Abstract

A unifying framework is presented for algorithms that use the bands of a multispectral image to segment the image at material (i.e., reflectance) boundaries while ignoring spatial inhomogeneities incurred by accidents of lighting and viewing geometry. The framework assumes that the visual stimulus (image field) from a uniformly colored object is the sum of a small number of terms, each term being the product of a spatial and a spectral part. Based on this assumption, several quantities depending on the reflected light can be computed that are spatially invariant within object boundaries. For an image field either from two light sources on a matte surface or from a single light source on a dielectric surface with highlights, the invariants are the components of the unit normal to the plane in color space spanned by the pixels from the object. In some limited cases the normal to the plane can be used to estimate spectral-reflectance parameters of the object. However, in general the connection of color-constancy theories with image segmentation by object color is a difficult problem. The concomitant constraints on segmentation and color-constancy algorithms are discussed in light of this fact.

© 1990 Optical Society of America

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References

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  1. R. A. Schowengerdt, Techniques for Image Processing and Classification in Remote Sensing (Academic, New York, 1983), pp. 154–159.
  2. J. M. Rubin, W. A. Richards, “Color vision and image intensity: when are changes material?” Biol. Cybern. 45, 215–226 (1982).
    [CrossRef]
  3. G. J. Klinker, “A physical approach to color image understanding,” Ph.D. dissertation, Doc. CMU-CS-88-161 (Carnegie-Mellon University, Pittsburgh, Pa., May1988).
  4. S. A. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10, 210–218 (1985).
    [CrossRef]
  5. P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988).
    [PubMed]
  6. P. P. Nikolaev, “Algorithms for color discrimination of objects according to the reactions of logarithmic receptors,” Biofizika 33, 517–521 (1988).
    [PubMed]
  7. Nikolaev defines the rank as 2M if none of the g functions is constant or linearly dependent on the others; if one of the g functions is constant or linearly dependent on the others, then his rank is 2M− 1. However, there seems to be no difficulty in adopting the simpler definition that the rank is M.
  8. Purists will note that we have defined nˆas perpendicular to all tristimulus vectors Q(x″, y″) from a particular reflecting object, without having a prior definition of distance in the space. This is permissible because the relation nˆ·Q(x″,y″)=0that apparently defines perpendicularity actually defines coplanarity of Q(x, y), Q(x′, y′), and Q(x″, y″) when the substitution [Eq. (4)] is made for nˆ. The dot product is not between two ordinary vectors in tristimulus space but between an ordinary vector and an axial vector nˆ(which in two dimensions is the cross product of ordinary vectors). That no metric structure is needed to define this particular dot product can be verified algebraically by noting that the relation nˆ·Q(x″,y″)=0is immune to any linear nonsingular transformation on all the Q vectors.
  9. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986).
    [CrossRef] [PubMed]
  10. G. Buchsbaum, “A spatial processor model for object color perception,” J. Franklin Inst. 310, 1–26 (1980).
    [CrossRef]
  11. M. H. Brill, “Computer-simulated object-color recognizer,” Progress Rep. 122 (MIT Research Laboratory of Electronics, Cambridge, Mass., 1980), pp. 214–221.
  12. M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
    [CrossRef]
  13. M. D’Zmura, P. Lennie, “Mechanisms of color constancy,” J. Opt. Soc. Am. A 3, 29–33 (1986).
  14. R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation and color constancy,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 245–255 (1990).
    [CrossRef]
  15. S. Tominaga, B. A. Wandell, “Standard surface-reflection model and illuminant estimation,” J. Opt. Soc. Am. A 6, 576–584 (1989).
    [CrossRef]

1989 (1)

1988 (2)

P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988).
[PubMed]

P. P. Nikolaev, “Algorithms for color discrimination of objects according to the reactions of logarithmic receptors,” Biofizika 33, 517–521 (1988).
[PubMed]

1986 (3)

1985 (1)

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

1982 (1)

J. M. Rubin, W. A. Richards, “Color vision and image intensity: when are changes material?” Biol. Cybern. 45, 215–226 (1982).
[CrossRef]

1980 (1)

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

Bajcsy, R.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation and color constancy,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 245–255 (1990).
[CrossRef]

Brill, M. H.

