Abstract

We present a theoretical analysis of what we believe to be a new color constancy method that inputs two chromaticities of an identical surface taken under two blackbody illuminations. By using the Planck formula for modeling spectra of outdoor illumination and by assuming that a narrowband camera sensitivity function is sufficiently narrow, surface colors can be estimated mathematically. Experiments with simulation and real data have been conducted to evaluate the effectiveness of the method. The results showed that although this method is a perfect vehicle for simulation data, it produces significant errors with real data. A thorough investigation of the cause of errors indicates how important the assumptions on both blackbody illuminations and narrowband camera sensitivities are to the method. Finally, we discuss the robustness of our method and the limitation of solving color constancy using the illumination constraint.

© 2007 Optical Society of America

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    [Crossref]
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    [Crossref]
  6. G. D. Finlayson and G. Schaefer, "Solving for color constancy using a constrained dichromatic reflection model," Int. J. Comput. Vis. 42, 127-144 (2001).
    [Crossref]
  7. R. T. Tan, K. Nishino, and K. Ikeuchi, "Color constancy through inverse intensity-chromaticity space," J. Opt. Soc. Am. A 21, 321-334 (2004).
    [Crossref]
  8. D. H. Brainard and W. T. Freeman, "Bayesian color constancy," J. Opt. Soc. Am. A 14, 1393-1411 (1997).
    [Crossref]
  9. G. D. Finlayson, S. D. Hordley, and P. M. Hubel, "Color by correlation: a simple, unifying, framework for color constancy," IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
    [Crossref]
  10. S. Tominaga, S. Ebisui, and B. A. Wandell, "Scene illuminant classification: brighter is better," J. Opt. Soc. Am. A 18, 55-64 (2001).
    [Crossref]
  11. S. Tominaga and B. A. Wandell, "Natural scene-illuminant estimation using the sensor correlation," Proc. IEEE 90, 42-56 (2002).
    [Crossref]
  12. G. D. Finlayson and S. D. Hordley, "Color constancy at a pixel," J. Opt. Soc. Am. A 18, 253-264 (2001).
    [Crossref]
  13. J. M. Geusebroek, R. Boomgaard, S. Smeulders, and T. Gevers, "A physical basis for color constancy," in Proceedings of the First European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 3-6.
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    [Crossref]
  15. Y. Ohta and Y. Hayashi, "Recovery of illuminant and surface colors from images based on the CIE daylight," in Proceedings of the Third European Conference on Computer Vision (CGIV, 1994), Vol. II, pp. 235-246.
  16. G. D. Finlayson, B. V. Funt, and K. Barnard, "Color constancy under varying illumination," in Proceedings of IEEE International Conference on Computer Vision (IEEE, 1995), pp. 720-725.
    [Crossref]
  17. K. Barnard, G. Finlayson, and B. Funt, "Color constancy for scenes with varying illumination," Comput. Vis. Image Underst. 65, 311-321 (1997).
    [Crossref]
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2004 (1)

2002 (2)

S. Tominaga and B. A. Wandell, "Natural scene-illuminant estimation using the sensor correlation," Proc. IEEE 90, 42-56 (2002).
[Crossref]

J. M. Geusebroek, R. Boomgaard, S. Smeulders, and T. Gevers, "A physical basis for color constancy," in Proceedings of the First European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 3-6.

2001 (4)

G. D. Finlayson and S. D. Hordley, "Color constancy at a pixel," J. Opt. Soc. Am. A 18, 253-264 (2001).
[Crossref]

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, "Color by correlation: a simple, unifying, framework for color constancy," IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[Crossref]

S. Tominaga, S. Ebisui, and B. A. Wandell, "Scene illuminant classification: brighter is better," J. Opt. Soc. Am. A 18, 55-64 (2001).
[Crossref]

G. D. Finlayson and G. Schaefer, "Solving for color constancy using a constrained dichromatic reflection model," Int. J. Comput. Vis. 42, 127-144 (2001).
[Crossref]

2000 (1)

1997 (2)

K. Barnard, G. Finlayson, and B. Funt, "Color constancy for scenes with varying illumination," Comput. Vis. Image Underst. 65, 311-321 (1997).
[Crossref]

D. H. Brainard and W. T. Freeman, "Bayesian color constancy," J. Opt. Soc. Am. A 14, 1393-1411 (1997).
[Crossref]

1995 (1)

G. D. Finlayson, B. V. Funt, and K. Barnard, "Color constancy under varying illumination," in Proceedings of IEEE International Conference on Computer Vision (IEEE, 1995), pp. 720-725.
[Crossref]

1994 (1)

Y. Ohta and Y. Hayashi, "Recovery of illuminant and surface colors from images based on the CIE daylight," in Proceedings of the Third European Conference on Computer Vision (CGIV, 1994), Vol. II, pp. 235-246.

