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In-Depth Information
Λ
=
N−
1
(
S
(
n
)
− C
(
n
))
W
(
n
)
.
n
=0
The probabilities
P
m
and
P
fa
can be calculated by (20) and (21). However,
N
m
(
η −
µ
=
(35)
1)
(It is easy to see that in a general case, when an attacker selects
ε
(
n
)=
ε
m
and
m
<m
then the corresponding value of
µ
, according to (35) is increased. If
an attacker selects
m
>m
the same value of
µ
as it appears in (35) is obtained).
Comparing
µ
given by (35) and
µ
0
by (28), corresponding to the case of no
filtering attack, we can conclude that the tile-based WM system produces a
m
times degradation of the WM system with respect to no filtering attack.
3 Simulation Results
Firstly we have used a 1-D simulation in order to check that, indeed, Var
Λ
0
≈
Var
Λ
1
for any “short” pulse response. We have found that the approximating
relation stated above holds and hence the theoretical formulas (20) and (21) are
correct unless
K
N
≥
1
2
. Thereafter we examined the effect of filtering attack on
a watermarked 2-D images. Two images,
, have been used to
embed WM in an additive manner according to eq. (1). The attack model was
taken as in (2):
fishboat
and
lena
U
e
−
n
1
+
n
2
h
(
n
)=
1
(36)
2
δ
2
where
U
=
n
e
−
n
1
+
n
2
2
δ
2
. Four different cases have been considered:
O
.
No filtering attack when both
W
and
ε
are white noise sequences.
A
.
Filtering attack when
W
is a white noise sequence given by (15) and
ε
is
also a white noise sequence.
B
.
Filtering attack when
W
is a colored noise sequence obtained after a passing
of white noise WM sequence through the filter with pulse response (36) and
ε
is still a white noise sequence.
C
.
Filtering attack when both
W
and
are colored noise sequences obtained
after a passing of the white noise WM and additive white noise attack se-
quences through the filter with pulse response (36).
ε
The correlation detector (6) has been used to decide whether a WM is present
or not. The results in the calculations of minimum values of WM lengths
N
to
guarantee
P
m
=
P
fa
=10
−
3
are presented in Table 1.
Here we can see, in Case A, that the use of filtering attack results in a
significant degradation of the WM system, especially for large
δ
, because the
number of WM elements required to provide
P
m
=
P
fa
=10
−
3
increases from
the order of tens to the order of hundreds. Simultaneously we can see in Figure 1
