties of the wavelet transform coefficients for the optimization of integration time in Fourier transform spectrometry,” Opt. Lett. 27, 643–645 (2002).
[CrossRef]

S. Hadjiloucas, L. S. Karatzas, J. W. Bowen, “Measurements of leaf water content using terahertz radiation,” IEEE Trans. Microwave Theory Tech. 47, 142–149 (1999).
[CrossRef]

Boyer, J. S.

J. S. Boyer, Measuring the Water Content of Plants and Soils (Academic, San Diego, Calif., 1995).

Bueno, M. I. M. S.

I. Facchin, C. Mello, M. I. M. S. Bueno, R. J. Poppi, “Simultaneous determination of lead and sulfur by energy-dispersive x-ray spectrometry. Comparison between artificial neural networks and other multivariate calibration methods,” X-Ray Spectrom. 28, 173–177 (1999).
[CrossRef]

Chen, D.

S. Qian, D. Chen, Joint Time–Frequency Analysis—Methods and Applications (Prentice-Hall, Upper Saddle River, N.J., 1996).

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

Dorney, T. D.

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Durik, M.

L. Smrcok, M. Durik, V. Jorik, “Wavelet denoising of powder diffraction patterns,” Powder Diffr. 14, 300–304 (1999).
[CrossRef]

Facchin, I.

I. Facchin, C. Mello, M. I. M. S. Bueno, R. J. Poppi, “Simultaneous determination of lead and sulfur by energy-dispersive x-ray spectrometry. Comparison between artificial neural networks and other multivariate calibration methods,” X-Ray Spectrom. 28, 173–177 (1999).
[CrossRef]

Feldmann, J.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Ferguson, B.

B. Ferguson, D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32, 943–953 (2001).
[CrossRef]

B. Ferguson, D. Abbott, “Wavelet de-noising of optical terahertz imaging data,” Fluctuat. Noise Lett. 1, L65–L70 (2001).
[CrossRef]

Fidell, L. S.

B. G. Tabachnick, L. S. Fidell, Using Multivariate Statistics, 4th ed. (Allyn and Bacon, Boston, Mass., 2001).

Fromm, J.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Galvão, R. K. H.

R. K. H. Galvão, S. Hadjiloucas, J. W. Bowen, “Use of the statistical properties of the wavelet transform coefficients for the optimization of integration time in Fourier transform spectrometry,” Opt. Lett. 27, 643–645 (2002).
[CrossRef]

R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
[CrossRef]

Goicoechea, H. C.

H. C. Goicoechea, A. C. Olivieri, “Wavelength selection by net analyte signals calculated with multivariate factor-based hybrid linear analysis (HLA). A theoretical and experimental comparison with partial least-squares (PLS),” Analyst (London) 124, 725–731 (1999).

Hadjiloucas, S.

R. K. H. Galvão, S. Hadjiloucas, J. W. Bowen, “Use of the statistical properties of the wavelet transform coefficients for the optimization of integration time in Fourier transform spectrometry,” Opt. Lett. 27, 643–645 (2002).
[CrossRef]

S. Hadjiloucas, L. S. Karatzas, J. W. Bowen, “Measurements of leaf water content using terahertz radiation,” IEEE Trans. Microwave Theory Tech. 47, 142–149 (1999).
[CrossRef]

Haferkorn, R.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Han, P. Y.

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Hecker, N. E.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Hempel, M.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Hunsche, S.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

R. H. Jacobsen, D. M. Mittleman, M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[CrossRef] [PubMed]

D. M. Mittleman, R. H. Jacobsen, M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

Jiang, Z.

Jorik, V.

L. Smrcok, M. Durik, V. Jorik, “Wavelet denoising of powder diffraction patterns,” Powder Diffr. 14, 300–304 (1999).
[CrossRef]

Karatzas, L. S.

S. Hadjiloucas, L. S. Karatzas, J. W. Bowen, “Measurements of leaf water content using terahertz radiation,” IEEE Trans. Microwave Theory Tech. 47, 142–149 (1999).
[CrossRef]

Keilmann, F.

D. W. van der Weide, J. Murakowski, F. Keilmann, “Gas-absorption spectroscopy with electronic terahertz techniques,” IEEE Trans. Microwave Theory Tech. 48, 740–743 (2000).
[CrossRef]

Kersting, R.

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

Knobloch, P.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Koch, M.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

M. Koch, “THz imaging: fundamentals and biological applications,” in Terahertz Spectroscopy and Applications II, J. M. Chamberlain, ed., Proc. SPIE3828, 202–208 (1999).
[CrossRef]

Kono, S.

