Abstract

We extend the applicability of the recently revised Kubelka–Munk (K–M) theory to inhomogeneous optical media by treating inhomogeneous ink penetration of the substrate. We propose a method for describing light propagation in either homogeneous or inhomogeneous layers using series representations for the K–M scattering and absorption coefficients as well as for intensities of the upward and downward light streams. The conventional and matrix expressions for spectral reflectance and transmittance values of optically homogeneous media in the K–M theory are shown to be special cases of the present framework. Three types of ink distribution—homogeneous, linear, and exponential—have been studied. Simulations of spectral reflectance predict a depression of reflectance peaks and reduction of absorption bands characteristic of hue shifts and significant reduction of saturation and, in turn, color gamut.

© 2004 Optical Society of America

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References

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  1. P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  2. P. Kubelka, “New contribution to the optics of intensely light-scattering materials. Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  3. J. L. Saunderson, “Calculation of the color pigmented plastics,” J. Opt. Soc. Am. 32, 727–736 (1942).
    [CrossRef]
  4. P. Kubelka, “New contribution to the optics of intensely light-scattering materials. Part II,” J. Opt. Soc. Am. 44, 330–335 (1954).
    [CrossRef]
  5. A. Mandelis, J. P. Grossman, “Perturbation theory approach to the generalized Kubelka–Munk problem in nonhomogeneous optical media,” Appl. Spectrosc. 46, 737–745 (1992).
    [CrossRef]
  6. L. Yang, B. Kruse, “Revised Kubelka–Munk theory. I. Theory and application,” J. Opt. Soc. Am. A 21, 1933–1941 (2004).
    [CrossRef]
  7. W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Theory II-Diffuse Reflectance (Wiley Interscience, New York, 1966).
  8. S. Rousu, “Differential absorption of offset ink constituents on coated paper,” Ph.D thesis (Laboratory of Paper Chemistry, A°bo Akademi University, Åbo, Finland, 2002).
  9. L. Yang, “Color reproduction of inkjet printing: Model and simulation,” J. Opt. Soc. Am. A 20, 1149–1154 (2003).
    [CrossRef]
  10. E. Allen, “Calculations for colorant formulations,” in Industrical Color Technology: Advance in Chemistry Series 107 (American Chemical Society, Washington, D.C., 1971), pp. 87–119.
  11. W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Color Measurements (Wiley Interscience, New York, 1966).
  12. D. B. Judd, G. Wyszecki, “Physics and psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, New York, 1975).
  13. P. Emmel, “Modèles de prédiction couleur appliqués á l’impression jet d’encre,” Thèse No. 1857 (École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1998).
  14. P. Emmel, R. D. Hersch, “Towards a color prediction model for printed patches,” IEEE Comput. Graphics Appl. 19, 54–60 (1999).
    [CrossRef]
  15. P. Emmel, R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).
  16. L. Yang, “Ink–paper interaction: a study in ink-jet color reproduction,” Ph.D thesis, dissertation No. 806 (Linköping University, Linköping, Sweden, 2003).
  17. O. Norberg, M. Andersson, “Focusing on paper properties in color characterization of printing situations,” in IS&T’s NIP18: International Conference on Digital Printing Technologies (Society for Imaging Science and Technology, Springfield, Va., 2002), pp. 774–776.

2004 (1)

2003 (1)

2000 (1)

P. Emmel, R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

1999 (1)

P. Emmel, R. D. Hersch, “Towards a color prediction model for printed patches,” IEEE Comput. Graphics Appl. 19, 54–60 (1999).
[CrossRef]

1992 (1)

1954 (1)

1948 (1)

1942 (1)

1931 (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Allen, E.

E. Allen, “Calculations for colorant formulations,” in Industrical Color Technology: Advance in Chemistry Series 107 (American Chemical Society, Washington, D.C., 1971), pp. 87–119.

Andersson, M.

