Download Chapter 3c, Physical models for color prediction

Digital Color Imaging
HANDBOOK
Edited by
Gaurav Sharma
School of Electrical Engineering and Computer Science
Kyungpook National Univ.
3.8 Models for halftoned samples
‹
Printing devices
– Certain fixed density or leaving the substrate unprinted, only
bilevel
– Obtaining intermediate tone level by means of the halftoning
techinque
– Giving an illusion of a full range of intermediate color levels to
the eye when looking from a sufficient distance
‹
Focusing on predicting the reflectance of halftoned samples
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3.8.1 The Murray-Davis equation
‹
The total reflectance spectrum R (λ ) of a halftoned sample
(3.102)
Where Rs (λ ) is reflectance spectrum of a solid sample
Rg (λ ) is reflectance spectrum of the bare substrate
a
‹
is fraction of area covered with ink (0 ≤ a ≤ 1)
Change the relation by Reflection density spectrum D(λ ) = − log10 R(λ )
(3.103)
(3.104)
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‹
Reflectance of halftoned samples
Fig. 3.27. Reflectance of halftoned samples having a fraction a of
area covered with black ink (continuous line), the murry-davis
model (dotted line), the Clapper-Yule model (dashed line).
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3.8.2 The classical Neugebauer theory
‹
Predicting the spectra of halftoned color prints produced (1937)
– Superposition of cyan, magenta, and yellow dot-screens
– Observed, under the microscope, that such a halftone print was in
fact a mosaic of eight colors
• Neugebauer primary : cyan, magenta, yellow, red, green, blue, and
black
– Assume that the dots in the different screens are almost
independent of each other
– Neugebauer’s model
(3.105)
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‹
Fraction of area occupied by the eight Neugebauer primaries
Fig. 3.28. Fraction of area occupied by the eight primaries
of the neugebauer model.
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3.8.3 Extended neugebauer theory
‹
Cellular Neugebauer method (Heuberger, 1992)
– Classical Neugebauer equation leads to color prediction errors of
about ∆E = 10
– Subdividing CMY color space into rectangular cells
~
– Using new reflectance spectra R
for each cells
j (λ ) (1 ≤ j ≤ 8)
– Error drops to ∆E = 3 , but requiring measuring 53 = 125 samples
‹
Generalization of Neugebauer theory with the number of inks k
(3.106)
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3.8.4 The Yule-Nielsen equation
‹
Consider the light scattering in the substrate
– Photon that penetrates the substrate in an area without ink may
emerge in an inked area, and vice versa
– Fraction of area a obtained from eq. 3.104 is greater than the real
area covered by ink, called optical dot gain or Yule-Nielsen effect
– Improve the prediction of the reflection density (1951)
(3.107)
(3.108)
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3.8.5 The Clapper-Yule equation
‹
In this model
– A fraction of light emerges at each reflection cycle
– The total reflectance is the sum of all those fractions
rs : surface reflection
ri : internal reflection
ρg : body reflectance of the substrate
T ( λ) : transmittance spectrum of the
ink under diffuse light
Fig. 3.28. In the Clapper-Yule model, fractions of light emerge
at each reflection cycle.
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‹
The emerging fraction of light
– Surface reflection
rs
– First emergence
– Second emergence
– Third emergence
– …n th emergence
‹
Clapper-Yule equation (1953)
(3.109)
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3.8.6 Advanced models
‹
Relation the empirical parameter n to physical quantities
‹
Point Spread Function
– Expresses the density of probability for a photon entering the
substrate at location (0,0) to emerge at the location (x,y)
(3.110)
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‹
Ruckdeschel and Hauser (1978)
– Assumed a Gaussian PSF
(3.111)
where σ is a characteristic scattering length of the photon in the substrate
– Yule-Nielsen factor
(3.112)
where L is a period of the screen
– Value of n approaches 1
• The substrate approaches a specular surface (Murray-Davis model)
– Value of n approaches 2
• The substrate becomes a perfect diffuser (Clapper-Yule model)
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‹
Rogers (1997)
– Characteristic scattering length
σ
is related to
• Absorption in the substrate and optical thickness of the substrate
– PSF is a series of convolutions whose term are the contribution of
the multiple internal reflections occurring in the substrate
(3.113)
– Yule-Nilesen factors n that are greater than 2.0
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‹
Gustavson
– Internal PSF is closely approximated by a function has a circular
symmetry
and a strong radial decay
(3.114)
where d controls the radial extent of the internal PSF
‹ Arney (1997)
– Less complex than the PSF convolution
– Scattering probability δ i, j for a photon
• Enters the substrate through a region covered by the Neugebauer
primary j of transmittance T j
• Emerge through a region covered by the Neugebauer primary i of
transmittance Ti
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– Reflectance of a halftone print
(3.115)
– In the particular case of traditional halftone screens
(3.116)
where w is an empirical parameter
–
w is related to σ
and
L
w = 1 − exp[−α (σ / L)]
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3.8.7 New mathematical framework for
color prediction of halftones
‹
Generalization the models
– Consider two Neugebauer primaries for the sake of simplicity
– Assume that the exchange of photons between inked and noninked
areas take place only in the substrate
– Assume that the ink layer behaves according to the Kubelka-Munk
model
Fig. 3.29. A schematic model of the printed surface.
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– Extend eq. 3.49 to take several inking levels in to account
(3.117)
– By intergrating eq. 3.117 between x=0 and x=X
(3.118)
• Definition of the matrix exponential is given in eq. 3.51
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– To take into consideration the multiple internal reflections (eq. 3.66)
(3.119)
– Optical dot gain affects only the boundary condition at x=0
• we assume that the exchange of photons takes place only in the
substrate
(3.120)
where δ u, v represents the overall probabilit y of a photon
entering ink level u and emerging from ink level v
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– New prediction model
(3.121)
First matrix : the Saunderson correction
Second matrix : the Kubelka - Munk model of the ink absorbing layer
Third matrix : the light scattering in the substrate
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– Change of basis matrix for emerging fluxes as function of the
incident fluxes
(3.122)
– Reflectance spectrum
(3.123)
where i0 = i1 = i and a1 be the inked fraction of area and
a0 = 1 − a1 be the non - inked fraction of area
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