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

Libon, I.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Lopez-Alonso, J. M.

Martens, H.

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

Mello, C.

I. Facchin, C. Mello, M. I. M. S. Bueno, R. J. Poppi, “Simultaneous determination of lead and sulfur by energy-dispersive x-ray spectrometry. Comparison between artificial neural networks and other multivariate calibration methods,” X-Ray Spectrom. 28, 173–177 (1999).
[CrossRef]

Mevik, B. H.

T. Naes, B. H. Mevik, “Understanding the collinearity problem in regression and discriminant analysis,” J. Chemom. 15, 413–426 (2001).
[CrossRef]

Misiti, M.

M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, Wavelet Toolbox User’s Guide (Mathworks, Natick, Mass., 1996).

Misiti, Y.

M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, Wavelet Toolbox User’s Guide (Mathworks, Natick, Mass., 1996).

Mittleman, D. M.

T. D. Dorney, R. G. Baraniuk, D. M. Mittleman, “Material parameter estimation using terahertz time-domain spectroscopy,” J. Opt. Soc. Am. A 18, 1562–1571 (2001).
[CrossRef]

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

R. H. Jacobsen, D. M. Mittleman, M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[CrossRef] [PubMed]

D. M. Mittleman, R. H. Jacobsen, M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

Murakowski, J.

D. W. van der Weide, J. Murakowski, F. Keilmann, “Gas-absorption spectroscopy with electronic terahertz techniques,” IEEE Trans. Microwave Theory Tech. 48, 740–743 (2000).
[CrossRef]

Naes, T.

T. Naes, B. H. Mevik, “Understanding the collinearity problem in regression and discriminant analysis,” J. Chemom. 15, 413–426 (2001).
[CrossRef]

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

Neelamani, R.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

Nobel, P. S.

P. S. Nobel, Physicochemical and Environmental Plant Physiology (Academic, San Diego, Calif., 1991).

Nuss, M. C.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

R. H. Jacobsen, D. M. Mittleman, M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. 21, 2011–2013 (1996).
[CrossRef] [PubMed]

D. M. Mittleman, R. H. Jacobsen, M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Olivieri, A. C.

H. C. Goicoechea, A. C. Olivieri, “Wavelength selection by net analyte signals calculated with multivariate factor-based hybrid linear analysis (HLA). A theoretical and experimental comparison with partial least-squares (PLS),” Analyst (London) 124, 725–731 (1999).

Oppenheim, G.

M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, Wavelet Toolbox User’s Guide (Mathworks, Natick, Mass., 1996).

Parker, T. J.

J. R. Birch, T. J. Parker, “Dispersive Fourier transform spectrometry,” in Infrared and Millimeter Waves, K. Button, ed. (Academic, New York, 1979), Vol. 2, Chap. 3.

Pen, U. L.

U. L. Pen, “Application of wavelets to filtering of noisy data,” Philos. Trans. R. Soc. London Ser. A 357, 2561–2571 (1999).
[CrossRef]

Pimentel, M. F.

R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
[CrossRef]

Poggi, J. M.

M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, Wavelet Toolbox User’s Guide (Mathworks, Natick, Mass., 1996).

Poppi, R. J.

I. Facchin, C. Mello, M. I. M. S. Bueno, R. J. Poppi, “Simultaneous determination of lead and sulfur by energy-dispersive x-ray spectrometry. Comparison between artificial neural networks and other multivariate calibration methods,” X-Ray Spectrom. 28, 173–177 (1999).
[CrossRef]

Qian, S.

S. Qian, D. Chen, Joint Time–Frequency Analysis—Methods and Applications (Prentice-Hall, Upper Saddle River, N.J., 1996).

Rehberg, E.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Saloma, C.

Sautter, I.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Schmalstieg, K.

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

Schmuttenmaer, C. A.

D. S. Venables, C. A. Schmuttenmaer, “Far-infrared spectra and associated dynamics in acetonitrile–water mixtures measured with femtosecond THz pulse spectroscopy,” J. Chem. Phys. 108, 4935–4944 (1998).
[CrossRef]

Smrcok, L.

L. Smrcok, M. Durik, V. Jorik, “Wavelet denoising of powder diffraction patterns,” Powder Diffr. 14, 300–304 (1999).
[CrossRef]

Soriano, M.

Sustek, J.

J. Sustek, “Method for choice of optimal analytical positions in spectrophotometric analysis of multicomponent systems,” Ann. Geophys. 46, 1676–1679 (1974).

Tabachnick, B. G.