O. Norberg, M. Andersson, “Focusing on paper properties in color characterization of printing situations,” in IS&T’s NIP18: International Conference on Digital Printing Technologies (Society for Imaging Science and Technology, Springfield, Va., 2002), pp. 774–776.

Emmel, P.

P. Emmel, R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

P. Emmel, R. D. Hersch, “Towards a color prediction model for printed patches,” IEEE Comput. Graphics Appl. 19, 54–60 (1999).
[CrossRef]

P. Emmel, “Modèles de prédiction couleur appliqués á l’impression jet d’encre,” Thèse No. 1857 (École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1998).

Grossman, J. P.

Hecht, H.

W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Color Measurements (Wiley Interscience, New York, 1966).

W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Theory II-Diffuse Reflectance (Wiley Interscience, New York, 1966).

Hersch, R. D.

P. Emmel, R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

P. Emmel, R. D. Hersch, “Towards a color prediction model for printed patches,” IEEE Comput. Graphics Appl. 19, 54–60 (1999).
[CrossRef]

Judd, D. B.

D. B. Judd, G. Wyszecki, “Physics and psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, New York, 1975).

Kruse, B.

Kubelka, P.

Mandelis, A.

Munk, F.

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Norberg, O.

O. Norberg, M. Andersson, “Focusing on paper properties in color characterization of printing situations,” in IS&T’s NIP18: International Conference on Digital Printing Technologies (Society for Imaging Science and Technology, Springfield, Va., 2002), pp. 774–776.

Rousu, S.

S. Rousu, “Differential absorption of offset ink constituents on coated paper,” Ph.D thesis (Laboratory of Paper Chemistry, A°bo Akademi University, Åbo, Finland, 2002).

Saunderson, J. L.

Wendlandt, W.

W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Theory II-Diffuse Reflectance (Wiley Interscience, New York, 1966).

W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Color Measurements (Wiley Interscience, New York, 1966).

Wyszecki, G.

D. B. Judd, G. Wyszecki, “Physics and psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, New York, 1975).

Yang, L.

Appl. Spectrosc. (1)

IEEE Comput. Graphics Appl. (1)

P. Emmel, R. D. Hersch, “Towards a color prediction model for printed patches,” IEEE Comput. Graphics Appl. 19, 54–60 (1999).
[CrossRef]

J. Imaging Sci. Technol. (1)

P. Emmel, R. D. Hersch, “A unified model for color prediction of halftoned prints,” J. Imaging Sci. Technol. 44, 351–359 (2000).

J. Opt. Soc. Am. (3)

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

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka, F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Other (8)

L. Yang, “Ink–paper interaction: a study in ink-jet color reproduction,” Ph.D thesis, dissertation No. 806 (Linköping University, Linköping, Sweden, 2003).

O. Norberg, M. Andersson, “Focusing on paper properties in color characterization of printing situations,” in IS&T’s NIP18: International Conference on Digital Printing Technologies (Society for Imaging Science and Technology, Springfield, Va., 2002), pp. 774–776.

E. Allen, “Calculations for colorant formulations,” in Industrical Color Technology: Advance in Chemistry Series 107 (American Chemical Society, Washington, D.C., 1971), pp. 87–119.

W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Color Measurements (Wiley Interscience, New York, 1966).

D. B. Judd, G. Wyszecki, “Physics and psychophysics of colorant layers,” in Color in Business, Science and Industry, 3rd ed. (Wiley, New York, 1975).

P. Emmel, “Modèles de prédiction couleur appliqués á l’impression jet d’encre,” Thèse No. 1857 (École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1998).

W. Wendlandt, H. Hecht, Reflectance Spectroscopy: Theory II-Diffuse Reflectance (Wiley Interscience, New York, 1966).

S. Rousu, “Differential absorption of offset ink constituents on coated paper,” Ph.D thesis (Laboratory of Paper Chemistry, A°bo Akademi University, Åbo, Finland, 2002).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a linear ink distribution (with backing). Ink penetration begins from the surface of the substrate at z=D and ends at z=0. The remaining part of the paper sheet (below z=0) acts as a backing for the inked part.