1992 (1)

1991 (1)

B. V. Funt, M. Drew, and J. Ho, "Color constancy from mutual reflection," Int. J. Comput. Vis. 6, 5-24 (1991).
[Crossref]

1990 (1)

H. C. Lee, "Illuminant color from shading," Proc. SPIE 1250, 233-244 (1990).

1989 (1)

1988 (1)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

1986 (2)

1971 (1)

1964 (1)

1933 (1)

D. B. Judd, "Sensibility to color-temperature change as a function of temperature," J. Opt. Soc. Am. 23, 127-134 (1933).
[Crossref]

Barnard, K.

K. Barnard, G. Finlayson, and B. Funt, "Color constancy for scenes with varying illumination," Comput. Vis. Image Underst. 65, 311-321 (1997).
[Crossref]

G. D. Finlayson, B. V. Funt, and K. Barnard, "Color constancy under varying illumination," in Proceedings of IEEE International Conference on Computer Vision (IEEE, 1995), pp. 720-725.
[Crossref]

Boomgaard, R.

J. M. Geusebroek, R. Boomgaard, S. Smeulders, and T. Gevers, "A physical basis for color constancy," in Proceedings of the First European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 3-6.

Brainard, D. H.

Drew, M.

B. V. Funt, M. Drew, and J. Ho, "Color constancy from mutual reflection," Int. J. Comput. Vis. 6, 5-24 (1991).
[Crossref]

D'Zmura, M.

Ebisui, S.

Finlayson, G.

K. Barnard, G. Finlayson, and B. Funt, "Color constancy for scenes with varying illumination," Comput. Vis. Image Underst. 65, 311-321 (1997).
[Crossref]

Finlayson, G. D.

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, "Color by correlation: a simple, unifying, framework for color constancy," IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[Crossref]

G. D. Finlayson and G. Schaefer, "Solving for color constancy using a constrained dichromatic reflection model," Int. J. Comput. Vis. 42, 127-144 (2001).
[Crossref]

G. D. Finlayson and S. D. Hordley, "Color constancy at a pixel," J. Opt. Soc. Am. A 18, 253-264 (2001).
[Crossref]

G. D. Finlayson, B. V. Funt, and K. Barnard, "Color constancy under varying illumination," in Proceedings of IEEE International Conference on Computer Vision (IEEE, 1995), pp. 720-725.
[Crossref]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Freeman, W. T.

Funt, B.

K. Barnard, G. Finlayson, and B. Funt, "Color constancy for scenes with varying illumination," Comput. Vis. Image Underst. 65, 311-321 (1997).
[Crossref]

Funt, B. V.

G. D. Finlayson, B. V. Funt, and K. Barnard, "Color constancy under varying illumination," in Proceedings of IEEE International Conference on Computer Vision (IEEE, 1995), pp. 720-725.
[Crossref]

B. V. Funt, M. Drew, and J. Ho, "Color constancy from mutual reflection," Int. J. Comput. Vis. 6, 5-24 (1991).
[Crossref]

Geusebroek, J. M.

J. M. Geusebroek, R. Boomgaard, S. Smeulders, and T. Gevers, "A physical basis for color constancy," in Proceedings of the First European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 3-6.

Gevers, T.

J. M. Geusebroek, R. Boomgaard, S. Smeulders, and T. Gevers, "A physical basis for color constancy," in Proceedings of the First European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 3-6.

Hayashi, Y.

Y. Ohta and Y. Hayashi, "Recovery of illuminant and surface colors from images based on the CIE daylight," in Proceedings of the Third European Conference on Computer Vision (CGIV, 1994), Vol. II, pp. 235-246.

Ho, J.

B. V. Funt, M. Drew, and J. Ho, "Color constancy from mutual reflection," Int. J. Comput. Vis. 6, 5-24 (1991).
[Crossref]

Hordley, S. D.

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, "Color by correlation: a simple, unifying, framework for color constancy," IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[Crossref]

G. D. Finlayson and S. D. Hordley, "Color constancy at a pixel," J. Opt. Soc. Am. A 18, 253-264 (2001).
[Crossref]

Hubel, P. M.

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, "Color by correlation: a simple, unifying, framework for color constancy," IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[Crossref]

Ikeuchi, K.

Judd, D. B.

Land, E. H.

Lee, H. C.

Lennie, P.

MacAdam, D. L.

Marchant, J. A.

McCann, J. J.

Nishino, K.

Ohta, Y.

Y. Ohta and Y. Hayashi, "Recovery of illuminant and surface colors from images based on the CIE daylight," in Proceedings of the Third European Conference on Computer Vision (CGIV, 1994), Vol. II, pp. 235-246.

Onyango, C. M.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Schaefer, G.

