Information Technology Reference
In-Depth Information
2 Calculation of the Probabilities
P
m
and
P
fa
General model
: The linear filter and additive noise attack on the stegomessage
S
=(
S
(
n
))
N−
1
n
=0
can be determined by the relation:
S
(
n
)=(
Sh
)(
n
)+
ε
(
n
)
,
n
=0
,...,N −
1
(2)
where
S
=(
S
(
n
))
N−
1
n
=0
is the attacked SM,
h
=(
h
(
n
))
N−
1
n
=0
is the pulse response
of the attack filter,
is the
convolution operator
:(
Sh
)(
n
)=
n
1
+
n
2
=
n
S
(
n
1
)
h
(
n
2
);
=(
ε
(
n
))
N−
1
and
n
=0
is the attack noise sequence. The
distortion constraints
based
on the MSE-criterion just after WM and after the attack are, respectively
ε
Var (
C
)
Var (
W
)
≥ η
w
(3)
Var (
C
)
Var (
S
−
C
)
=
Var (
C
)
Var (
C
(
h
−
δ
)
≥ η
a
(4)
)+
W
h
+
ε
=(
δ
(
n
))
N−
1
where Var is the variance,
1,
η
w
is the
signal-to-noise ratio
after WM, and
η
a
is the
signal-to-noise ratio
after
the attack to the watermarked message.
For simplicity and without loss of generality we may assume that strict in-
equalities hold in both eq. (3) and (4). The main notion in our model, which is in
line with the model in [1], is the assumption that
C
,
W
and
δ
n
=0
,
δ
(0) = 1 and
δ
(
n
) = 0 for all
n ≥
are discrete-time
mutually independent stochastic processes. Then (4) can be rewritten as
ε
Var (
C
)
)
=
η
a
(5)
Var (
C
(
h
−
δ
)) + Var (
W
h
)+Var(
ε
We will use a
correlation detector
since it is enough robust against changes of
probability distributions of
C
,
W
and
ε
and because it is more suitable to
perform the theoretical analysis.
Assuming that the WM detector knows
C
,
W
and
h
the following WM-
detection rules
are established:
-
If
Λ ≥ λ
then WM is detected in the presented
S
, and
-
if
Λ<λ
then WM is not detected in
S
, where
Λ
=
N−
1
(
S
(
n
)
−
(
Ch
)(
n
))(
Wh
)(
n
)
.
(6)
n
=0
It is easy to see from eqs. (1), (2) and (6) that whenever the WM is indeed
embedded into CM, the value of
Λ
coincides with