Fig. 2
Fig. 2

Schematic diagram of a linear ink distribution in a freely suspended layer. Ink penetration begins from the surface of the substrate at z=D and ends at z=0.

Fig. 3
Fig. 3

Spectral dependence of depths for status transition of primary colors (linear ink distribution).

Fig. 4
Fig. 4

Convergence of the computed spectral reflectance values of primary inks with respect to different order of series expansion. The corresponding values computed analytically with Eqs. (30) and (31) are shown with dots (not visible under the n=10 curve).

Fig. 5
Fig. 5

Computed spectral reflectance of primary inks with and without considering ink–paper mixing in the case of homogeneous ink distribution.

Fig. 6
Fig. 6

Computed spectral reflectance of primary inks with and without considering ink–paper mixing in the case of linear ink distribution.

Fig. 7
Fig. 7

Ink distribution with respect to depth of ink penetration z in cases of homogeneous and linear ink distribution.

Fig. 8
Fig. 8

Computed spectral reflectance of inked sheets with either homogeneous or linear ink distribution. The sheets have the same thickness and contain the same total amounts of inks.

Equations (109)

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dIdz=(S+K)I-SJ,
dJdz=-(S+K)J+SI
K(z)=μαa,S(z)=μαs/2.
μ(z)=[s(z)/a(z)]1/2s(z)a(z)1,otherwise,
Ja(0)=Ia(0)Rg.
Ib(D)=I0(1-r0)+Jb(D)r1,
Ia(D)R=Ia(D)r0+Jb(D)(1-r1),
ρ(Z)=ρzD,(0<zD).
ai=Ki/μiαi,si=2Si/μiαi,
ap=Kp/μpαp,sp=2Sp/μpαp,
μi(λ)=2Si(λ)Ki(λ)1/2,2Si(λ)Ki(λ)1,otherwise
μp(λ)=2Sp(λ)Kp(λ)1/2,2Sp(λ)Kp(λ)1,otherwise
aip=ap+ρ(Z)ai,
sip=sp+ρ(Z)si,
μip(λ)=sp(λ)+ρ(Z)si(λ)ap(λ)+ρ(Z)ai(λ)1/2,sip(λ)>aip(λ)1,otherwise.
Kip(Z)=μipαipaip=μipμpαipαp Kp+ρ(Z) μipμiαipαi Ki=μiph(Kp, Ki, Z),
Sip(Z)=μipαipsip/2=μipμpαipαp Sp+ρ(Z) μipμiαipαi Si=μiph(Sp, Si, Z),