B. G. Tabachnick, L. S. Fidell, Using Multivariate Statistics, 4th ed. (Allyn and Bacon, Boston, Mass., 2001).

Tani, M.

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

Usami, M.

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

van der Weide, D. W.

D. W. van der Weide, J. Murakowski, F. Keilmann, “Gas-absorption spectroscopy with electronic terahertz techniques,” IEEE Trans. Microwave Theory Tech. 48, 740–743 (2000).
[CrossRef]

Venables, D. S.

D. S. Venables, C. A. Schmuttenmaer, “Far-infrared spectra and associated dynamics in acetonitrile–water mixtures measured with femtosecond THz pulse spectroscopy,” J. Chem. Phys. 108, 4935–4944 (1998).
[CrossRef]

Visani, V.

R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
[CrossRef]

Yoneyama, T.

R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
[CrossRef]

Zhang, X.-C.

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

Z. Jiang, X.-C. Zhang, “Single-shot spatial-temporal THz field imaging,” Opt. Lett. 23, 1114–1116 (1998).
[CrossRef]

X.-C. Zhang, “Next rays? T. ray!” presented at the Plenary Session of the 26th International Conference on Infrared and Millimeter Waves, Toulouse, France, September 2001.

Anal. Chim. Acta (1)

R. K. H. Galvão, M. F. Pimentel, M. C. U. Araujo, T. Yoneyama, V. Visani, “Aspects of the successive projections algorithm for variable selection in multivariate calibration applied to plasma emission spectrometry,” Anal. Chim. Acta 443, 107–115 (2001).
[CrossRef]

Analyst (London) (1)

H. C. Goicoechea, A. C. Olivieri, “Wavelength selection by net analyte signals calculated with multivariate factor-based hybrid linear analysis (HLA). A theoretical and experimental comparison with partial least-squares (PLS),” Analyst (London) 124, 725–731 (1999).

Ann. Geophys. (1)

J. Sustek, “Method for choice of optimal analytical positions in spectrophotometric analysis of multicomponent systems,” Ann. Geophys. 46, 1676–1679 (1974).

Appl. Opt. (2)

Appl. Phys. B (1)

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[CrossRef]

Fluctuat. Noise Lett. (1)

B. Ferguson, D. Abbott, “Wavelet de-noising of optical terahertz imaging data,” Fluctuat. Noise Lett. 1, L65–L70 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. M. Mittleman, R. H. Jacobsen, M. C. Nuss, “T-ray imaging,” IEEE J. Sel. Top. Quantum Electron. 2, 679–692 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

S. Hadjiloucas, L. S. Karatzas, J. W. Bowen, “Measurements of leaf water content using terahertz radiation,” IEEE Trans. Microwave Theory Tech. 47, 142–149 (1999).
[CrossRef]

D. W. van der Weide, J. Murakowski, F. Keilmann, “Gas-absorption spectroscopy with electronic terahertz techniques,” IEEE Trans. Microwave Theory Tech. 48, 740–743 (2000).
[CrossRef]

Infrared Phys. (1)

J. R. Birch, “Pseudocoherence in dispersive Fourier transform spectroscopy,” Infrared Phys. 28, 345–352 (1998).
[CrossRef]

J. Chemom. (1)

T. Naes, B. H. Mevik, “Understanding the collinearity problem in regression and discriminant analysis,” J. Chemom. 15, 413–426 (2001).
[CrossRef]

J. Appl. Phys. (1)

P. Y. Han, M. Tani, M. Usami, S. Kono, R. Kersting, X.-C. Zhang, “A direct comparison between terahertz time-domain spectroscopy and far-infrared Fourier transform spectroscopy,” J. Appl. Phys. 89, 2357–2359 (2001).
[CrossRef]

J. Chem. Phys. (1)

D. S. Venables, C. A. Schmuttenmaer, “Far-infrared spectra and associated dynamics in acetonitrile–water mixtures measured with femtosecond THz pulse spectroscopy,” J. Chem. Phys. 108, 4935–4944 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

Microelectron. J. (1)

B. Ferguson, D. Abbott, “De-noising techniques for terahertz responses of biological samples,” Microelectron. J. 32, 943–953 (2001).
[CrossRef]

Opt. Lett. (3)

Philos. Trans. R. Soc. London Ser. A (1)

U. L. Pen, “Application of wavelets to filtering of noisy data,” Philos. Trans. R. Soc. London Ser. A 357, 2561–2571 (1999).
[CrossRef]

Powder Diffr. (1)

