Abstract

Spectral sharpening is a method for developing camera or other optical-device sensor functions that are more narrowband than those in hardware, by means of a linear transform of sensor functions. The utility of such a transform is that many computer vision and color-correction algorithms perform better in a sharpened space, and thus such a space can be used as an intermediate representation for carrying out calculations. In this paper we consider how one may sharpen sensor functions such that the transformed sensors are all positive. We show that constrained optimization can be used to produce positive sensors in two fundamentally different ways: by constraining the coefficients in the transform or by constraining the functions directly. In the former method, we prove that convexity can be used to constrain the solution exactly. In a sense, we are continuing the work of MacAdam and of Pearson and Yule, who formed positive combinations of the color-matching functions. However, the advantage of the spectral sharpening approach is that not only can we produce positive curves, but the process is “steerable” in that we can produce positive curves with as good or better properties for sharpening within a given set of sharpening intervals. At base, however, it is positive colors in the transformed space that are the prime objective. Therefore we also carry out sharpening of sensor curves governed not by positivity of the curves themselves but of colors resulting from them. Curves that result have negative lobes but generate positive colors. We find that this type of constrained sharpening generates the best results, which are almost as good as for unconstrained sharpening but without the penalty of negative colors. All methods discussed may be used with any number of sensors.

© 2000 Optical Society of America

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References

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  1. G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
    [CrossRef]
  2. K. Barnard, B. V. Funt, “Experiments in sensor sharpening for color constancy,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 43–46.
  3. M. S. Drew, M. H. Brill, “Color from shape from color: a simple formalism with known light sources,” J. Opt. Soc. Am. A 17, 1371–1381 (2000).
    [CrossRef]
  4. B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
    [CrossRef]
  5. M. L. Pearson, J. A. Yule, “Transformations of color mixture functions without negative portions,” J. Color Appearance 2, 30–35 (1973). Inspiration for their work is attributed to MacAdam.
  6. P. L. Vora, J. E. Farrell, J. D. Tietz, D. H. Brainard, “Digital color cameras. 2. spectral response,” (Hewlett–Packard Laboratories, Palo Alto, Calif., 1997). http://www.hpl.hp.com/techreports/97/HPL-97-54.html .
  7. G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
    [CrossRef]
  8. M. H. Brill, G. D. Finlayson, P. M. Hubel, W. Thornton, “Prime wavelengths and color imaging,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 33–42.
  9. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).
  10. S. M. Newhall, D. Nickerson, D. B. Judd, “Final report of the OSA subcommittee on the spacing of the Munsell colors,” J. Opt. Soc. Am.33, 385–418 (1943).
    [CrossRef]
  11. M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).
  12. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).
  13. D. A. Forsyth, “A novel approach to color constancy,” in Proceedings of the International Conference on Computer Vision ’88 (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 9–18.
  14. G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1034–1038 (1996).
    [CrossRef]

2000 (1)

1996 (2)

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1034–1038 (1996).
[CrossRef]

G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
[CrossRef]

1995 (1)

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

1994 (2)

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

1973 (1)

M. L. Pearson, J. A. Yule, “Transformations of color mixture functions without negative portions,” J. Color Appearance 2, 30–35 (1973). Inspiration for their work is attributed to MacAdam.

Barnard, K.

K. Barnard, B. V. Funt, “Experiments in sensor sharpening for color constancy,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 43–46.

Brainard, D. H.

P. L. Vora, J. E. Farrell, J. D. Tietz, D. H. Brainard, “Digital color cameras. 2. spectral response,” (Hewlett–Packard Laboratories, Palo Alto, Calif., 1997). http://www.hpl.hp.com/techreports/97/HPL-97-54.html .

Brill, M. H.

M. S. Drew, M. H. Brill, “Color from shape from color: a simple formalism with known light sources,” J. Opt. Soc. Am. A 17, 1371–1381 (2000).
[CrossRef]

M. H. Brill, G. D. Finlayson, P. M. Hubel, W. Thornton, “Prime wavelengths and color imaging,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 33–42.

Drew, M. S.

Farrell, J. E.

P. L. Vora, J. E. Farrell, J. D. Tietz, D. H. Brainard, “Digital color cameras. 2. spectral response,” (Hewlett–Packard Laboratories, Palo Alto, Calif., 1997). http://www.hpl.hp.com/techreports/97/HPL-97-54.html .

Finlayson, G. D.

G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
[CrossRef]

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1034–1038 (1996).
[CrossRef]

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

M. H. Brill, G. D. Finlayson, P. M. Hubel, W. Thornton, “Prime wavelengths and color imaging,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 33–42.

Forsyth, D. A.

D. A. Forsyth, “A novel approach to color constancy,” in Proceedings of the International Conference on Computer Vision ’88 (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 9–18.