h(x, y, Z)=1μpαipαp x+ρ(Z)μiαipαi y,
sip/aip=2Sip/Kip=2h(Sp, Si, Z)/h(Kp, Ki, Z),
μip(λ)=2h(Sp(λ), Si(λ), Z)h(Kp(λ), Ki(λ), Z)1/2,2h(Sp, Si, Z)>h(Kp, Ki, Z)1,otherwise.
Kip(λ)=[2h(Sp, Si, Z)h(Kp, Ki, Z)]1/2,2h(Sp, Si, Z) >h(Kp, Ki, Z)h(Kp, Ki, Z),otherwise,
Sip(λ)=2h2(Sp, Si, Z)/Kip,2h(Sp, Si, Z) >h(Kp, Ki, Z)h(Sp, Si, Z),otherwise.
f(Z)=Kip(λ)=[2h(Sp, Si, Z)h(Kp, Ki, Z)]1/2,2h(Sp, Si, Z) >h(Kp, Ki, Z)h(Kp, Ki, Z),otherwise,
g(Z)=Sip(λ)=2h2(Sp, Si, Z)/Kip,2h(Sp, Si, Z) >h(Kp, Ki, Z)h(Sp, Si, Z),otherwise.
f(Z)=f(0)++1n! f(n)(0)Zn+=ϕ0(0)++ϕn(0)Zn+,
ϕn(0)=1n! f(n)(0),
g(Z)=g(0)++1m! g(m)(0)Zm+=ψ0(0)++ψm(0)Zm+,
ψm(0)=1m! g(m)(0),
Kip(Z)=l=0ϕlZl,
Sip(Z)=l=0ψlZl.
I=n=0anZn,
J=m=0bmZm.
1Dn=1nanZn-1=l=0m=0(ϕl+ψl)amZl+m-l=0m=0ψlbmZl+m,
1Dn=1nbnZn-1=-l=0m=0(ϕl+ψl)bmZl+m+l=0m=0ψlamZl+m.
an=Dnl=0n-1(ϕl+ψl)an-l-1-l=0n-1ψlbn-l-1,
bn=Dn-l=0n-1(ϕl+ψl)bn-l-1+l=0n-1ψlan-l-1.
b0=Rga0.
n=0an=I0(1-r0)+r1n=1bn,
I0R=I0r0+(1-r1)n=0bn.
R=r0+(1-r0)(1-r1) m=0bmm=0(am-bmr1).
b0r2a0,
I0T=a0(1-r2),
T=a0(1-r0)(1-r2)/m=0(am-bmr1).
ρ(ξ)=μiμpαiαp(2Sp-Kp)(Ki-2Si).
ρ(Z)=ρ1Z,
ξ=1ρ1μiμpαiαp(2Sp-Kp)(Ki-2Si).
ρ(Z)=ρ1,(0<Z1).
h(x, y, Z)=1μpαipμp x+ρ1μiαipαi y
K=ϕ0=[2h(Sp, Si, 1)h(Kp, Ki, 1)]1/2,2h(Sp, Si, 1)>h(Kp, Ki, 1)h(Kp, Ki, 1),otherwise,
S=ψ0=2h2(Sp, Si, 1)/ϕ0,2h(Sp, Si, 1)>h(Kp, Ki, 1)h(Sp, Si, 1),otherwise,
ϕn=0,ψn=0(n1).
an=Dn [(K+S)an-1-Sbn-1],
bn=Dn [-(K+S)bn-1+San-1].
anbn=Dn(K+S)-SS-(K+S) an-1bn-1.
anbn=Dnn!(K+S)-SS-(K+S)na0b0.
n=0ann=0bn=exp(K+S)-SS-(K+S)D a0b0=tuvw a0b0.
I(Z=1)=n=0an,
J(Z=1)=n=0bn,
Rbulk=v+Rgwt+Rgu=(R-Rg)exp(-2bSD)-R(1-RRg)R(R-Rg)exp(-2bSD)-(1-RRg),
R=1+K/S-(K2/S2+2K/S)1/2,
b=(1-R2)/2R.
R=r0+(1-r0)(1-r1) Rbulk1-r1Rbulk=r0+(1-r0)(1-r1)×(R-Rg)exp(-2bSD)-R(1-RRg)(R-r1)(R-Rg)exp(-2bSD)-(1-Rr1)(1-RRg).
T=(1-r0)(1-r1)×(1-R2)exp(-bSD)(1-Rr1)2-(R-r1)2exp(-2bSD).
ρ(Z)=CZ+ρ0,