L. Smrcok, M. Durik, V. Jorik, “Wavelet denoising of powder diffraction patterns,” Powder Diffr. 14, 300–304 (1999).
[CrossRef]

X-Ray Spectrom. (1)

I. Facchin, C. Mello, M. I. M. S. Bueno, R. J. Poppi, “Simultaneous determination of lead and sulfur by energy-dispersive x-ray spectrometry. Comparison between artificial neural networks and other multivariate calibration methods,” X-Ray Spectrom. 28, 173–177 (1999).
[CrossRef]

Other (13)

J. R. Birch, T. J. Parker, “Dispersive Fourier transform spectrometry,” in Infrared and Millimeter Waves, K. Button, ed. (Academic, New York, 1979), Vol. 2, Chap. 3.

J. E. Bertie, “Optical constants,” in Handbook of Vibrational Spectroscopy, J. Chalmers, P. R. Griffiths, eds. (Wiley, New York, 2001).

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

X.-C. Zhang, “Next rays? T. ray!” presented at the Plenary Session of the 26th International Conference on Infrared and Millimeter Waves, Toulouse, France, September 2001.

M. Koch, “THz imaging: fundamentals and biological applications,” in Terahertz Spectroscopy and Applications II, J. M. Chamberlain, ed., Proc. SPIE3828, 202–208 (1999).
[CrossRef]

M. Koch, K. Schmalstieg, P. Knobloch, S. Hunsche, M. C. Nuss, E. Rehberg, I. Sautter, J. Fromm, M. Hempel, R. Haferkorn, I. Libon, N. E. Hecker, J. Feldmann, “THz imaging of biological samples,” in Terahertz Sources and Systems, Vol. 27 of NATO Science Series II, R. E. Miles, P. Harrison, D. Lippens, eds., (Kluwer Academic, Dordrecht, The Netherlands, 2000), pp. 241–258.

J. S. Boyer, Measuring the Water Content of Plants and Soils (Academic, San Diego, Calif., 1995).

P. S. Nobel, Physicochemical and Environmental Plant Physiology (Academic, San Diego, Calif., 1991).

S. Qian, D. Chen, Joint Time–Frequency Analysis—Methods and Applications (Prentice-Hall, Upper Saddle River, N.J., 1996).

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).

M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, Wavelet Toolbox User’s Guide (Mathworks, Natick, Mass., 1996).

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

B. G. Tabachnick, L. S. Fidell, Using Multivariate Statistics, 4th ed. (Allyn and Bacon, Boston, Mass., 2001).

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Figures (12)

Fig. 1
Fig. 1

Co-averaged (left) and noisy (right) interferograms in the (a) time, (b) Fourier, and (c) wavelet domains.

Fig. 2
Fig. 2

(a) Evaluation of matrix effect, (b) instrument response function at two wavelengths, (c) PC axes, (d) data ranges along the original axes and along the first PC axis, (e) rotation along PC axes, (f) reduction of singular components for PC analysis.

Fig. 3
Fig. 3

Measured attenuation of wet, dry, and oven-dry Fatsia Japonica leaves.

Fig. 4
Fig. 4

(a) Distribution of the power in the data among the three PCs (PC1, PC2, PC3) and (b) the same information presented in a triangle plot.

Fig. 5
Fig. 5

Loadings of (a) PC1 and (b) PC2.

Fig. 6
Fig. 6

Separating the data by using the first two PCs. The oven-dry leaf is distinctly different from the two others, as can be seen from the distances in the PC1–PC2 plane (note that the axes are not on the same scale).

Fig. 7
Fig. 7

Mean-centering and scaling the data (a) makes the differences between the samples more apparent, (b) redistributes the explained variance in the samples to PC1 and PC2 only, and (c) makes the separation of the data in PC2 more pronounced.

Fig. 8
Fig. 8

(a) Fourier-transformed spectra and (b) fraction of power in each PC before co-averaging. With this level of noise, the spectra of the dry and wet leaves cannot be distinguished.

Fig. 9
Fig. 9

Effect of increased noise level in the spectra to the PCs. The points represent degradations in the spectrum simulated by adding artificial white Gaussian noise with standard deviation of 0.1–1.

Fig. 10
Fig. 10

Sequence of consecutive linear transformations of the PCs for two samples A and B, which lead to a transform close to the optimal one calculated by using Lagrange multipliers.

Fig. 11
Fig. 11

Number of PCs used to describe the observed variance in the spectra and their noise content.

