Abstract

A new (to our knowledge) theory of component pattern analysis in multispectral images is developed by using the methods of principal component analysis and nonlinear optimization with a nonnegativity constraint. Given images of a scene in different color bands, we estimate both the spectral curves of components included in the image and the spatial pattern corresponding to each spectral curve. In this method, neither spatial nor spectral features of the components are necessary, but the physical rule of nonnegative absorptivity and density nonnegativity is used for any material of any optical frequency at any position in the image. Experimental results of component analysis with real microscopic image data are shown to demonstrate the effectiveness of the proposed method.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. For example, J. Sklansky, G. N. Wassel, Pattern Classifiers and Trainable Machines (Springer-Verlag, New York, 1981).
    [CrossRef]
  2. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  3. G. Nagy, “Digital image-processing activities in remote sensing for earth resources,” Proc. IEEE 60, 1177–1200 (1972).
    [CrossRef]
  4. L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
    [CrossRef] [PubMed]
  5. W. H. Lawton, E. A. Sylvestre, “Self-modeling curve resolution,” Technometrics 13, 617–633 (1971).
    [CrossRef]
  6. K. Sasaki, S. Kawata, S. Minami, “Constrained nonlinear method for estimating component spectra from multicomponent mixtures,” Appl. Opt. 22, 3599–3603 (1983).
    [CrossRef] [PubMed]
  7. S. Kawata, H. Komeda, K. Sasaki, S. Minami, “Advanced algorithm for determining component spectra based on principal component analysis,” Appl. Spectrosc. 39, 610–614 (1985).
    [CrossRef]
  8. W. C. Chiou, “NASA image-based geological expert system development project for hyperspectral image analysis,” Appl. Opt. 24, 2085–2091 (1985).
    [CrossRef] [PubMed]
  9. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Princeton, N. J., 1974).
  10. E. R. Malinowski, “Determination of the number of factors and experimental error in data matrix,” Anal. Chem. 49, 612–617 (1977).
    [CrossRef]
  11. H. Akaike, “A new look at the statistical model identification,”IEEE Trans. Autom. Control AC-19, 716–723 (1974).
    [CrossRef]
  12. J. Kowalik, M. R. Osborne, Methods for Unconstrained Optimization Problems (Elsevier, New York, 1968).

1985 (2)

1983 (1)

1981 (1)

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

1977 (1)

E. R. Malinowski, “Determination of the number of factors and experimental error in data matrix,” Anal. Chem. 49, 612–617 (1977).
[CrossRef]

1974 (1)

H. Akaike, “A new look at the statistical model identification,”IEEE Trans. Autom. Control AC-19, 716–723 (1974).
[CrossRef]

1972 (1)

G. Nagy, “Digital image-processing activities in remote sensing for earth resources,” Proc. IEEE 60, 1177–1200 (1972).
[CrossRef]

1971 (1)

W. H. Lawton, E. A. Sylvestre, “Self-modeling curve resolution,” Technometrics 13, 617–633 (1971).
[CrossRef]

Akaike, H.

H. Akaike, “A new look at the statistical model identification,”IEEE Trans. Autom. Control AC-19, 716–723 (1974).
[CrossRef]

Alvarez, R. E.

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Brody, W. R.

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Chiou, W. C.

Hall, A. L.

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Princeton, N. J., 1974).

Kawata, S.

Komeda, H.

Kowalik, J.

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

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Princeton, N. J., 1974).

Lawton, W. H.

W. H. Lawton, E. A. Sylvestre, “Self-modeling curve resolution,” Technometrics 13, 617–633 (1971).
[CrossRef]

Lehmann, L. A.

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Macovski, A.

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Malinowski, E. R.

E. R. Malinowski, “Determination of the number of factors and experimental error in data matrix,” Anal. Chem. 49, 612–617 (1977).
[CrossRef]

Minami, S.

Nagy, G.

G. Nagy, “Digital image-processing activities in remote sensing for earth resources,” Proc. IEEE 60, 1177–1200 (1972).
[CrossRef]

Osborne, M. R.

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

Pelc, N. J.

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Riederer, S. J.

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Sasaki, K.

Sklansky, J.

For example, J. Sklansky, G. N. Wassel, Pattern Classifiers and Trainable Machines (Springer-Verlag, New York, 1981).
[CrossRef]

Sylvestre, E. A.

W. H. Lawton, E. A. Sylvestre, “Self-modeling curve resolution,” Technometrics 13, 617–633 (1971).
[CrossRef]

Wassel, G. N.

