Abstract

In the standard reflectance model for inhomogeneous materials it is assumed that light is reflected by two independent mechanisms. One component is reflected at the interface of the material and air. Light reflected by this mechanism does not interact with surface colorant, and its spectral composition is assumed to equal that of the incident light. The second component is reflected after entering and interacting with the subsurface structure of the material. This interaction substantially changes the spectral composition of the reflected light. We adopt a vector analysis technique for testing the standard reflectance model. Further, we develop a computational method to determine the components of the observed spectra, and we obtain an estimate of the illuminant without using a reference white standard. Finally, we evaluate the accuracy of the standard model and the feasibility of the illuminant spectral estimation by using several test objectives.

© 1989 Optical Society of America

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References

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  1. G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vision 2, 7–32 (1988).
    [CrossRef]
  2. H.-C. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
    [CrossRef] [PubMed]
  3. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [CrossRef]
  4. G. Buchsbaum, “A spatial processor model for object color perception,”J. Franklin Inst. 310, 1–26 (1980).
    [CrossRef]
  5. M. H. Brill, G. West, “Contributions to the theory of invariance of color under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
    [CrossRef]
  6. L. T. Maloney, B. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  7. R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graph. 15, 307–316 (1981).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).
  9. S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
    [CrossRef]
  10. H. Lee, E. J. Breneman, C. P. Schulte, “An experimental study of a color reflection model,” (Kodak Research Laboratories, Rochester, N.Y., 1986).
  11. All that is required, in fact, is that the specular reflectance function be common and known among the different surfaces. However, it is convenient, and not too inaccurate, to adopt the assumption that the function is constant with respect to wavelength.
  12. G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

1988 (1)

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vision 2, 7–32 (1988).
[CrossRef]

1986 (3)

1985 (1)

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

1981 (2)

M. H. Brill, G. West, “Contributions to the theory of invariance of color under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graph. 15, 307–316 (1981).
[CrossRef]

1980 (1)

G. Buchsbaum, “A spatial processor model for object color perception,”J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).

Breneman, E. J.

H. Lee, E. J. Breneman, C. P. Schulte, “An experimental study of a color reflection model,” (Kodak Research Laboratories, Rochester, N.Y., 1986).

Brill, M. H.

M. H. Brill, G. West, “Contributions to the theory of invariance of color under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

Buchsbaum, G.

G. Buchsbaum, “A spatial processor model for object color perception,”J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

Cook, R. L.

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graph. 15, 307–316 (1981).
[CrossRef]

D’Zmura, M.

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Kanade, T.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vision 2, 7–32 (1988).
[CrossRef]

Klinker, G. J.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vision 2, 7–32 (1988).
[CrossRef]

Lee, H.

H. Lee, E. J. Breneman, C. P. Schulte, “An experimental study of a color reflection model,” (Kodak Research Laboratories, Rochester, N.Y., 1986).

Lee, H.-C.

Lennie, P.

Maloney, L. T.

Schulte, C. P.

H. Lee, E. J. Breneman, C. P. Schulte, “An experimental study of a color reflection model,” (Kodak Research Laboratories, Rochester, N.Y., 1986).

Shafer, S. A.

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vision 2, 7–32 (1988).
[CrossRef]

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

Torrance, K. E.

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graph. 15, 307–316 (1981).
[CrossRef]

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

Wandell, B.

West, G.

M. H. Brill, G. West, “Contributions to the theory of invariance of color under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).

Color Res. Appl. (1)

S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
[CrossRef]

Comput. Graph. (1)

R. L. Cook, K. E. Torrance, “A reflectance model for computer graphics,” Comput. Graph. 15, 307–316 (1981).
[CrossRef]

Int. J. Comput. Vision (1)

G. J. Klinker, S. A. Shafer, T. Kanade, “The measurement of highlights in color images,” Int. J. Comput. Vision 2, 7–32 (1988).
[CrossRef]

J. Franklin Inst. (1)

G. Buchsbaum, “A spatial processor model for object color perception,”J. Franklin Inst. 310, 1–26 (1980).
[CrossRef]

J. Math. Biol. (1)

M. H. Brill, G. West, “Contributions to the theory of invariance of color under the condition of varying illumination,” J. Math. Biol. 11, 337–350 (1981).
[CrossRef]

J. Opt. Soc. Am. A (3)

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1983).

H. Lee, E. J. Breneman, C. P. Schulte, “An experimental study of a color reflection model,” (Kodak Research Laboratories, Rochester, N.Y., 1986).

