Abstract

The scattering of light within paper can affect the color of a halftone image. Because of scattering, a photon may enter and emerge from the paper in different regions of the halftone microstructure. The microstructure of a halftone print consists of a number of dots of ink of varying color and size. The color of the halftone image is the partitive mixture of the colors of the microstructure—the colors of the dots, the colors of the dot overlaps, and the color of the bare paper. In the present study the tristimulus values of the color of a halftone print are calculated in terms of the halftone microstructure. The analysis includes the effects of the scattering of light within the paper, an effect known as optical dot gain or the Yule–Nielsen effect. The tristimulus values are expressed as the trace of the product of two matrices—one a matrix that expresses the different colors of the microstructure that contribute to the partitive mixture and is a function of the ink transmittances and the paper reflectance and the other a matrix that expresses the amount of each color that contributes. The relative amount of each color is equal to the probability of the scattering process that gives rise to that color. These probabilities are calculated in terms of the dot shapes and sizes and in terms of the photon migration in the paper. As a result of the scattering, several “new” colors contribute to the partitive mixture.

© 1998 Optical Society of America

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References

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  1. G. L. Rogers, “Neugebauer revisited: random dots in halftone screening,” Color Res. Appl. 23, 104–113 (1998).
    [CrossRef]
  2. G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 45, 643–656 (1997).
  3. S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).
  4. J. S. Arney, C. D. Arney, P. G. Engeldrum, “Modeling the Yule–Nielsen halftone effect,” J. Imaging Sci. Technol. 40, 233–238 (1996).
  5. J. S. Arney, P. G. Engeldrum, H. Zeng, “An expanded Murray–Davies model of tone reproduction in halftone imaging,” J. Imaging Sci. Technol. 39, 502–508 (1995).
  6. F. R. Ruckdeschel, O. G. Hauser, “Yule–Nielsen effect in printing: a physical analysis,” Appl. Opt. 17, 3376–3383 (1978).
    [CrossRef] [PubMed]
  7. R. W. G. Hunt, The Reproduction of Color in Photography, Printing and Television, 4th ed. (Fountain Press, Tolworth, UK, 1987); W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, New York, 1991).
  8. H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE—Society of Photo-optical Instrumentation Engineers, Bellingham, Wash., 1997).
  9. M. Mahy, P. Delabastita, “Inversion of the Neugebauer equations,” Color Res. Appl. 21, 404–411 (1996); H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mahrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).
    [CrossRef]
  10. J. A. C. Yule, W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Tech. Assoc. Graphic Arts Proc. 3, 65–76 (1951).
  11. P. G. Engeldrum, “The color gamut limits of halftone printing with and without the paper spread function,” J. Imaging Sci. Technol. 40, 239–244 (1996).
  12. P. G. Engeldrum, “The color between the dots,” J. Imaging Sci. Technol. 38, 545–551 (1994).
  13. Y. Shiraiwa, T. Mizuno, “Equation to predict colors of halftone prints considering the optical property of paper,” J. Imaging Sci. Technol. 37, 385–391 (1993).
  14. R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).
  15. T. N. Pappas, D. L. Neuhoff, “Model-based halftoning,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 244–255 (1991).
    [CrossRef]
  16. J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974).
  17. P. Oittinen, “Limits of microscopic print quality,” in Advances in Printing Science and Technology, W. H. Banks, ed. (Pentech, London, 1982), Vol. 16; P. G. Engeldrum, B. Pridham, “Application of turbid medium theory to paper spread function measurements,” Tech. Assoc. Graphic Arts Proc. 47, 339–347 (1995).
  18. J. S. Arney, C. D. Arney, M. Katsube, “An MTF analysis of papers,” J. Imaging Sci. Technol. 40, 19–25 (1996).
  19. G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42(4) (1998).
  20. J. S. Arney, “A probability description of the Yule–Nielsen Effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).
  21. F. W. Billmeyer, M. Saltzman, Principles of Color Technology, 2nd ed. (Wiley, New York, 1981).

1998 (2)

G. L. Rogers, “Neugebauer revisited: random dots in halftone screening,” Color Res. Appl. 23, 104–113 (1998).
[CrossRef]

G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42(4) (1998).

1997 (3)

J. S. Arney, “A probability description of the Yule–Nielsen Effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).

G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 45, 643–656 (1997).