Funt, B. V.

G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
[CrossRef]

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

G. D. Finlayson, M. S. Drew, B. V. Funt, “Spectral sharpening: sensor transformations for improved color constancy,” J. Opt. Soc. Am. A 11, 1553–1563 (1994).
[CrossRef]

K. Barnard, B. V. Funt, “Experiments in sensor sharpening for color constancy,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 43–46.

Gershon, R.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

Hubel, P. M.

M. H. Brill, G. D. Finlayson, P. M. Hubel, W. Thornton, “Prime wavelengths and color imaging,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 33–42.

Iwan, L. S.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Judd, D. B.

S. M. Newhall, D. Nickerson, D. B. Judd, “Final report of the OSA subcommittee on the spacing of the Munsell colors,” J. Opt. Soc. Am.33, 385–418 (1943).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

Newhall, S. M.

S. M. Newhall, D. Nickerson, D. B. Judd, “Final report of the OSA subcommittee on the spacing of the Munsell colors,” J. Opt. Soc. Am.33, 385–418 (1943).
[CrossRef]

Nickerson, D.

S. M. Newhall, D. Nickerson, D. B. Judd, “Final report of the OSA subcommittee on the spacing of the Munsell colors,” J. Opt. Soc. Am.33, 385–418 (1943).
[CrossRef]

Pearson, M. L.

M. L. Pearson, J. A. Yule, “Transformations of color mixture functions without negative portions,” J. Color Appearance 2, 30–35 (1973). Inspiration for their work is attributed to MacAdam.

Stiles, W. S.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

Thornton, W.

M. H. Brill, G. D. Finlayson, P. M. Hubel, W. Thornton, “Prime wavelengths and color imaging,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 33–42.

Tietz, J. D.

P. L. Vora, J. E. Farrell, J. D. Tietz, D. H. Brainard, “Digital color cameras. 2. spectral response,” (Hewlett–Packard Laboratories, Palo Alto, Calif., 1997). http://www.hpl.hp.com/techreports/97/HPL-97-54.html .

Vora, P. L.

P. L. Vora, J. E. Farrell, J. D. Tietz, D. H. Brainard, “Digital color cameras. 2. spectral response,” (Hewlett–Packard Laboratories, Palo Alto, Calif., 1997). http://www.hpl.hp.com/techreports/97/HPL-97-54.html .

Vrhel, M. J.

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

Wyszecki, G.

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

Yule, J. A.

M. L. Pearson, J. A. Yule, “Transformations of color mixture functions without negative portions,” J. Color Appearance 2, 30–35 (1973). Inspiration for their work is attributed to MacAdam.

Color Res. Appl. (2)

G. D. Finlayson, B. V. Funt, “Coefficient channels: derivation and relationship to other theoretical studies,” Color Res. Appl. 21, 87–96 (1996).
[CrossRef]

M. J. Vrhel, R. Gershon, L. S. Iwan, “Measurement and analysis of object reflectance spectra,” Color Res. Appl. 19, 4–9 (1994).

IEEE Trans. Pattern Anal. Mach. Intell. (2)

B. V. Funt, G. D. Finlayson, “Color constant color indexing,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 522–529 (1995).
[CrossRef]

G. D. Finlayson, “Color in perspective,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 1034–1038 (1996).
[CrossRef]

J. Color Appearance (1)

M. L. Pearson, J. A. Yule, “Transformations of color mixture functions without negative portions,” J. Color Appearance 2, 30–35 (1973). Inspiration for their work is attributed to MacAdam.

J. Opt. Soc. Am. A (2)

Other (7)

K. Barnard, B. V. Funt, “Experiments in sensor sharpening for color constancy,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 43–46.

P. L. Vora, J. E. Farrell, J. D. Tietz, D. H. Brainard, “Digital color cameras. 2. spectral response,” (Hewlett–Packard Laboratories, Palo Alto, Calif., 1997). http://www.hpl.hp.com/techreports/97/HPL-97-54.html .

G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas, 2nd ed. (Wiley, New York, 1982).

D. A. Forsyth, “A novel approach to color constancy,” in Proceedings of the International Conference on Computer Vision ’88 (IEEE Computer Society Press, Los Alamitos, Calif., 1998), pp. 9–18.

M. H. Brill, G. D. Finlayson, P. M. Hubel, W. Thornton, “Prime wavelengths and color imaging,” in 6th Color Imaging Conference: Color, Science, Systems and Applications (Society for Imaging Science & Technology/Society for Information Display, Springfield, Va., 1998), pp. 33–42.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

S. M. Newhall, D. Nickerson, D. B. Judd, “Final report of the OSA subcommittee on the spacing of the Munsell colors,” J. Opt. Soc. Am.33, 385–418 (1943).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Original sensors for Kodak DCS-420 digital camera.

