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Mourad: Improved calibration of optical characteristics of paper by an adapted paper-MTF model
added dependencies on the spatial frequencies and . 7 For
the PDE system at hand, the generalized two-dimensional
Fourier transform is appropriate, since the fluxes are defined
on the real plane R 2 and decay greatly as x +y →. 24,32,33
Solving the system of transformed PDEs at z=D yields the
spectral reflectance MTF
where
H R , = Fh R x,y = A
B
,
A = a 12 + a 21 − c pb e cD − a 12 + a 21 + c pb e −cD ,
5.1
5.2
B =−a 21 + c + a 12 pa + pb + a 21 − c pa pb e cD + a 21
− c + a 12 pa + pb + a 21 + c pa pb e −cD . 5.3
The coefficients a 12 =a 12 , a 21 =a 21 , and c=c depend on
the lateral frequencies , and are given by Eqs.
(6.1)–(6.6):
where
2 2
c =a 21 − a 12 , 6.1
a 12 =
a 21 =
s 3
s 1
,
s 2
s 1
,
6.2
6.3
s 1 = − b 2 −2 l + b +2 l + b +4 2 2 − b 2
2 + 2 +16 4 2 2 ,
s 2 = − b 2 −2 l + b +2 l + b −4 l 2
+4 2 − b + b −2 l 2 2 + 2
6.4
+16 4 2 2 , 6.5
s 3 = − b 2 −2 l + b b +2 l + b −4 l 2
+4 2 − b b + b −2 l 2 2 + 2
+16 4 b 2 2 . 6.6
Likewise, the microspectral transmittance distribution is
modeled by
Tx,y = pa x,yF −1 †H T ,F bp x,y‡,
with the spectral transmittance MTF H T ,
7
H T , = Fh T x,y =− 2c
B
.
Here, bp x,y describes the fraction of light transmitted into
the substrate through the bottom layer. Equation (7) implies
that the optical spreading has no direct observable effect on
transmittance measurements of single-side, upward-oriented
printed paper sheets. This is consistent with microscopic
transmittance images published by Koopipat et al. 13
The two spatial formulas, Eqs. (3) and (7), are the foundation
used in predicting the spectral reflectance of arbitrary
halftone prints presented and discussed in the Model Application
and Model Discussion sections below. The following
section considers the calibration of the optical parameters.
MODEL CALIBRATION
Our next objective is to determine the optical dot gain for
arbitrary halftones and dithering frequencies from a few
macroscopic spectral measurements. More precisely, we need
to calculate the isolated optical dot gain in order to distinguish
between the different effects leading to printing
nonlinearities. In order to meet this requirement, Eq. (3)
describes the spectral reflectance Rx,y as a function of the
bulk parameters D, , l , and b together with the surface
refractive coefficients ap , ap , pa , pa , and pb . Unfortunately,
only the paper thickness, D, is directly measurable
with common instruments. Therefore, we determine the remaining
parameters in such a way as to best match the calculated
results to the measured spectra of a small set of test
patches. In order to avoid any printing irregularity and to
increase the accuracy of the parameter estimation, the test
patches are chosen to be as unambiguous as possible. In
particular, we chose only solid patches of the primary
colors—in our case cyan, magenta, yellow, and black prints,
abbreviated to CMYK—plus a sample of paper-white (W).
For this work, we consider only single side prints. As well as
avoiding the printing irregularities, the uniformity of the
solid patches also reduces the matrix multiplication of Eq.
(3) to a simple scalar multiplication. This is because
Fx,yof a uniform patch is only different from zero at
zero frequencies ,0,0. Hence, for a solid patch,
Eqs. (3) and (7) reduce to
R solid = ap + pa H R 0,0 ap ,
T solid = pa H T 0,0 bp .
8
9
10
In other words, the form of the PSF h R x,y is not involved
in the calibration process.
Traditionally, the KM theory scattering and absorption
coefficients K and S are determined using two distinct reflectance
measurements: one measurement over a black
backing and another over a white backing, both backings of
known reflectances. However, in our case, more measurements
are required because we need to determine not only
the scattering and absorption coefficients but also the inner
transmittance of the primary inks used in addition to the
286 J. Imaging Sci. Technol. 514/Jul.-Aug. 2007