S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).

1996 (4)

J. S. Arney, C. D. Arney, P. G. Engeldrum, “Modeling the Yule–Nielsen halftone effect,” J. Imaging Sci. Technol. 40, 233–238 (1996).

M. Mahy, P. Delabastita, “Inversion of the Neugebauer equations,” Color Res. Appl. 21, 404–411 (1996); H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mahrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).
[CrossRef]

P. G. Engeldrum, “The color gamut limits of halftone printing with and without the paper spread function,” J. Imaging Sci. Technol. 40, 239–244 (1996).

J. S. Arney, C. D. Arney, M. Katsube, “An MTF analysis of papers,” J. Imaging Sci. Technol. 40, 19–25 (1996).

1995 (1)

J. S. Arney, P. G. Engeldrum, H. Zeng, “An expanded Murray–Davies model of tone reproduction in halftone imaging,” J. Imaging Sci. Technol. 39, 502–508 (1995).

1994 (1)

P. G. Engeldrum, “The color between the dots,” J. Imaging Sci. Technol. 38, 545–551 (1994).

1993 (1)

Y. Shiraiwa, T. Mizuno, “Equation to predict colors of halftone prints considering the optical property of paper,” J. Imaging Sci. Technol. 37, 385–391 (1993).

1978 (1)

1951 (1)

J. A. C. Yule, W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Tech. Assoc. Graphic Arts Proc. 3, 65–76 (1951).

Arney, C. D.

J. S. Arney, C. D. Arney, M. Katsube, “An MTF analysis of papers,” J. Imaging Sci. Technol. 40, 19–25 (1996).

J. S. Arney, C. D. Arney, P. G. Engeldrum, “Modeling the Yule–Nielsen halftone effect,” J. Imaging Sci. Technol. 40, 233–238 (1996).

Arney, J. S.

J. S. Arney, “A probability description of the Yule–Nielsen Effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).

J. S. Arney, C. D. Arney, P. G. Engeldrum, “Modeling the Yule–Nielsen halftone effect,” J. Imaging Sci. Technol. 40, 233–238 (1996).

J. S. Arney, C. D. Arney, M. Katsube, “An MTF analysis of papers,” J. Imaging Sci. Technol. 40, 19–25 (1996).

J. S. Arney, P. G. Engeldrum, H. Zeng, “An expanded Murray–Davies model of tone reproduction in halftone imaging,” J. Imaging Sci. Technol. 39, 502–508 (1995).

Billmeyer, F. W.

F. W. Billmeyer, M. Saltzman, Principles of Color Technology, 2nd ed. (Wiley, New York, 1981).

Dainty, J. C.

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974).

Delabastita, P.

M. Mahy, P. Delabastita, “Inversion of the Neugebauer equations,” Color Res. Appl. 21, 404–411 (1996); H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mahrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).
[CrossRef]

Engeldrum, P. G.

J. S. Arney, C. D. Arney, P. G. Engeldrum, “Modeling the Yule–Nielsen halftone effect,” J. Imaging Sci. Technol. 40, 233–238 (1996).

P. G. Engeldrum, “The color gamut limits of halftone printing with and without the paper spread function,” J. Imaging Sci. Technol. 40, 239–244 (1996).

J. S. Arney, P. G. Engeldrum, H. Zeng, “An expanded Murray–Davies model of tone reproduction in halftone imaging,” J. Imaging Sci. Technol. 39, 502–508 (1995).

P. G. Engeldrum, “The color between the dots,” J. Imaging Sci. Technol. 38, 545–551 (1994).

Gustavson, S.

S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).

Hauser, O. G.

Hunt, R. W. G.

R. W. G. Hunt, The Reproduction of Color in Photography, Printing and Television, 4th ed. (Fountain Press, Tolworth, UK, 1987); W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, New York, 1991).

Kang, H. R.

H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE—Society of Photo-optical Instrumentation Engineers, Bellingham, Wash., 1997).

Katsube, M.

J. S. Arney, C. D. Arney, M. Katsube, “An MTF analysis of papers,” J. Imaging Sci. Technol. 40, 19–25 (1996).

Mahy, M.

