Information Technology Reference
In-Depth Information
Willems and van Dijk [8,13] defined a rate-distortion function to evaluate
the performance of message embedding schemes. They show that it is not
useful to consider schemes that have |
c
i
-
s
i
| > 1 for some component
i
.
According to them, the squared error cannot be larger than one; hence, this
measure for message embedding is called ±1 steganography. They give the
rate-distortion function as the upper bound on the embedding rate of ±1
steganography subject to the constraint of expected distortion
D
exp
(
)
+
for
0
≤ ≤
D
2
/
3
HD
D
,
(
)
=
exp
exp
exp
rD
.
exp
for
D
>
23
/
log,
2
3
exp
The rate-distortion function tells us the large embedding rate, given a
certain distortion level. A plot of
r
(
D
exp
) is shown in Figure 10.1. We can see
that to achieve an embedding rate of 1, we need an average distortion of at
least 0.22.
To evaluate embedding efficiency, we rewrite the rate-distortion function
as an upper bound on embedding efficiency
E
with a given embedding
rate
R
R
rD
E
=
(
)
,
0
≤≤
R
log
3
,
2
−1
exp
where
r
-1
is the inverse function of
r
. Figure 10.2 illustrates a theoretically
achievable bound on the embedding efficiency for ±1 steganography.
1.8
1.6
1.4
1.2
1
Upper bound
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distortion
Figure 10.1
Rate-distortion function for squared error and ±1 embedding changes.
