Image Processing Reference
In-Depth Information
The above integral has a different closed-form solution for different values of the
parameter p. The Springett
-
Melnyk PIDC equations for different values of p are
given below.
If p ¼
1, then
V ( X ) V r
V i V r
e h 0
V ( X ) V i þ V c ln
þ
C X ¼
0
(
10
:
17
)
If p
1, then
"
#
p 1
p 1
V c
V c
V ( X ) V r
V c
V i V r
e h 0
C X ¼
V ( X ) V i þ
þ
0
(
10
:
18
)
1
p
where V i is the initial voltage. It is not possible to obtain an explicit general form for
V(X) in terms of p. However, for the special case of p ¼
2, it is possible to obtain an
explicit form. This case is sometimes referred to as the quadratic Melnyk (QM) PIDC
model and is given by
q
b
2
V ( X ) ¼ V r þ b þ
þ V 2
c
(
10
:
19
)
V 2
c
e h 0
1
where
b ¼
2 [ V i V r
V i V r
C X ]
It is customary to de
ne a coef
cient of exposure as S. Hence,
e h 0
C ¼
e h 0 L
e
S ¼
(
10
:
20
)
The initial slope of the PIDC, that is, dV =
0 is called the sensitivity of the
photoconductor. It is calculated from Equation 10.14 by using V ¼ V i at X ¼
dX at X ¼
0:
X ¼ 0 ¼
d V
d X
S
p
(
10
:
21
)
V c
V i V c
1
þ
Generally, in photoconductors of practical interest, Vi. c < V i . This implies that the
initial slope of the PIDC is practically independent on the initial voltage Vi, i , thus all
PIDCs start with the same slope regardless of the initial voltage:
X ¼ 0
d V
d X
S
p
(
10
:
22
)
V V i
1
þ
This de
ciency in the PIDC model is overcome by modifying the exposure param-
eter S, so that the model captures the functional form of a PIDC throughout the
operating region. The new parameter introduced is S ¼ LS 0 , where S 0 ¼ e h 0 =e
and
the ratio L = L 0 can be considered equal to 1 for our purpose. It is important to
note, however, that the ratio L = L 0 is signi
cant when photoconductors of varying
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