Abstract

For inhomogeneous materials, the standard reflectance model suggests that under all viewing geometries surface reflectance functions can be described as the sum of a constant function of wavelength (specular) and a diffuse function that is characteristic of the material. As the viewing geometry varies, the relative contribution of these two terms varies. In a previous study [ J. Opt. Soc. Am. A 6, 576 ( 1989)] we described how to use light reflected from inhomogeneous materials, measured in different viewing geometries, to estimate the relative spectral power distribution of the ambient light. Here we show that two restrictions, that (a) surface reflectance functions are all nonnegative and (b) surface reflectance functions are the positive weighted sum of subsurface (diffuse) and interface (specular) components, may be used to estimate the subsurface component of the surface reflectance function. A band of surface spectral reflectances is recovered, as possible solutions for the subsurface estimates.

© 1990 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. S. Tominaga, B. A. Wandell, “The standard surface reflectance model and illuminant estimation,” J. Opt. Soc. Am. A 6, 576–584 (1989).
    [CrossRef]
  3. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 1662–1672 (1986).
    [CrossRef]
  4. H. Lee, “Method for computing the scene-illuminant chromaticity from specular highlights,” J. Opt. Soc. Am. A 3, 1694–1699 (1986).
    [CrossRef] [PubMed]
  5. B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 4, 1770–1779 (1979).
    [CrossRef]
  6. W. H. Lawton, E. A. Sylvestre, “Self modeling curve resolution,” Technometrics 13, 617–633 (1971).
    [CrossRef]
  7. 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]

1989 (1)

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]

1987 (1)

1986 (2)

1979 (1)

B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 4, 1770–1779 (1979).
[CrossRef]

1971 (1)

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

D’Zmura, M.

Horn, B. K. P.

B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 4, 1770–1779 (1979).
[CrossRef]

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]

Kawata, S.

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]

Lawton, W. H.

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

Lee, H.

Lennie, P.

Minami, S.

Sasaki, K.

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]

Sjoberg, R. W.

B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 4, 1770–1779 (1979).
[CrossRef]

Sylvestre, E. A.

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

Tominaga, S.

Wandell, B. A.

Appl. Opt. (1)

B. K. P. Horn, R. W. Sjoberg, “Calculating the reflectance map,” Appl. Opt. 4, 1770–1779 (1979).
[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. Opt. Soc. Am. A (4)

Technometrics (1)

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

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Figures (10)

Fig. 1
Fig. 1

Pictorial diagram of the (cI, cI) coordinate system in which the observed surface spectral reflectances are projected onto the quarter of the unit circle within the positive quadrant. Each dot indicates the observation point. A + on the quarter circle indicates the upper limit of the permissible directions for describing the surface reflectance vectors. The possible solution band is shown by the bold arc on the circle.

Fig. 2
Fig. 2

Schematic representation of the experimental procedure.

Fig. 3
Fig. 3

Estimation results of the illuminant spectral power distributions of two light sources.2 Crosses represent the estimate of the flood lamp, and filled squares represent that of the tungsten halogen lamp of a slide projector.

Fig. 4
Fig. 4

Normalized curves of the observed surface spectral reflectances of the green plastic ashtray.

Fig. 5
Fig. 5

Basis curves of the observed surface reflectances. The nearly straight line indicates a constant spectrum of the interface reflectance function.

Fig. 6
Fig. 6

Projection points of the surface reflectances. The eight points are indicated with filled squares. The upper limit of the solution band is marked with a +.

Fig. 7
Fig. 7

An estimate of the band of the subsurface reflectance functions of the green plastic ashtray.

Fig. 8
Fig. 8

Normalized curves of the observed surface spectral reflectances of the lemon.

Fig. 9
Fig. 9

Estimate of the band of the subsurface reflectance functions of the lemon.

Fig. 10
Fig. 10

Comparison between the estimated physical limits of the subsurface reflectance function of the green ashtray in two cases, in which a flood lamp and a slide projector were used as the light sources.

Equations (11)

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Y ( θ , λ ) = c I ( θ ) S I ( λ ) E ( λ ) + c S ( θ ) S S ( λ ) E ( λ ) ,
Y ( θ , λ ) = c I ( θ ) L I ( λ ) + c S ( θ ) L S ( λ ) ,
Y ( θ , λ ) E ( λ ) = S ( θ , λ ) = c I ( θ ) S I ( λ ) + c S ( θ ) S S ( λ ) .
s i = p i 1 u 1 + p i 2 u 2 ( i = 1 , 2 , , m ) .
1 = s i 2 = p i 1 u 1 + p i 2 u 2 2 = p i 1 u 1 2 + p i 2 u 2 2 = p i 1 2 + p i 2 2 .
[ S I , S I ] = [ u 1 , u 2 ] T .
T = 1 ( t 1 2 + t 2 2 ) 1 / 2 [ t 1 t 2 t 2 t 1 ] .
r 2 r 1 max 1 i m ( c I i c I i ) ,
r 1 r 2 max 1 λ n ( s I λ s I λ ) ,
r 2 r 1 [ max 1 λ n ( s I λ s I λ ) ] 1 .
max 1 i m ( c I i c I i ) r 2 r 1 [ max 1 λ n ( s I λ s I λ ) ] 1 .

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