G. D. Finlayson and G. Schaefer, "Solving for color constancy using a constrained dichromatic reflection model," Int. J. Comput. Vis. 42, 127-144 (2001).
[Crossref]

Smeulders, S.

J. M. Geusebroek, R. Boomgaard, S. Smeulders, and T. Gevers, "A physical basis for color constancy," in Proceedings of the First European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 3-6.

Tan, R. T.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Tominaga, S.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Wandel, B. A.

Wandell, B. A.

S. Tominaga and B. A. Wandell, "Natural scene-illuminant estimation using the sensor correlation," Proc. IEEE 90, 42-56 (2002).
[Crossref]

S. Tominaga, S. Ebisui, and B. A. Wandell, "Scene illuminant classification: brighter is better," J. Opt. Soc. Am. A 18, 55-64 (2001).
[Crossref]

Wyszecky, G.

Comput. Vis. Image Underst. (1)

K. Barnard, G. Finlayson, and B. Funt, "Color constancy for scenes with varying illumination," Comput. Vis. Image Underst. 65, 311-321 (1997).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

G. D. Finlayson, S. D. Hordley, and P. M. Hubel, "Color by correlation: a simple, unifying, framework for color constancy," IEEE Trans. Pattern Anal. Mach. Intell. 23, 1209-1221 (2001).
[Crossref]

Int. J. Comput. Vis. (2)

B. V. Funt, M. Drew, and J. Ho, "Color constancy from mutual reflection," Int. J. Comput. Vis. 6, 5-24 (1991).
[Crossref]

G. D. Finlayson and G. Schaefer, "Solving for color constancy using a constrained dichromatic reflection model," Int. J. Comput. Vis. 42, 127-144 (2001).
[Crossref]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (9)

Proc. IEEE (1)

S. Tominaga and B. A. Wandell, "Natural scene-illuminant estimation using the sensor correlation," Proc. IEEE 90, 42-56 (2002).
[Crossref]

Proc. SPIE (1)

H. C. Lee, "Illuminant color from shading," Proc. SPIE 1250, 233-244 (1990).

Other (4)

J. M. Geusebroek, R. Boomgaard, S. Smeulders, and T. Gevers, "A physical basis for color constancy," in Proceedings of the First European Conference on Colour in Graphics, Image and Vision (CGIV, 2002), pp. 3-6.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Y. Ohta and Y. Hayashi, "Recovery of illuminant and surface colors from images based on the CIE daylight," in Proceedings of the Third European Conference on Computer Vision (CGIV, 1994), Vol. II, pp. 235-246.

G. D. Finlayson, B. V. Funt, and K. Barnard, "Color constancy under varying illumination," in Proceedings of IEEE International Conference on Computer Vision (IEEE, 1995), pp. 720-725.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Problem statement. We estimate color temperatures from image chromaticities taken under two blackbody illuminations.

Fig. 2
Fig. 2

Shape of the evaluation function Θ r , given an arbitrary T 1 . A solution of T 2 always exists and can be calculated by bracketing. We let the initial value of t 2 be one side of the brackets and find the other side by going up or down the slope until the sign of Θ r changes.

Fig. 3
Fig. 3

Shape of T 2 r T 2 g around the true solution. If T 1 increases/decreases, T 2 r T 2 g increases/decreases. Therefore, the solution can be calculated by bracketing.

Fig. 4
Fig. 4

Camera sensitivities used in the first experiments in Subsection 4B. They are the sensitivities of bandpass filters (MellesGriot 03FIV119, 03FIV111, 03FIV004).

Fig. 5
Fig. 5

Plots of the estimation error against the violation of the blackbody assumption; “ΔT1” and “ΔT2” express the estimation error. JND chromaticity difference (5.5 mired) is also shown. To limit the estimation error to lower than 5.5 mired, the violation should be under about 0.1%.

Fig. 6
Fig. 6

Data used in the fourth experiments in Subsection 4B. (a) Virtual camera sensitivities. Here “5” and “20” are Gaussian functions whose standard deviation is 5 and 20 nm , respectively. (b) Reflectances. Three reflectances are “dark skin,” “light skin,” and “Green” of the Macbeth ColorChecker. “Dark skin” varies linearly around wavelengths of a camera sensitivity compared with “green” and “light skin.”

Fig. 7
Fig. 7

Plot of the estimation error against the violation of the narrowband sensitivity assumption. The more the bandwidth of a camera sensitivity (the standard deviation σ of a Gaussian function) grows, the larger the estimation error becomes. Three reflectances, “dark skin,” “light skin,” and “green” were tested. The speed of the error growth depends on the reflectance.

