Abstract

In this paper, we propose a method which uniquely determines a set of single curves, each as an estimate of a component spectrum. No reference spectrum from a library is necessary; the spectrum set of mixtures of unknown components with various concentrations is used for the estimation of component spectral curves. The method is based on entropy minimization. In comparison with an earlier method [ Appl. Opt. 22, 3599 ( 1983)], which gives the bands of the possible component spectra, this method has the advantage of providing a unique estimation of the component spectra, which helps chemists with quantitative analysis and further mathematical processing. Two experimental results demonstrate the effectiveness of the method: for infrared absorption spectra of xylene isomers and visible absorption spectra of dyes.

© 1984 Optical Society of America

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References

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  1. E. R. Malinowski, “Determination of the Number of Factors and Experimental Error in a Data Matrix,” Anal. Chem. 49, 612 (1977).
    [CrossRef]
  2. P. C. Gillette, J. L. Koenig, “Noise Reduction via Factor Analysis in FT-IR Spectra,” Appl. Spectrosc. 36, 535 (1982).
    [CrossRef]
  3. P. C. Gillette, J. B. Lando, J. L. Koenig, “Computer-Assisted Spectral Identification of Unknown Mixtures,” Appl. Spectrosc. 36, 661 (1982).
    [CrossRef]
  4. W. H. Lawton, E. A. Sylvestre, “Self-Modeling Curve Resolution,” Technometrics 13, 617 (1971).
    [CrossRef]
  5. K. Sasaki, S. Kawata, S. Minami, “Constrained Nonlinear Method for Estimating Component Spectra from Multicomponent Mixtures,” Appl. Opt. 22, 3599 (1983).
    [CrossRef] [PubMed]
  6. J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).
  7. E. R. Malinowski, “Theory of Error in Factor Analysis,” Anal. Chem. 49, 606 (1977).
    [CrossRef]
  8. J. A. Blackburn, “Computer Program for Multicomponent Spectrum Analysis Using Least-Squares Method,” Anal. Chem. 37, 1000 (1965).
    [CrossRef]
  9. A. Savitzky, M. J. E. Golay, “Smoothing and Differentiation of Data by Simplified Least Squares Procedures,” Anal. Chem. 36, 1627 (1964).
    [CrossRef]

1983 (1)

1982 (2)

1977 (2)

E. R. Malinowski, “Determination of the Number of Factors and Experimental Error in a Data Matrix,” Anal. Chem. 49, 612 (1977).
[CrossRef]

E. R. Malinowski, “Theory of Error in Factor Analysis,” Anal. Chem. 49, 606 (1977).
[CrossRef]

1971 (1)

W. H. Lawton, E. A. Sylvestre, “Self-Modeling Curve Resolution,” Technometrics 13, 617 (1971).
[CrossRef]

1965 (1)

J. A. Blackburn, “Computer Program for Multicomponent Spectrum Analysis Using Least-Squares Method,” Anal. Chem. 37, 1000 (1965).
[CrossRef]

1964 (1)

A. Savitzky, M. J. E. Golay, “Smoothing and Differentiation of Data by Simplified Least Squares Procedures,” Anal. Chem. 36, 1627 (1964).
[CrossRef]

Blackburn, J. A.

J. A. Blackburn, “Computer Program for Multicomponent Spectrum Analysis Using Least-Squares Method,” Anal. Chem. 37, 1000 (1965).
[CrossRef]

Gillette, P. C.

Golay, M. J. E.

A. Savitzky, M. J. E. Golay, “Smoothing and Differentiation of Data by Simplified Least Squares Procedures,” Anal. Chem. 36, 1627 (1964).
[CrossRef]

Kawata, S.

Koenig, J. L.

Kowalik, J.

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

Lando, J. B.

Lawton, W. H.

W. H. Lawton, E. A. Sylvestre, “Self-Modeling Curve Resolution,” Technometrics 13, 617 (1971).
[CrossRef]

Malinowski, E. R.

E. R. Malinowski, “Determination of the Number of Factors and Experimental Error in a Data Matrix,” Anal. Chem. 49, 612 (1977).
[CrossRef]

E. R. Malinowski, “Theory of Error in Factor Analysis,” Anal. Chem. 49, 606 (1977).
[CrossRef]

Minami, S.