Fig. 2
Fig. 2

Sensors sharpened by sensor-based L2L2 unconstrained sharpening.

Fig. 3
Fig. 3

Sensors sharpened by sensor-based L2L1 unconstrained sharpening. Original sensors are also shown, by the dotted curves.  

Fig. 4
Fig. 4

Convexity drives an L1L1 constrained-coefficient solution to the boundary of the set of possible coefficients and thence to the original sensor itself. Shown here is the simplest, p=3, case. The surface S represents the contribution of a combination of sensors to the sharpening interval.

Fig. 5
Fig. 5

Sensors sharpened by constrained L1L1 sharpening with the sensor result constrained to nonnegativity. Original sensors are also shown, by the dotted curves.

Fig. 6
Fig. 6

Sensors sharpened by L2L2 sharpening with the sensor result constrained to be nonnegative. Original sensors are also shown, by the dotted curves.

Fig. 7
Fig. 7

Sensors sharpened by constrained L2L1 sharpening with coefficients constrained to be nonnegative. Original sensors are also shown, by the dotted curves.

Fig. 8
Fig. 8

Sensors sharpened by L2L1 sharpening with the output sensor constrained to nonnegativity. Original sensors are also shown, by the dotted curves.

Fig. 9
Fig. 9

Sensors sharpened by a data-driven L2L2 sharpening with transformed convex hull points constrained to nonnegativity. Original sensors are also shown, by the dotted curves.

Tables (4)

Tables Icon

Table 1 Energy Concentration in Sharpening Intervals for Original DCS-420 Sensors and for Sensors Sharpened with Unconstrained L2L2 Optimization a

Tables Icon

Table 2 Cross talk κ for Original DCS-420 Sensors and for Sensors Sharpened with Unconstrained L2L2 Optimization

Tables Icon

Table 3 Ratio of Energy Concentration in Sharpening Intervals for Sensors Sharpened with Unconstrained L2L1 Optimization Compared with Original DCS-420 Sensors

Tables Icon

Table 4 Ratio of Energy Concentration in Sharpening Intervals for Sharpened Sensors Compared with Original DCS-420 Sensors

Equations (44)

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

ρ=T-1D E,ETρ E,S.
Tρ=D E,ETρ E,S.
r(λ)g(λ)b(λ)=t11t12t13t21t22t23t31t32t33 r(λ)g(λ)b(λ).
ρ=D E,EρE,s.
Q=QT.
minλϕk[Q(λ)t]2+μλω[Q(λ)t]2-1,
k=1  p,
Λ(α)=λαQt(λ)Q(λ)=QtΔαQ.
Λ(ϕk)t+μ[Λ(ω)t]=0.
λω[Q(λ)t]2=ttΛ(ω)t=1.
[Λ(ω)]-1Λ(ϕk)t=-μt.
=100 λψk|qk(λ)|2λω|qk(λ)|2
κ=cos-1|qitqj|qiqj,
minλϕk[Q(λ)t]2-μλωQ(λ)t-1,
k=1  p.
Λ(ϕk)t=μf (ω),
t=μ[Λ(ϕk)]-1f (ω).
f (ω)tt=1.
minλϕk[Q(λ)t]
withconstraints λω[Q(λ)t]=1L1 normalizationt0 nonnegativecoefficients.
minλϕk[Q(λ)t]
withconstraintsω[Q(λ)t]=1L1 normalizationQ(λ)t0 nonnegativesensorresult.
Λ(ω)=QtQ.
χq2tq1=t2tΛ(ω)t1.
q=aq1+bq2
a2+b2+2abχ1.
a,b,[0  1],a+b1.
u1=Δψkq1,u2=Δψkq2.
au1+bu22(au1+bu2)2,
minλϕk[Q(λ)t]2
withconstraintsλω[Q(λ)t]2=1,L2normalizationt0,nonnegativecoefficients.
minλϕk[Q(λ)t]2
withconstraintsλω[Q(λ)t]2=1,L2normalizationQ(λ)t0,nonnegativesensorresult.
minλϕk[Q(λ)t]2
withconstraintsλω[Q(λ)t]=1,L1 normalizationt0,nonnegativecoefficients.
minλϕk[Q(λ)t]2
withconstraintsλω[Q(λ)t]=1,L1 normalizationQ(λ)t0,nonnegativesensorresult
C(λ)=E(λ)S(λ),
ρ=λωC(λ)Q(λ).
R=CQ.
Q=QT,
R=CQT=RT.
minλϕk[Q(λ)tk]ν
withcontraintsλω[Q(λ)tk]ν=1,Lν normalizationR˜tk0,nonnegativeRGBvalues,

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