Abstract

In Part I [ J. Opt. Soc. Am. A 4, 2101 ( 1987)] of this series, we developed a method for estimating both spatial patterns and spectral curves of components in a multispectral scene. This method does not need spatial and spectral information about the components but only multispread imagery data. The estimation is given as a feasible solution set satisfying the nonnegativity constraint for density and spectral response for all components at all pixels. In this paper, we estimate unique solutions for both the component patterns and the spectra from the feasible solution set. The solution is given by optimizing an entropy minimization criterion. This criterion enhances the spectral or spatial features of individual components. Two experimental results are shown to demonstrate the effectiveness of this method with biological and cytochemical specimens. The limitations of this method for unique pattern estimation are also discussed.

© 1989 Optical Society of America

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References

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  1. S. Kawata, K. Sasaki, S. Minami, “Component analysis of spatial and spectral patterns in multispectral images. I. Basis,” J. Opt. Soc. Am. A 4, 2101–2106 (1987).
    [Crossref] [PubMed]
  2. J. Kowalik, M. R. Osborne, Method for Unconstrained Optimization Problems (Elsevier, New York, 1968).
  3. 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]
  4. F. P. Altman, “Studies on the reduction of tetrazolium salts. III. The products of chemical and enzymic reduction,” Histochemie 38, 155–171 (1974).
    [Crossref] [PubMed]
  5. K. Chikamori, K. Araki, M. Yamada, “Pattern analysis of heterogeneous distribution of succinate dehydrogenase in single rat hepatic lobules,” Cell. Mol. Biol. 31, 217–222 (1985).
  6. K. Sasaki, S. Kawata, S. Minami, “Estimation of component spectral curves from unknown mixture spectra,” Appl. Opt. 23, 1955–1959 (1984).
    [Crossref] [PubMed]

1987 (1)

1985 (2)

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]

K. Chikamori, K. Araki, M. Yamada, “Pattern analysis of heterogeneous distribution of succinate dehydrogenase in single rat hepatic lobules,” Cell. Mol. Biol. 31, 217–222 (1985).

1984 (1)

1974 (1)

F. P. Altman, “Studies on the reduction of tetrazolium salts. III. The products of chemical and enzymic reduction,” Histochemie 38, 155–171 (1974).
[Crossref] [PubMed]

Altman, F. P.

F. P. Altman, “Studies on the reduction of tetrazolium salts. III. The products of chemical and enzymic reduction,” Histochemie 38, 155–171 (1974).
[Crossref] [PubMed]

Araki, K.

K. Chikamori, K. Araki, M. Yamada, “Pattern analysis of heterogeneous distribution of succinate dehydrogenase in single rat hepatic lobules,” Cell. Mol. Biol. 31, 217–222 (1985).

Chikamori, K.

K. Chikamori, K. Araki, M. Yamada, “Pattern analysis of heterogeneous distribution of succinate dehydrogenase in single rat hepatic lobules,” Cell. Mol. Biol. 31, 217–222 (1985).

Kawata, S.

Komeda, H.

Kowalik, J.

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

Minami, S.

Osborne, M. R.

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

Sasaki, K.

Yamada, M.

K. Chikamori, K. Araki, M. Yamada, “Pattern analysis of heterogeneous distribution of succinate dehydrogenase in single rat hepatic lobules,” Cell. Mol. Biol. 31, 217–222 (1985).

Appl. Opt. (1)

Appl. Spectrosc. (1)

Cell. Mol. Biol. (1)

K. Chikamori, K. Araki, M. Yamada, “Pattern analysis of heterogeneous distribution of succinate dehydrogenase in single rat hepatic lobules,” Cell. Mol. Biol. 31, 217–222 (1985).

Histochemie (1)

F. P. Altman, “Studies on the reduction of tetrazolium salts. III. The products of chemical and enzymic reduction,” Histochemie 38, 155–171 (1974).
[Crossref] [PubMed]

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

Other (1)

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

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Figures (10)

Fig. 1
Fig. 1

Schematic illustration of Eq. (1).

Fig. 2
Fig. 2

Microscopic image of paramecium dyed by hematoxylin and fast green.

Fig. 3
Fig. 3

Estimated spectra of two dyes. Shaded bands represent the feasible solution sets obtained with the nonnegativity constraints, and solid curves represent the estimates obtained by the entropy minimization method.

Fig. 4
Fig. 4

Estimated spatial patterns of two dyes: (a) and (b) correspond to the spectra A and B; respectively, in Fig. 3.

Fig. 5
Fig. 5

Microscopic image of a tissue section of rat liver dyed by NBT.

Fig. 6
Fig. 6

Multispectral images of the specimen shown in Fig. 5, observed at (a) 450 nm, (b) 550 nm, and (c) 650 nm.

Fig. 7
Fig. 7

Plot of eigenvalues of the multispectral images.

Fig. 8
Fig. 8

Estimated spectra for three components.

Fig. 9
Fig. 9

Estimated density patterns and profiles along a line: (a), (b), and (c) correspond to the spectra A, B, and C in Fig. 8, respectively.

Fig. 10
Fig. 10

Example of the estimation giving a solution different from the true one. (a) Computer-simulated component spectra. (b) Estimated spectra obtained by the entropy minimization technique.

Equations (15)

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[ I ] = [ S ] [ P ] ,
[ I ] = [ U ] [ Λ ] [ V ] ,
[ P ] = [ T ] [ V ] ,
[ S ] = [ U ] [ Λ ] [ T ] - 1 ,
[ T ] [ V ] [ 0 ] ,
[ U ] [ Λ ] [ T ] - 1 [ 0 ] .
H ( [ S ] ) = - i = 1 M j = 1 L a i j ln a i j min ,
a i j = S i j j = 1 L s i j
H ( [ P ] ) = - i = 1 M j = 1 N b i j ln b i j min ,
b i j = p i j j = 1 N p i j .
R ( [ S ] , [ P ] ) = H ( [ S ] , [ P ] ) + γ Q ( [ S ] ,             [ P ] ) min ,
Q = m = 1 M i = 1 N F ( p m i ) + j = 1 L m = 1 M F ( s j m ) ,
F ( x ) = { 0 ( x 0 ) x 2 ( x < 0 ) .
max 1 j N ( λ 2 u j 2 λ 1 u j 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 j N ( λ 2 u j 2 λ 1 u j 1 ) ,

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