Image Processing Reference
In-Depth Information
PIDC for most of the practical photoconductors used in digital electrophoto-
graphic printers based on electrophotography. Often, the quadratic model fails to
capture the performance at
the knee of the PIDC. To resolve this issue,
it
is
necessary to use a more general model (Equation 10.18) to achieve a good
t. It
is also interesting to cite an analytical solution we were able to
find for the case
p
¼
1. It is easy to show that the solution can be expressed in terms of the well-
studied Lambert function W(x), which is, by de
nition, the solution of Equation
10.24 with respect to W:
We
W
¼
x
(
10
:
24
)
Then one can show that the solution of Equation 10.17 is given by
S
V
c
V
(
X
) ¼
V
r
þ
V
c
WF
exp
X
(
10
:
25
)
where
1
V
c
exp
V
c
ln
V
i
V
r
ð
Þ
V
i
V
r
V
c
F
¼
e
h
0
C
S
¼
One can also
4by
solving the polynomial equations (with respect to V(X)) that originate from Equation
10.18. For the rest of the cases, numerical solutions are recommended.
It is also interesting to observe that the exposure model developed in this section
is based on the assumption that the photoconductor is positively charged and this
charge decreases with exposure. This is true in the case of charged area development
(CAD) but not true in the case of discharged area development (DAD) (for de
find closed-form solutions for the cases when p
¼
3 and p
¼
n-
itions, see Section 10.2.3). In that case, one can simply consider the following
generalized relation that works for both cases:
V
(
X
) ¼ sgn
V
ðÞexpose
ð
V
jj
, X
Þ
(
10
:
26
)
where sgn(x) is the signum function de
ned by
n
1
x
0
sgn(
x
) ¼
(
10
:
27
)
x
<
1
0
expose(.,.) denotes the exposure model developed above
jj
stands for the absolute value
Figure 10.9 shows PIDC curves for a photoconductor for different values of initial
voltage Vi
i
whose parameters are shown in Tables 10.3 and 10.4.
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