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texture and shape, which are extremely difficult to imitate. Thereby large space of
anti-counterfeiting in color printing can be generated by fractal graphics [2].
Under this background, an anti-counterfeiting approach based on escape time
algorithm of Julia-set is proposed in the paper for improving anti-counterfeiting
capacity in color printing. By setting different parameters of fractal graphics,
anti-counterfeiting space can be generated, which is large enough and very difficult to
search, then anti-counterfeiting in color printing is implemented. After many
experiments with the proposed algorithm, anti-counterfeiting space generated by RGB
color, graph size, shape and adjusting parameters makes imitation more difficult.
2 Fractal Theory Based on Julia-set
Basic function of Julia-set is shown in (1), in which
Z
and
C
are all complex
numbers. Different initial
C
has great influence on fractal graphics of Julia-set. Let
x
denote the real part of
Z
, and
y
denote the image part of
Z
. Let
p
denote the real
part of
C
, and
q
denote the image part of
C
. Then
=
can be
obtained. Different
p
and
q
can generate very different fractal graphics of Julia-set
[3].
and
C
p
+
qi
Z
=
x
+
yi
(1)
F
(
Z
)
=
Z
2
+
C
2.1 Escape Time Algorithm
Escape time algorithm is an effective theoretical algorithm of constructing fractal
graphics. The key of this algorithm lies in the construction of escape time function,
x
denotes initial
x
, then corresponding
y
can be obtained, i.e.
. If
y
=
f
(
x
)
x
denote new
x
, then new
y
can be written as
. Let
.
y
=
x
=
f
(
0
x
)
y
=
x
=
f
(
1
x
)
1
2
After repeated and iterative computation for
n
times, i.e.
,
y
=
x
=
f
(
x
),
n
=
0
2
L
n
+
1
n
x
,
x
,
x
,
,
x
L
n
L
the sequence of
can be obtained. Some points of the sequence
are limitary, that is, tracks of these points can not exceed a certain range. But other
points of the sequence are limitless, that is, these points can escape to the infinite. If the
different points are displayed on computer screen with different colors, then fractal
graphics are obtained. That's the basic idea of escape time algorithm [4].
0
1
2
2.2 Escape Time Algorithm Based on Julia-set
Julia-set is an iterative process with complex numbers by
.
From the point
of escape time algorithm, the interior of Julia-set is convergent for one or several
points, and the exterior of Julia-set is emanative to the infinite with escape time. Escape
bound is Julia-set. Escape time algorithm based on Julia-set can generate complicated
fractal graphics in complex plane [5]
.
Let
2
Z
→
Z
+
C
a
×
b
denote the resolution of fractal graphics, i.e. graph size.
types of
K
+
1
colors can be displayed, which are denoted as
. Let
t
denote the escape time.
0
L
,
K
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