For example, J. Sklansky, G. N. Wassel, Pattern Classifiers and Trainable Machines (Springer-Verlag, New York, 1981).
[CrossRef]

Anal. Chem. (1)

E. R. Malinowski, “Determination of the number of factors and experimental error in data matrix,” Anal. Chem. 49, 612–617 (1977).
[CrossRef]

Appl. Opt. (2)

Appl. Spectrosc. (1)

IEEE Trans. Autom. Control (1)

H. Akaike, “A new look at the statistical model identification,”IEEE Trans. Autom. Control AC-19, 716–723 (1974).
[CrossRef]

Med. Phys. (1)

L. A. Lehmann, R. E. Alvarez, A. Macovski, W. R. Brody, N. J. Pelc, S. J. Riederer, A. L. Hall, “Generalized image combination in dual KVP digital radiography,” Med. Phys. 8, 659–667 (1981).
[CrossRef] [PubMed]

Proc. IEEE (1)

G. Nagy, “Digital image-processing activities in remote sensing for earth resources,” Proc. IEEE 60, 1177–1200 (1972).
[CrossRef]

Technometrics (1)

W. H. Lawton, E. A. Sylvestre, “Self-modeling curve resolution,” Technometrics 13, 617–633 (1971).
[CrossRef]

Other (4)

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Princeton, N. J., 1974).

For example, J. Sklansky, G. N. Wassel, Pattern Classifiers and Trainable Machines (Springer-Verlag, New York, 1981).
[CrossRef]

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Illustration of multispectral images. The right-hand side of the figure is an image set measured at N wavelengths, and the left-hand side is a spectrum set at individual image pixels.

Fig. 2
Fig. 2

Imaging model for multicomponent patterns. Multispectral images are given by linear combinations of M component patterns weighted by their corresponding spectral responses.

Fig. 3
Fig. 3

Schematic illustration Eq. (1).

Fig. 4
Fig. 4

Schematic illustration of Eq. (2).

Fig. 5
Fig. 5

Explanatory illustration of the feasible region in [T] space determined by nonnegativity constraints and the normalization for a three-component system. The cone A is the constraint of inequality (7), and the three concave boundaries B1, B2, and B3 are made by inequality (8). Spherical surface C is given by Eq. (9).

Fig. 6
Fig. 6

Multispectral images of Paramecium sp. observed at (a) 450, (b) 550, and (c) 650 nm.

Fig. 7
Fig. 7

Estimated bands of two-component spectra (shaded areas a and b).

Fig. 8
Fig. 8

Estimated bands of two-component patterns. The solution bands are limited by patterns (a) and (b) and by patterns (c) and (d), which correspond to the spectra of curves a and b, respectively, in Fig. 7.

Tables (1)

Tables Icon

Table 1 Eigenvalues of [I]t[I]

Equations (19)

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

[ I ] = [ S ] [ P ] ,
[ I ] = [ U ] [ Λ ] [ V ] ,
[ P ] = [ T ] [ V ] ,
[ S ] = [ U ] [ Λ ] [ T ] - 1 .
[ P ] [ 0 ] ,
[ S ] [ 0 ] ,
[ T ] [ V ] [ 0 ] ,
[ U ] [ Λ ] [ T ] - 1 [ 0 ] ,
diag ( [ T ] [ T ] t ) = [ E ] ,
t 11 v 1 i + t 12 v 2 i 0 ,             i = 1 , 2 , , L ,
t 21 v 1 i + t 22 v 2 i 0 ,             i = 1 , 2 , , L .
- [ max 1 i L ( v 2 i v 1 i ) ] - 1 t 12 t 11 - [ min 1 i L ( v 2 i v 1 i ) ] - 1
- [ max 1 i L ( v 2 i v 1 i ) ] - 1 t 22 t 21 - [ min 1 i L ( v 2 i v 1 i ) ] - 1 .
u j 1 λ 1 t 22 - u j 2 λ 2 t 21 t 11 t 22 - t 12 t 21 0 ,             j = 1 , , N ,
- u j 1 λ 1 t 12 + u j 2 λ 2 t 11 t 11 t 22 - t 12 t 21 0 ,             j = 1 , , N .
t 22 t 21 min 1 j N ( λ 2 u j 2 λ 1 u j 1 ) ,
t 12 t 11 max 1 j N ( λ 2 u j 2 λ 1 u j 1 ) .
t 11 2 + t 12 2 = 1
t 21 2 + t 22 2 = 1.

Metrics