M. Mahy, P. Delabastita, “Inversion of the Neugebauer equations,” Color Res. Appl. 21, 404–411 (1996); H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mahrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).
[CrossRef]

Mizuno, T.

Y. Shiraiwa, T. Mizuno, “Equation to predict colors of halftone prints considering the optical property of paper,” J. Imaging Sci. Technol. 37, 385–391 (1993).

Neuhoff, D. L.

T. N. Pappas, D. L. Neuhoff, “Model-based halftoning,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 244–255 (1991).
[CrossRef]

Nielsen, W. J.

J. A. C. Yule, W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Tech. Assoc. Graphic Arts Proc. 3, 65–76 (1951).

Oittinen, P.

P. Oittinen, “Limits of microscopic print quality,” in Advances in Printing Science and Technology, W. H. Banks, ed. (Pentech, London, 1982), Vol. 16; P. G. Engeldrum, B. Pridham, “Application of turbid medium theory to paper spread function measurements,” Tech. Assoc. Graphic Arts Proc. 47, 339–347 (1995).

Pappas, T. N.

T. N. Pappas, D. L. Neuhoff, “Model-based halftoning,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 244–255 (1991).
[CrossRef]

Rogers, G. L.

G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42(4) (1998).

G. L. Rogers, “Neugebauer revisited: random dots in halftone screening,” Color Res. Appl. 23, 104–113 (1998).
[CrossRef]

G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 45, 643–656 (1997).

Ruckdeschel, F. R.

Saltzman, M.

F. W. Billmeyer, M. Saltzman, Principles of Color Technology, 2nd ed. (Wiley, New York, 1981).

Shaw, R.

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974).

Shiraiwa, Y.

Y. Shiraiwa, T. Mizuno, “Equation to predict colors of halftone prints considering the optical property of paper,” J. Imaging Sci. Technol. 37, 385–391 (1993).

Ulichney, R.

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

Yule, J. A. C.

J. A. C. Yule, W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Tech. Assoc. Graphic Arts Proc. 3, 65–76 (1951).

Zeng, H.

J. S. Arney, P. G. Engeldrum, H. Zeng, “An expanded Murray–Davies model of tone reproduction in halftone imaging,” J. Imaging Sci. Technol. 39, 502–508 (1995).

Appl. Opt. (1)

Color Res. Appl. (2)

M. Mahy, P. Delabastita, “Inversion of the Neugebauer equations,” Color Res. Appl. 21, 404–411 (1996); H. E. J. Neugebauer, “Die theoretischen Grundlagen des Mahrfarbenbuchdrucks,” Z. Tech. Phys. (Leipzig) 36, 75–89 (1937).
[CrossRef]

G. L. Rogers, “Neugebauer revisited: random dots in halftone screening,” Color Res. Appl. 23, 104–113 (1998).
[CrossRef]

J. Imaging Sci. Technol. (10)

G. L. Rogers, “Optical dot gain in a halftone print,” J. Imaging Sci. Technol. 45, 643–656 (1997).

S. Gustavson, “Color gamut of halftone reproduction,” J. Imaging Sci. Technol. 41, 283–290 (1997).

J. S. Arney, C. D. Arney, P. G. Engeldrum, “Modeling the Yule–Nielsen halftone effect,” J. Imaging Sci. Technol. 40, 233–238 (1996).

J. S. Arney, P. G. Engeldrum, H. Zeng, “An expanded Murray–Davies model of tone reproduction in halftone imaging,” J. Imaging Sci. Technol. 39, 502–508 (1995).

P. G. Engeldrum, “The color gamut limits of halftone printing with and without the paper spread function,” J. Imaging Sci. Technol. 40, 239–244 (1996).

P. G. Engeldrum, “The color between the dots,” J. Imaging Sci. Technol. 38, 545–551 (1994).

Y. Shiraiwa, T. Mizuno, “Equation to predict colors of halftone prints considering the optical property of paper,” J. Imaging Sci. Technol. 37, 385–391 (1993).

J. S. Arney, C. D. Arney, M. Katsube, “An MTF analysis of papers,” J. Imaging Sci. Technol. 40, 19–25 (1996).

G. L. Rogers, “Optical dot gain: lateral scattering probabilities,” J. Imaging Sci. Technol. 42(4) (1998).

J. S. Arney, “A probability description of the Yule–Nielsen Effect,” J. Imaging Sci. Technol. 41, 633–640 (1997).

Tech. Assoc. Graphic Arts Proc. (1)

J. A. C. Yule, W. J. Nielsen, “The penetration of light into paper and its effect on halftone reproduction,” Tech. Assoc. Graphic Arts Proc. 3, 65–76 (1951).