Fig. 12
Fig. 12

Simulated test spectra with relative water content in the range 65%–95% (using the wet leaf as a standard). The inset shows the root mean square error of prediction (RMSEP) of models built from individual spectral bins (univariate calibration). The horizontal line in the inset marks the RMSEP obtained when regression is carried out after PCA.

Tables (4)

Tables Icon

Table 1 Assignment of a Single Index to Pairs of Indices (f, p) of the Coefficients for the Windowed Fourier Transform a

Tables Icon

Table 2 Overview of Linear Transforms

Tables Icon

Table 3 Advantages and Shortcomings of Linear Transforms

Tables Icon

Table 4 Mean Difference between the Wet and Dry Spectra Shown in Fig. 3 , as Calculated by Using Average Values Across All Frequencies, the Calculated Difference Using PCA, a and the Optimal Difference as Calculated by Using Lagrange Multipliers

Equations (33)

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

xλ=i=1ntivi(λ).
ui1, ui2=λ=1nvi1(λ)vi2*(λ)=1,i1=i20,i1i2,
ti=x, ui=λ=1nxλvi*(λ).
vi(λ)=1,λ=i0,λi.
vi(λ)=exp(j2πfiλ)=cos(2πfiλ)+j sin(2πfiλ),
fi=(i-1)/n.
t(f, p)=λ=1nx(λ)vf,p*(λ),
vf,p(λ)=w(λ-p)exp[j2πf(λ-p)],
νa,b(λ)=ψa,b(λ)=1a ψλ-ba,
a=a0j,b=kb0a0j,
X=USVT,
x1×n=yk1×nyk=xykkT=xkTy=xkT(kkT)-1.
X=x11x12xnm.
P=(XTX)-1(y-yˆ)T(y-yˆ)m-n.
SEi=[mσxi2(1-Di2)]-1(y-yˆ)T(y-yˆ)m-n1/2,
S=s110000s220000smm0m×n,S¯=s11000s22000srrr×r,
Xm×n=Tm×nΨn×n,
Lˆ(λ)=4nˆ(λ)[1+nˆ(λ)]2exp[i2π(nˆ-1)d/λ]1-{[1-nˆ(λ)]/[1+nˆ(λ)]}2exp(4iπnˆd/λ),
log(GM)=logi=1mPCi1/m=1mi=1mlogPCi,
t=λ=1nx(λ)v(λ),
J=tDRY-tWET=λ=1n[xDRY(λ)-xWET(λ)]v(λ),
Lv(λ)=xDRY(λ)-xWET(λ)+2qv(λ)=02qv(λ)-2Q(λ)=0,
Lq=λv2(λ)-1=0λQ2(λ)/q2=1,
v(λ)=xDRY(λ)-xWET(λ){λ=1n[xDRY(λ)-xWET(λ)]2}1/2.
pC(x)=1(2π)n/2(detΣC)1/2×exp[-12 (x-μC)(ΣC)-1(x-μC)T],
(x-μC)(ΣC)-1(x-μC)T=(rC)2,
rC=σ-1[(x-μC)(x-μC)T]1/2,
Z=w1t1+w2t2,
E[t]=t¯=λE[x¯(λ)v(λ)]+λE[η(λ)v(λ)]
t¯=λE[x¯(λ)v(λ)].
E[(t-t¯)2]=E[t2-2tt¯+t¯2]=E[t2]-2tE¯[t]+t¯2=E[t2]-2t¯2+t¯2=E[t2]-t¯2=Eλ[x¯(λ)+η(λ)]v(λ)2-t¯2=Eλ[x¯(λ)v(λ)]+λ[(η(λ)v(λ)]2-t¯2=Eλ[x¯(λ)v(λ)]2+2λ[(x¯(λ)v(λ)]λ[(η(λ)v(λ)]+λ[(η(λ)v(λ)]2-t¯2=t¯2+2tE¯λ[(η(λ)v(λ)]+Eλ[(η(λ)v(λ)]2-t¯2=2tE¯λ[(η(λ)v(λ)]+Eλ[(η(λ)v(λ)]2=2t¯λE[(η(λ)]v(λ)+Eλ[(η(λ)v(λ)]2=0+Eλ[(η(λ)v(λ)]2=Eλ[(η(λ)v(λ)]2.
Eλ[(η(λ)v(λ)]2
=Eλ[η2(λ)v2(λ)]+λ1λ2[η(λ1)v(λ1)η(λ2)v(λ2)]=λ{E[η2(λ)]v2(λ)}+λ1λ2{E[η(λ1)η(λ2)]v(λ1)v(λ2)}=λσ2v2(λ)+0=σ2λv2(λ)=σ2.

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