Osborne, M. R.

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

Sasaki, K.

Savitzky, A.

A. Savitzky, M. J. E. Golay, “Smoothing and Differentiation of Data by Simplified Least Squares Procedures,” Anal. Chem. 36, 1627 (1964).
[CrossRef]

Sylvestre, E. A.

W. H. Lawton, E. A. Sylvestre, “Self-Modeling Curve Resolution,” Technometrics 13, 617 (1971).
[CrossRef]

Anal. Chem. (4)

E. R. Malinowski, “Determination of the Number of Factors and Experimental Error in a Data Matrix,” Anal. Chem. 49, 612 (1977).
[CrossRef]

E. R. Malinowski, “Theory of Error in Factor Analysis,” Anal. Chem. 49, 606 (1977).
[CrossRef]

J. A. Blackburn, “Computer Program for Multicomponent Spectrum Analysis Using Least-Squares Method,” Anal. Chem. 37, 1000 (1965).
[CrossRef]

A. Savitzky, M. J. E. Golay, “Smoothing and Differentiation of Data by Simplified Least Squares Procedures,” Anal. Chem. 36, 1627 (1964).
[CrossRef]

Appl. Opt. (1)

Appl. Spectrosc. (2)

Technometrics (1)

W. H. Lawton, E. A. Sylvestre, “Self-Modeling Curve Resolution,” Technometrics 13, 617 (1971).
[CrossRef]

Other (1)

J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (American Elsevier, New York, 1968).

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Figures (8)

Fig. 1
Fig. 1

Computer simulation of a component spectrum determination by use of the entropy criterion. (a) Set of computer-simulated two-component mixture spectra (solid lines). Two component spectra shown by the dotted lines are the Gaussians whose peaks are at γp and γq, respectively, plus a constant bias. (b) Estimated result of the left-side Gaussian component by the band-estimation method (solid lines I and II)5 and the true component spectrum (dotted line T). (c) Second derivative-of the true component spectrum (dotted line T″) and those of the two boundary curves (solid lines I″ and II″) with respect to λ.

Fig. 2
Fig. 2

Infrared absorbance spectra of fifteen mixtures of xylene isomers with various relative concentrations.

Fig. 3
Fig. 3

Component spectra estimated by the proposed method from the data set shown in Fig. 2.

Fig. 4
Fig. 4

Pure component spectra of (a) o-xylene, (b) m-xylene, and (c) p-xylene.

Fig. 5
Fig. 5

Bands of three component spectra satisfying the constraints, each bounded by two curves.

Fig. 6
Fig. 6

Visible absorbance spectra of fifteen mixtures of three dyes with various relative concentrations.

Fig. 7
Fig. 7

Component spectra estimated by the proposed method from the data set shown in Fig. 6.

Fig. 8
Fig. 8

Pure component spectra of (a) bromocresol green (BCG), (b) methyl orange (MO), and (c) indigo carmine (IC).

Equations (15)

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x i = j = 1 N c i j s j ,             i = 1 , , M ,
s j = k = 1 N t j k v k ,             j = 1 , , N ,
[ R ] = i = 1 M x i x i t / M .
s j l ( t 11 , , t N N ) 0             for all j , l ,
c i j ( t 11 , , t N N ) 0             for all i , j ;
c i j = k = 1 N t j k ' ( v k t · x i ) ,
l = 1 L s j l = 1             for all j .
h j = - l = 1 L p j l · log p j l min ,             j = 1 , , N ,
p j l = s ^ j l l = 1 L s ^ j l ,
H = j = 1 N h j min .
U = H + P min .
P ( t ^ 11 , , t ^ N N , γ ) = γ · { j = 1 N l = 1 L F ( s ^ j l ) s ^ j l 2 + i = 1 M j = 1 N F ( c ^ i j ) c ^ i j 2 } ,
F ( y ) = { 0 ( y 0 ) , 1 ( y < 0 ) .
l = 1 L s ^ j l 2 = 1 ,             j = 1 , , N .
k = 1 N t ^ j k 2 = 1 ,             j = 1 , , N .

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