All that is required, in fact, is that the specular reflectance function be common and known among the different surfaces. However, it is convenient, and not too inaccurate, to adopt the assumption that the function is constant with respect to wavelength.

G. H. Golub, C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983).

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Figures (15)

Fig. 1
Fig. 1

Light reflected from a surface is modeled as the sum of two components. Each component represents a different light path.

Fig. 2
Fig. 2

Color signal plane by a linear combination of the interface reflection component LI(λ) and the subsurface reflection component LS(λ).

Fig. 3
Fig. 3

The intersection of two color-signal planes P(1) and P(2).

Fig. 4
Fig. 4

A red cup (left) and a green ashtray (right) of plastic illuminated by a flood lamp.

Fig. 5
Fig. 5

Normalized curves of the measured spectra of the red plastic cup.

Fig. 6
Fig. 6

Normalized curves of the measured spectra of the green plastic ashtray.

Fig. 7
Fig. 7

An apple (left) and a lemon (right) illuminated by a tungsten halogen lamp from a slide projector.

Fig. 8
Fig. 8

Normalized curves of the measured spectra of the apple.

Fig. 9
Fig. 9

Normalized curves of the measured spectra of the lemon.

Fig. 10
Fig. 10

Basis curves of the measured spectra of the red plastic cup.

Fig. 11
Fig. 11

Basis curves of the measured spectra of the green plastic ashtray.

Fig. 12
Fig. 12

Estimation results of the illuminant spectral power distribution of a flood lamp. Filled squares represent the estimate from the two plastic objects, and open circles represent the measurement by the standard white.

Fig. 13
Fig. 13

Basis curves of the measured spectra of the apple.

Fig. 14
Fig. 14

Basis curves of the measured spectra of the lemon.

Fig. 15
Fig. 15

Estimation results of the illuminant spectral power distribution of a slide projector. Filled squares represent the estimate from the two fruits, and open circles represent the measurement by the standard white.

Equations (19)

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Y ( θ , λ ) = Y I ( θ , λ ) + Y S ( θ , λ ) ,
Y ( θ , λ ) = c I ( θ ) L I ( λ ) + c S ( θ ) L S ( λ ) .
Y ( θ , λ ) = c I ( θ ) S I ( λ ) E ( λ ) + c S ( θ ) S S ( λ ) E ( λ ) .
Y 1 ( θ , λ ) = c I 1 ( θ ) E ( λ ) + c S 1 ( θ ) S S 1 ( λ ) E ( λ ) ,
Y 2 ( θ , λ ) = c I 2 ( θ ) E ( λ ) + c S 2 ( θ ) S S 2 ( λ ) E ( λ ) .
Y = [ y 1 , y 2 , , y m ] .
Y = U Σ V T ,
Y = σ 1 u 1 v 1 T + σ 2 u 2 v 2 T + + σ m u m v m T ,
I ( K ) = i = 1 K σ i 2 i = 1 m σ i 2 = i = 1 K σ i 2 m .
y i = c i 1 u 1 + c i 2 u 2             ( i = 1 , 2 , , m ) ,
P = { y y = c 1 u 1 + c 2 u 2 , c i R } .
c 1 u 1 ( 1 ) + c 2 u 2 ( 1 ) = c 1 u 1 ( 2 ) + c 2 u 2 ( 2 ) .
[ u 1 ( 1 ) , u 2 ( 1 ) , u 1 ( 2 ) , u 2 ( 2 ) ] [ c 1 c 2 - c 1 - c 2 ] = 0.
[ u 1 ( 1 ) , u 2 ( 1 ) , u 1 ( 2 ) , u 2 ( 2 ) ] = [ a 1 , a 2 , a 3 , a 4 ] [ λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 0 0 0 0 λ 4 - - - - ] × [ b 1 , b 2 , b 3 , b 4 ] T ,
e 1 = 2 [ b 41 u 1 ( 1 ) + b 42 u 2 ( 1 ) ]
e 2 = - 2 [ b 43 u 1 ( 2 ) + b 44 u 2 ( 2 ) ] ,
e ^ = ( e 1 + e 2 ) 2 .
e ^ = 0.4207 u 1 ( 1 ) + 0.2702 u 2 ( 1 ) + 0.4567 u 2 ( 2 ) - 0.2036 u 2 ( 2 ) .
e ^ = 0.4226 u 1 ( 1 ) + 0.2672 u 2 ( 1 ) + 0.4655 u 2 ( 2 ) - 0.1825 u 2 ( 2 ) .

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