M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
[CrossRef]

M. H. Brill, “Computer-simulated object-color recognizer,” Progress Rep. 122 (MIT Research Laboratory of Electronics, Cambridge, Mass., 1980), pp. 214–221.

Buchsbaum, G.

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

D’Zmura, M.

Klinker, G. J.

G. J. Klinker, “A physical approach to color image understanding,” Ph.D. dissertation, Doc. CMU-CS-88-161 (Carnegie-Mellon University, Pittsburgh, Pa., May1988).

Lee, S. W.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation and color constancy,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 245–255 (1990).
[CrossRef]

Lennie, P.

Leonardis, A.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation and color constancy,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 245–255 (1990).
[CrossRef]

Maloney, L. T.

Nikolaev, P. P.

P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988).
[PubMed]

P. P. Nikolaev, “Algorithms for color discrimination of objects according to the reactions of logarithmic receptors,” Biofizika 33, 517–521 (1988).
[PubMed]

Richards, W. A.

J. M. Rubin, W. A. Richards, “Color vision and image intensity: when are changes material?” Biol. Cybern. 45, 215–226 (1982).
[CrossRef]

Rubin, J. M.

J. M. Rubin, W. A. Richards, “Color vision and image intensity: when are changes material?” Biol. Cybern. 45, 215–226 (1982).
[CrossRef]

Schowengerdt, R. A.

R. A. Schowengerdt, Techniques for Image Processing and Classification in Remote Sensing (Academic, New York, 1983), pp. 154–159.

Shafer, S. A.

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

Tominaga, S.

Wandell, B. A.

West, G.

M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
[CrossRef]

Biofizika (2)

P. P. Nikolaev, “Monocular color discrimination of nonplanar objects under various illumination conditions,” Biofizika 33, 140–144 (1988).
[PubMed]

P. P. Nikolaev, “Algorithms for color discrimination of objects according to the reactions of logarithmic receptors,” Biofizika 33, 517–521 (1988).
[PubMed]

Biol. Cybern. (1)

J. M. Rubin, W. A. Richards, “Color vision and image intensity: when are changes material?” Biol. Cybern. 45, 215–226 (1982).
[CrossRef]

Color Res. Appl. (2)

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

M. H. Brill, G. West, “Chromatic adaptation and color constancy: a possible dichotomy,” Color Res. Appl. 11, 196–204 (1986).
[CrossRef]

J. Franklin Inst. (1)

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

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

Other (6)

M. H. Brill, “Computer-simulated object-color recognizer,” Progress Rep. 122 (MIT Research Laboratory of Electronics, Cambridge, Mass., 1980), pp. 214–221.

R. Bajcsy, S. W. Lee, A. Leonardis, “Color image segmentation and color constancy,” in Perceiving, Measuring, and Using Color, M. H. Brill, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1250, 245–255 (1990).
[CrossRef]

R. A. Schowengerdt, Techniques for Image Processing and Classification in Remote Sensing (Academic, New York, 1983), pp. 154–159.

G. J. Klinker, “A physical approach to color image understanding,” Ph.D. dissertation, Doc. CMU-CS-88-161 (Carnegie-Mellon University, Pittsburgh, Pa., May1988).

Nikolaev defines the rank as 2M if none of the g functions is constant or linearly dependent on the others; if one of the g functions is constant or linearly dependent on the others, then his rank is 2M− 1. However, there seems to be no difficulty in adopting the simpler definition that the rank is M.

Purists will note that we have defined nˆas perpendicular to all tristimulus vectors Q(x″, y″) from a particular reflecting object, without having a prior definition of distance in the space. This is permissible because the relation nˆ·Q(x″,y″)=0that apparently defines perpendicularity actually defines coplanarity of Q(x, y), Q(x′, y′), and Q(x″, y″) when the substitution [Eq. (4)] is made for nˆ. The dot product is not between two ordinary vectors in tristimulus space but between an ordinary vector and an axial vector nˆ(which in two dimensions is the cross product of ordinary vectors). That no metric structure is needed to define this particular dot product can be verified algebraically by noting that the relation nˆ·Q(x″,y″)=0is immune to any linear nonsingular transformation on all the Q vectors.