Other (7)

R. W. G. Hunt, The Reproduction of Color in Photography, Printing and Television, 4th ed. (Fountain Press, Tolworth, UK, 1987); W. F. Schreiber, Fundamentals of Electronic Imaging Systems (Springer-Verlag, New York, 1991).

H. R. Kang, Color Technology for Electronic Imaging Devices (SPIE—Society of Photo-optical Instrumentation Engineers, Bellingham, Wash., 1997).

F. W. Billmeyer, M. Saltzman, Principles of Color Technology, 2nd ed. (Wiley, New York, 1981).

R. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).

T. N. Pappas, D. L. Neuhoff, “Model-based halftoning,” in Human Vision, Visual Processing, and Digital Display II, B. E. Rogowitz, M. H. Brill, J. P. Allebach, eds., Proc. SPIE1453, 244–255 (1991).
[CrossRef]

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974).

P. Oittinen, “Limits of microscopic print quality,” in Advances in Printing Science and Technology, W. H. Banks, ed. (Pentech, London, 1982), Vol. 16; P. G. Engeldrum, B. Pridham, “Application of turbid medium theory to paper spread function measurements,” Tech. Assoc. Graphic Arts Proc. 47, 339–347 (1995).

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Figures (3)

Fig. 1
Fig. 1

Difference in color between s=0.5 (scattering) and s=1.0 (no scattering) for cyan and yellow inks, with μcyan=0.5 and μyellow varied from 0 to 1. Color differences are in CIE Lab* units.

Fig. 2
Fig. 2

Gamut of the cyan and yellow inks and the colors obtained with μcyan=0.5 and μyellow varied from 0 to 1 for s=1 (no scattering) and s=0.5 (scattering). Contour lines are lines of constant L* spaced 2.5 units apart.

Fig. 3
Fig. 3

Color difference DE* for s=0.0, s=0.25, s=0.5, and s=0.75 with μcyan=0.5 and μyellow varied from 0 to 1.

Tables (1)

Tables Icon

Table 1 Bracket Integrals for Two Inks

Equations (93)

