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Let
,
x
=
y
=
−
1
.
5
and
x
=
y
=
1
. Then
Δ
and
Δ
y
are
C
=
p
+
qi
min
min
max
max
shown by (2) and (3).
(2)
Δ
x
=
(
x
−
x
)/(
a
−
1
max
min
(3)
Δ
y
=
(
y
−
y
)/(
b
−
1
max
min
Let
denote the point of fractal graphics. For
and
n
x
=
0
2
,
a
−
1
(
n
x
n
,
)
L
y
,
implement circulated steps as follows.
n
y
=
0
2
,
b
−
1
L
Step 1. Make
,
and
.
x
=
x
+
n
×
Δ
x
y
=
y
+
n
×
Δ
y
t
=
0
0
min
x
0
min
y
by
according to iterative process shown as (4)
Step 2. Compute
(
x
,
y
)
(
x
t
y
,
)
t
+
1
t
+
1
t
and (5).
(4)
2
2
x
=
x
−
y
+
p
t
+
1
t
t
(5)
y
=
2
x
y
+
q
t
+
1
t
t
2
2
.
Step 3. Compute
r
=
x
+
y
t
t
If
,
select color
t
and then go to step 4.
>
r
t
If
,
select black color and then go to step 4.
t
=
K
If
and
,
then go to step 2.
r
≤
t
t
<
K
Step 4. Make point
display color
t
and then transfer to next point.
(
n
x
n
,
)
y
3 Analysis on Anti-counterfeiting Principle
Some anti-counterfeiting technologies of color printing can not prevent imitation
efficiently. The main reason lies in the ineffective anti-counterfeiting space. Fractal
graphics based on escape time algorithm of Julia-set have great anti-counterfeiting
space which is very difficult to search for purpose of imitation. A little bit changing of
parameters can lead to very different shape changing of fractal graphics. Therefore,
fractal graphics based on escape time algorithm of Julia-set can be applied in
anti-counterfeiting of color printing and can provide higher anti-counterfeiting capacity
for color printing due to great anti-counterfeiting space.
3.1 Anti-counterfeiting Space of RGB Color
Random function is adopted during the implementation process of fractal graphics
based on escape time algorithm of Julia-set. Every pixel of fractal graphics can generate
gray value which has three random colors, i.e. R, G and B. The range of every random
color is from 0 to 255. Therefore, random color for every pixel can generate different
fractal graphics with same shape and different colors as shown in Fig. 1.
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