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Figures (7)

Fig. 1
Fig. 1

One origin of rank-2 image fields: matte objects under two spectrally different light sources.

Fig. 2
Fig. 2

Plane in tristimulus space spanned by pixels of a matte object under two light sources.

Fig. 3
Fig. 3

Rank-2 image fields involving a single light source: (a) a dielectric object with a highlight, (b) a secondary reflection from one matte object to another.

Fig. 4
Fig. 4

Rank-3 image fields from three light sources on matte objects.

Fig. 5
Fig. 5

Role of reference colors in estimating spectral reflectance parameters for a highlighted object under one light source.

Fig. 6
Fig. 6

Geometric representation of the D’Zmura–Lennie method. The quantity a2/a3, characteristic of the spectral reflectance and independent of the illumination, is determined from angle ϕ.

Fig. 7
Fig. 7

Method of Tominaga and Wandell for determining the illuminant tristimulus vector. Each depicted plane contains the tristimulus vectors of light reflected from a single object with a highlight.

Equations (16)

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F ( λ , x , y ) = g 1 ( x , y ) S 1 ( λ ) r ( λ ) + g 2 ( x , y ) S 2 ( λ ) r ( λ ) .
F ( λ , x , y ) = g 1 ( x , y ) H 1 ( λ ) + g 2 ( x , y ) H 2 ( λ ) .
F ( λ , x , y ) = g 1 ( x , y ) H 1 ( λ ) + + g M ( x , y ) H M ( λ ) .
F ( λ , x , y ) = g 1 ( x , y ) S 1 ( λ ) r k ( λ ) + g 2 ( x , y ) S 2 ( λ ) r k ( λ ) ,
Q j ( x , y ) = g 1 ( x , y ) S 1 r k q j + g 2 ( x , y ) S 2 r k q j ,
N = [ | Q 2 Q 3 Q 2 Q 3 | , | Q 3 Q 1 Q 3 Q 1 | , | Q 1 Q 2 Q 1 Q 2 | ] ,
n 1 n 3 = | Q 2 Q 3 Q 2 Q 3 | / | Q 1 Q 2 Q 1 Q 2 | = | S 1 r q 2 S 1 r q 3 S 2 r q 2 S 2 r q 3 | / | S 1 r q 1 S 1 r q 2 S 2 r q 1 S 2 r q 2 | .
| g 1 g 2 g 1 g 2 |
n 2 n 3 = | S 1 r q 3 S 1 r q 1 S 2 r q 3 S 2 r q 1 | / | S 1 r q 1 S 1 r q 2 S 2 r q 1 S 2 r q 2 | .
F H ( λ , x , y ) = g 1 ( x , y ) S ( λ ) r ( λ ) + g 2 ( x , y ) S ( λ ) ,
F S ( λ , x , y ) = g 1 ( x , y ) S ( λ ) r ( λ ) + g 2 ( x , y ) S ( λ ) R ( λ ) r ( λ ) ,
F C ( λ , x , y ) = g ( x , y ) [ b 1 ( x , y ) ρ 1 ( λ ) + b 2 ( x , y ) ρ 2 ( λ ) ] S ( λ ) ,
| Q i Q j Q k Q i Q j Q k Q i Q j Q k | ,
Q H j ( x , y ) = g 1 ( x , y ) [ a 2 ρ 2 S q j + a 3 ρ 3 S q j ] + [ g 2 ( x , y ) + a 1 g 1 ( x , y ) ] S q j .
T 1 Q H = [ g 2 ( x , y ) + a 1 g 1 ( x , y ) g 1 ( x , y ) a 2 g 1 ( x , y ) a 3 ] .
a 2 / a 3 = ( T 1 Q H ) 2 / ( T 1 Q H ) 3 .

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