Abstract

A least-squares filter for restoration of blurred color images was developed directly from spectral and spatial correlations and on the basis of a mathematical model that assumes that spectroscopic images of an object are specified by spectral vectors and that a set of such orthonormal vectors forms a multidimensional color space. Primary colors are expressed by a linear combination of spectral vectors, and color images are given by the projection of spectroscopic images onto the primary color or observation vectors. Detailed discussions and a computer simulation are presented.

© 1988 Optical Society of America

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References

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  1. A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
    [CrossRef]
  2. B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed., Vol. 6 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 179–248.
  3. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 378–425.
  4. H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 126–146.
  5. B. R. Hunt, O. Kulber, “Karhunen–Loeve multispectral image restoration. Part I: theory,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 592–600 (1984).
    [CrossRef]
  6. G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967).
  7. N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
    [CrossRef]

1985 (1)

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

1984 (1)

B. R. Hunt, O. Kulber, “Karhunen–Loeve multispectral image restoration. Part I: theory,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 592–600 (1984).
[CrossRef]

Andrews, H.

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 126–146.

Frieden, B. R.

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed., Vol. 6 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 179–248.

Honda, T.

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

Hunt, B.

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 126–146.

Hunt, B. R.

B. R. Hunt, O. Kulber, “Karhunen–Loeve multispectral image restoration. Part I: theory,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 592–600 (1984).
[CrossRef]

Ikeda, S.

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Kulber, O.

B. R. Hunt, O. Kulber, “Karhunen–Loeve multispectral image restoration. Part I: theory,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 592–600 (1984).
[CrossRef]

Ohyama, N.

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

Ono, R.

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 378–425.

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967).

Suzuki, K.

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

Tsujiuchi, J.

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967).

IEEE Trans. Acoust. Speech Signal Process. (1)

B. R. Hunt, O. Kulber, “Karhunen–Loeve multispectral image restoration. Part I: theory,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 592–600 (1984).
[CrossRef]

Opt. Commun. (1)

N. Ohyama, K. Suzuki, T. Honda, J. Tsujiuchi, R. Ono, S. Ikeda, “Digital processing of endoscopic color images,” Opt. Commun. 55, 4, 242–247 (1985).
[CrossRef]

Other (5)

G. Wyszecki, W. S. Stiles, Color Science—Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967).

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed., Vol. 6 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975), pp. 179–248.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978), pp. 378–425.

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977), pp. 126–146.

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Figures (3)

Fig. 1
Fig. 1

A diagram of R, G, and B primary color vectors.

Fig. 2
Fig. 2

Luminance images calculated by Eq. (27): (A) an original image and (B) an image degraded through one-dimensional linear blur and addition of Gaussian noise. The original R, G, and B images are first blurred by averaging 13 pixels; then different noises are superimposed upon them.

Fig. 3
Fig. 3

Restored images by the four methods: (A) only the luminance component is deblurred; (B) independent treatment of R, G, and B images sequentially restored by the conventional Wiener filters; (C) image restored using the scalar filter given by Eq. (26); and (D) image restored through vector filtering defined by Eq. (18).

Tables (2)

Tables Icon

Table 1 Examples of Dot Products and Angles between Primary Color Vectors

Tables Icon

Table 2 Root-Mean-Square Errors of the Restored Images

Equations (36)