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R(x, y)=RpT(x, y)H(x-x, y-y)×T(x, y)dxdy,
T(x, y)=[1-(1-τ1)C1(x, y)]×[1-(1-τ2)C2(x, y)],
Ci(x, y)=circx2+y2di * gi,
circ(ρ/di)=1,ρdi0,ρ>di,
g1(x, y)=n,m=-N/2N/2δ(x-nr)δ(y-nr).
g2(x, y)=n,m=-N/2N/2δ(x-nr cos θ+mr sin θ)×δ(y-nr sin θ-mr cos θ).
R¯=1(Nr)2R(x, y)dxdy.
UHV1(Nr)2U(x, y)H(x-x, y-y)×V(x, y)dxdydxdy,
R¯=RpTHT.
tk=100τk,
ck(x, y)=1Ck(x, y)00,
u(1)=10-11.
Tk(x, y)=ck(x, y)u(1)tk.
T(x, y)=T1(x, y)T2(x, y).
T(x, y)=c(x, y)u(2)t,
c=c1c2,u(2)=u(1)u(1),t=t1t2.
T(x, y)=T1(x, y)T2(x, y)Tn(x, y).
R¯=RpT˜HT,
R¯=nmR¯nm.
H=c˜Hc
P=u˜(2)Hu(2),
R¯=Rpt˜Pt.
H=HHC1HC2HC1C2C1HC1HC1C1HC2C1HC1C2C2HC2HC1C2HC2C2HC1C2C1C2HC1C2HC1C1C2HC2C1C2HC1C2.
UHV=VHU;
Zi=nm|Jinm|2H˜nm,
nmPnm=1.
μkZk=P(k|k),
Pk=μkP(k|k),
P22=C2HC2-C2HC1C2-C1C2HC2+C1C2HC1C2.
P23=C2HC1C2-C1C2HC1C2,
P32=C1C2HC2-C1C2HC1C2,
P33=C1C2HC1C2,
C2HC2=μ22Z2=P2.
P2=P22+P23+P32+P33.
μiμiZi1,
Zi=μi-s,
1Z121μ1μ2,
Z12Z1Z2.
H=1μ1μ2μ1μ2μ1μ12-sμ1μ2μ12-sμ2μ2μ1μ2μ22-sμ1μ22-sμ1μ2μ12-sμ2μ1μ22-s(μ1μ2)2-s.
X=S(λ)R¯(λ)x¯(λ)dλ,
X=ijklPklS(λ)tki(λ)tlj(λ)Rp(λ)x¯(λ)dλ.
X=ijPijS(λ)tjj(λ)tii(λ)Rp(λ)x¯(λ)dλ.
Xji=S(λ)tjj(λ)tii(λ)Rp(λ)x¯(λ)dλ.
X=ijPijXji.
X=Tr(PX),Y=Tr(PY),Z=Tr(PZ).
H=1μ1μ2μ1μ2μ1μ1μ1μ2μ1μ2μ2μ1μ2μ2μ1μ2μ1μ2μ1μ2μ1μ2μ1μ2,
P=1-μ1-μ2+μ1μ20000μ1-μ1μ20000μ2-μ1μ20000μ1μ2.
X=(1-μ1)(1-μ2)Xp+μ1(1-μ2)X1+(1-μ1)μ2X2+μ1μ2X12,
UHV=1(Nr)2U˜*(k)H˜(k)V˜(k)d2k,
H˜(k)=F{H(x, y)},
F{gi}=1r2nmδ(k-knm(i)),
knm(1)=nr, mr,
knm(2)=nrcos θ-mrsin θ, nrsin θ+mrcos θ.
Fcircx2+y2di=dikJ1(2πkdi),
F{Ci}=μinmJinmδ(k-knm(i)),
μi=πdi2/r2
Jinm=J1(2πn2+m2di/r)πn2+m2di/r,
Ji00=1,nmJinm=1/μi.
C1(x, y)C2(x, y)=μ1μ2nmnmJ1nmJ2nm×exp[-2πi(knm(1)+knm(2))·r],
F{C1C2}=μ1μ2nmnmJ1nmJ2nm× exp[2πi(k-knm(1)-knm(2))·r]d2r.
F{C1C2}=μ1μ2nmnmJ1nmJ2nm×δ(k-knm(1)-knm(2)).
1N2δ(n-n cos θ+m sin θ)×δ(m-n sin θ-m cos θ),
1Nδ(n-m)=δnm,
δnm=1,n=m0,nm.
HC1=μ1,HC2=μ2.
δ(knm(1)+knm(2)),
HC1C2=μ1μ2.
δ(k-knm(i))H˜(|k|)δ(k-knm(j))d2k.
H˜nmδ(knm(i)-knm(j)),
H˜nm=H˜(|knm|)=H˜(n2+m2/r)
n=m=n=m=0.
C1HC2=μ1μ2.
n=n,m=m.
C1HC1=μ12Z1,C2HC2=μ22Z2,
Zi=nm|Jinm|2H˜nm.
δ(k-knm(i))H˜(|k|)δ(k-knm(1)-knm(2))d2k.
H˜nmδ(knm(i)-knm(1)-knm(2)).
n=n,m=m,n=0,m=0
ifi=1,
n=n,m=m,n=0,m=0
ifi=2.
C1HC1C2=μ12μ2Z1,C2HC1C2=μ1μ22Z2.
δ(k-knm(1)-knm(2))H˜(|k|)×δ(k-knm(1)-knm(2))d2k.
H˜nmnmδ(knm(1)-knm(1)+knm(2)-knm(2)),
H˜nmnm=H˜(|knm(1)+knm(2)|).
n=n,m=m,n=n,m=m.
Z12=nmnm|J1nm|2|J2nm|2H˜nmnm,
C1C2HC1C2=μ12μ22Z12.
Z1Z2=nmnm|J1nm|2|J2nm|2H˜nmH˜nm,
Z12=nmnm|J1nm|2|J2nm|2H˜nmnm,
H˜nmH˜00=H˜nm00=H˜00nm.
Z12Z1Z2.
Jinm=w˜i(knm(i))w˜i(0).

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