Tables (5)

Tables Icon

Table 1 Average Estimation Error in 126 Experiments Using Simulation Data

Tables Icon

Table 2 Average and Standard Deviation of Estimation Errors in 500 Experiments with Different Initial Values

Tables Icon

Table 3 Average Estimation Error in 168 Experiments Using Real Data

Tables Icon

Table 4 Difference between Blackbody and Real Spectra a

Tables Icon

Table 5 Difference between Real Illumination Colors and Colors Calculated from the Straight-Line Illumination Model

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

i c = s c e c , c = { r , g } .
i r = I R I B , i g = I G I B .
I c = τ Ω S ( λ ) E ( λ ) q c ( λ ) d λ , c = { R , G , B } ,
τ S c E c , ( S c = S ( λ c ) , E c = E ( λ c ) ) ,
e r ( T ) = M ( λ R , T ) M ( λ B , T ) , e g ( T ) = M ( λ G , T ) M ( λ B , T ) ,
M ( λ , T ) = c 1 λ 5 [ exp ( c 2 λ T ) 1 ] 1 ,
e r ( T ) = k r Φ B ( T ) Φ R ( T ) ( k r = λ B 5 λ R 5 ) ,
e g ( T ) = k g Φ B ( T ) Φ G ( T ) ( k g = λ B 5 λ G 5 ) ,
Φ R ( T ) = exp ( T λ R ) 1 ,
Φ G ( T ) = exp ( T λ G ) 1 ,
Φ B ( T ) = exp ( T λ B ) 1 .
Θ r ( T 1 , T 2 ) = i r 1 Φ R ( T 1 ) Φ B ( T 2 ) i r 2 Φ R ( T 2 ) Φ B ( T 1 ) = 0 ,
Θ g ( T 1 , T 2 ) = i g 1 Φ G ( T 1 ) Φ B ( T 2 ) i g 2 Φ G ( T 2 ) Φ B ( T 1 ) = 0 .
i r 1 e r 1 i r 2 e r 2 = 0 ,
i g 1 e g 1 i g 2 e g 2 = 0 ,
1 e r = Φ R ( T ) k r Φ B ( T ) ,
1 e g = Φ G ( T ) k g Φ B ( T ) .
i r 2 Φ B ( T 1 ) λ R i r 1 Φ R ( T 1 ) λ B > 0 .
i g 2 Φ B ( T 1 ) λ G i g 1 Φ G ( T 1 ) λ B > 0 .
Θ r T 2 = i r 1 λ B Φ R ( T 1 ) exp ( T 2 λ B ) i r 2 λ R exp ( T 2 λ R ) Φ B ( T 1 ) .
T 2 = λ R λ B λ R λ B ( log ( i r 2 λ R Φ B ( T 1 ) ) log ( i r 1 λ B Φ R ( T 1 ) ) ) .
2 Θ r T 2 2 = ( 1 λ B 1 λ R ) i r 1 λ B Φ R ( T 1 ) exp ( T 2 λ B ) + i r 1 λ B Θ r T 2 .
2 Θ r T 2 2 = ( 1 λ B 1 λ R ) i r 1 λ B Φ R ( T 1 ) exp ( T 2 λ B ) .
Θ r = Θ r ( t 1 , t 2 r ) + ( i r 1 Φ B ( t 2 r ) λ R exp ( t 1 λ R ) i r 2 Φ R ( t 2 r ) λ B exp ( t 1 λ B ) ) Δ t 1 + ( i r 1 Φ R ( t 1 ) λ B exp ( t 2 r λ B ) i r 2 Φ B ( t 1 ) λ R exp ( t 2 r λ R ) ) Δ t 2 r ,
Θ r = i r 1 Φ R ( t 1 ) Φ B ( t 2 r ) ( H r ( t 1 ) Δ t 1 H r ( t 2 r ) Δ t 2 r ) ,
H r ( t ) = exp ( t λ R ) λ R Φ R ( t ) exp ( t λ B ) λ B Φ B ( t ) .
Θ g = i g 1 Φ G ( t 1 ) Φ B ( t 2 g ) ( H g ( t 1 ) Δ t 1 H g ( t 2 g ) Δ t 2 g ) .
T 2 r T 2 g = ( H r ( t 1 ) H r ( t 2 r ) H g ( t 1 ) H g ( t 2 g ) ) Δ t 1 + ( t 2 r t 2 g ) .
T 2 r T 2 g = ( H r ( T ̂ 1 ) H r ( T ̂ 2 ) H g ( T ̂ 1 ) H g ( T ̂ 2 ) ) Δ t 1 .
I ( T 1 , T 2 ) = H r ( T 1 ) H g ( T 2 ) H g ( T 1 ) H r ( T 2 )
M ( λ , T ) c 1 λ 5 exp ( c 2 λ T ) 1 .
e r = m e g A ,
1 e g = m ( 1 e r ) + c .

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