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O ( r , λ ) = λ O λ ( r ) e λ ,
i = λ t i ( λ ) e λ .
f ( r ) = i = 1 3 f i ( r ) e i ,
f i ( r ) = O ( r , λ ) i = [ λ 1 O λ 1 ( r ) e λ 1 ] [ λ 2 t i ( λ 2 ) e λ 2 ] = λ 1 λ 2 O λ 1 ( r ) t i λ 2 e λ 1 e λ 2 .
f i ( r ) = λ O λ ( r ) t i ( λ ) .
θ i j = arccos ( e i e j ) .
g ( r ) = f ( r ) Δ h ( r r ) d r + n ( r ) ,
f ( r ) Δ h ( r r ) = i = 1 3 f i ( r ) h i ( r r ) e i .
h ( r ) = i = 1 3 h i ( r ) e i , n ( r ) = i = 1 3 n i ( r ) e i ,
e 2 = E [ | f ( r ) f ̂ ( r ) | 2 ] ,
f ̂ ( r ) = g ( r ) Δ m ( r r ) d r .
E { i = 1 3 a i j [ f i ( r ) m i ( r r ) g i ( r ) d r ] g j ( s ) } = 0 , j = 1 , 2 , 3 ,
a i j = e i e j .
i = 1 3 a i j E [ f i ( r ) g j ( s ) ] = i = 1 3 a i j { m i ( r r ) E [ g i ( r ) g j ( s ) ] d r } , j = 1 , 2 , 3.
i = 1 3 a i j R f i g j ( r s ) = i = 1 3 a i j [ m i ( r r ) R g i g j ( r s ) d r ] , j = 1 , 2 , 3 ,
i = 1 3 R f i g j ( v ) = i = 1 3 a i j [ m i ( v w ) R g i g j ( w ) d w ] , j = 1 , 2 , 3.
i = 1 3 a i j S f i g j ( u ) = i = 1 3 a i j M i ( u ) S g i g j ( u ) , j = 1 , 2 , 3 ,
[ M 1 ( u ) M 2 ( u ) M 3 ( u ) ] = [ a 11 S g 1 g 1 ( u ) a 21 S g 2 g 1 ( u ) a 31 S g 3 g 1 ( u ) a 12 S g 1 g 2 ( u ) a 22 S g 2 g 2 ( u ) a 32 S g 3 g 2 ( u ) a 13 S g 1 g 3 ( u ) a 23 S g 2 g 3 ( u ) a 33 S g 3 g 3 ( u ) ] 1 [ i = 1 3 a i 1 S f i g 1 ( u ) i = 1 3 a i 2 S f 1 g 2 ( u ) i = 1 3 a i 3 S f i g 3 ( u ) ] .
h 1 ( r ) = h 2 ( r ) = h 3 ( r ) ,
g ( r ) = h ( r r ) f ( r ) d r + n ( r ) .
E { [ f ( r ) m ( r r ) g ( r ) d r ] g ( s ) } = 0.
i = 1 3 j = 1 3 a i j S f i g j ( u ) = M ( u ) [ i = 1 3 j = 1 3 a i j S g i g j ( u ) ] .
S g i g j ( u ) = | H ( u ) | 2 S f i f j ( u ) + S n i n j ( u )
S f i g j ( u ) = H * ( u ) S f i f j ( u ) + S f i n j ( u ) .
S f i n j ( u ) = 0.
M ( u ) = H * ( u ) / { | H ( u ) | 2 + [ i = 1 3 j = 1 3 a i j S n i n j ( u ) / i = 1 3 j = 1 3 a i j S f i f j ( u ) ] } .
L ( r ) = 0.299 f 1 ( r ) + 0.587 f 2 ( r ) + 0.114 f 3 ( r ) ,
e = [ i j | f ( i , j ) f ̂ ( i , j ) | 2 / N 2 ] 1 / 2 ,
[ X Y Z ] = [ 0.606 0.174 0.200 0.299 0.587 0.114 0.000 0.066 1.115 ] [ R G B ] .
[ 1 2 3 ] = { A } [ 1 2 3 ] ,
e 2 = E [ | f ( r ) m ( r r ) Δ g ( r ) d r | 2 ] ,
e 2 = E [ | f ( r ) m ( r r ) Δ g ( r ) d r | 2 ] .
e 2 = E { | f ( r ) m ( r r ) Δ g ( r ) d r + [ m ( r r ) m ( r r ) ] Δ g ( r ) d r | 2 } .
e 2 = E [ | f ( r ) m ( r r ) Δ g ( r ) d r | 2 ] + E { | [ m ( r r ) m ( r r ) ] Δ g ( r ) d r | 2 } + 2 E ( [ f ( r ) m ( r r ) Δ g ( r ) d r ] × { [ m ( r r ) m ( r r ) ] Δ g ( r ) d r } ) .
e 2 = e 2 + E { | [ m ( r r ) m ( r r ) ] Δ g ( r ) d r | 2 } + 2 E ( [ f ( r ) m ( r r ) Δ g ( r ) d r ] × { g ( s ) Δ [ m ( r s ) m ( r s ) ] } ) d s .
e 2 = e 2 + E { | [ m ( r r ) m ( r r ) ] Δ g ( r ) d r | 2 } .

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