Z=Z/ξ,(0<Z<ξ)(Z-ξ)/(1-ξ),(ξ<Z<1).
h(x, y, Z)=1μpαipαp x+CZ+ρ0μiαipαi y.
ϕ0=h(Kp, Ki, 0),ψ0=h(Sp, Si, 0),
ϕ1=h(Kp, Ki, 0)=Cμiαipαi Ki,
ψ1=h(Sp, Si, 0)=Cμiαipαi Si,
ϕn=0,(n>1),ψn=0,(n>1),
ϕ0=[2h(Kp, Ki, 0)h(Sp, Si, 0)]1/2,
ψ0=2h2(Sp, Si, 0)/ϕ0,
ϕ1=[h(Kp, Ki, 0)h(Sp, Si, 0)+h(Kp, Ki, 0)h(Sp, Si, 0)]/ϕ0,
ψ1=[4h(Sp, Si, 0)h(Sp, Si, 0)-ψ0ϕ1]/ϕ0,
ϕn=m=0nh(m)(Kp, Ki, 0)h(n-m)(Sp, Si, 0)m!(n-m)!-m=1n-1ϕmϕn-m2/ϕ0,
ψn=m=0n2h(m)(Sp, Si, 0)h(n-m)(Sp, Si, 0)m!(n-m)!-m=1nϕmψn-m/ϕ0,
h(x, y, 0)=1μpαipαp x+Cμiαipαi y,
h(x, y, 0)=Cμiαipαi y,
h(n)(x, y, 0)=0,(n2).
ρ(Z)=ρ1exp[β(z-D)]=ρ1exp[γ(Z-1)],
0zD,
ρ(Z)=C exp[δ(Z-1)],0Z1,
h(x, y, 0)=1μpαipαp x+exp(-δ) Cμiαipαi y,
h(n)(x, y, 0)=exp(-δ) δnCμiαipαi y,(n1).
ϕ0=1upαipαp Kp+exp(-δ) Cuiαipαi Ki,
ψ0=1upαipαp Sp+exp(-δ) Cuiαipαi Si,
ϕn=exp(-δ) δnn!Cuiαipαi Ki,
ψn=exp(-δ) δnn!Cuiαipαi Si.
f(Z)=Kip(Z)=[2h(Kp, Ki, Z)h(Sp, Si, Z)]1/2,
g(Z)=Sip(Z)=2h2(Sp, Si, Z)/f(Z),
h(x, y, Z)=1μpαipαp x+ρ(Z)μiαipαi y.
f2(Z)=2h(Kp, Ki, Z)h(Sp, Si, Z),
f(Z)g(Z)=2h2(Sp, Si, Z).
[f(Z)g(Z)](n)=m=0nn!m!(n-m)! f(m)(Z)g(n-m)(Z)=f(Z)g(n)(Z)+m=1nn!m!(n-m)! f(m)×(Z)g(n-m)(Z).
2[h2(Sp, Si, Z)](n)
=2m=0nn!m!(n-m)! h(m)(Sp, Si, Z)h(n-m)(Sp, Si, Z).
g(n)(Z)=m=0n2n!m!(n-m)! h(m)(Sp, Si, Z)h(n-m)×(Sp, Si, Z)-m=1nn!m!(n-m)!×f(m)(Z)g(n-m)(Z)/f(Z).
f(n)(Z)=m=0nn!m!(n-m)! h(m)(Kp, Ki, Z)h(n-m)×(Sp, Si, Z)-12m=1n-1n!m!(n-m)!×f(m)(Z)f(n-m)(Z)/f(Z).
ϕ0=f(Z),ψ0=g(Z),
ϕm=f(m)m! (Z),ψn=g(n)n! (Z),
ϕn=m=0n1m!(n-m)! h(m)(Kp, Ki, Z)h(n-m)×(Sp, Si, Z)-12m=1n-1ϕ(m)ϕ(n-m)/ϕ0,
ψn=m=0n2m!(n-m)! h(m)(Sp, Si, Z)h(n-m)×(Sp, Si, Z)-m=1nϕ(m)ψ(n-m)/ϕ0.
ρ(Z)=CZ+ρ0
h(x, y, 0)=1μpαipαp x+ρ0μiαipαi y,
h(1)(x, y, 0)=Cμiαipαi y,
h(n)(x, y, 0)=0,(n2).
ρ(Z)=C exp[δ(Z-1)],(0Z1),
h(x, y, 0)=1μpαipαp x+exp(-δ) Cμiαipαi y,
h(n)(x, y, 0)=exp(-δ) δnCμiαipαi y,(n1).

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