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ratio of blocks with decreasing
f
. In general,
R
M
+
S
M
< 1. Similarly, another two pa-
rameters
R
−
M
and
S
−
M
can be defined when
F
−
1
is applied to a part of each block.
If the received image does not contain secret data,
F
1
and
F
−
1
should equally increase
the
f
value of blocks in a statistical manner. So,
R
≈
R
>
S
≈
S
.
(4)
M
−
M
M
−
M
When secret bits are embedded, the difference between
R
M
and
S
M
decreases whereas
the difference between
R
−
M
and
S
−
M
increases. Thus,
.
R
−
S
>
R
−
S
(5)
−
M
−
M
M
M
Therefore, an attacker can use the relation among the four parameters to detect the
presence of secret information.
2.3
Gray-Level Plane Crossing (GPC) Analysis
In this subsection, an alternative steganalytic approach against LSB embedding is
described. This, together with the χ
2
and RS analysis, will be used in the following to
examine the anti-steganalysis performance of a new LHA approach proposed in this
paper. By viewing an image as a landscape in a 3D space with the
z
coordinate repre-
senting the pixel gray-level, each pixel in the image is identified as a triplet (
x
,
y
,
z
).
Define two interlaced families of odd and even gray-level planes,
P
O
and
P
E
, parallel to
the XY plane, each containing planes between
z
= 2
i
+1 and 2
i
+2, and 2
i
and 2
i
+1,
respectively, where
i
=0, 1, … , 127. The odd planes in
P
O
may be designated
z
= 1.5,
z
= 3.5,
z
= 5.5,
,
z
=
254.5. The numbers of planes in the two families crossed by all lines connection ad-
jacent pixels are summed up, and denoted
N
O
and
N
E
respectively.
Clearly, if the received image does not contain any secret data,
N
O
and
N
E
should
roughly equal. LSB embedding will not change
N
O
because swapping between 2
i
and
2
i
+1 does not cross any plane in
P
O
. In contrary,
N
E
will be raised since each modified
pixel traverses one plane in
P
E
and the smoothness between adjacent pixels is reduced.
For example, if the two adjacent gray values of original image are equal, and one of
them is modified by LSB embedding,
N
E
will increase. Therefore a parameter
R
=
N
E
/
N
O
can be used to detect the presence of inserted data. If
R
is greater than a given
threshold
T
, the image is judged as containing a secret message. In order to enhance
sensibility of
R
, the number of crossings is not counted when the difference between
adjacent gray values is greater than a predefined value
D
, say, 4 or 5.
Fig.1 shows the relationship between
R
and the amount of secret bits in 3 test im-
ages, all sized 512×512. Note that
R
increases approximately linearly with the payload
L
, the ratio between the embedded bit number and the total number of host pixels. Also,
the less the high-frequency components in a cover image, the more sensitive the value
of
R
is with respect to the payload. The usefulness of the
R
-
L
relationship is demon-
strated in the following experiment. A total of 385 images captured with a digital
camera were used to establish statistical distributions of
R
at different payloads. Fig.2
shows the results where the ordinate is the image number corresponding to
R
on the
abscissa. When smoothed and normalized, these curves can be used to represent PDF of
…
,
z
= 255.5, and the even planes in
P
E
as
z
= 0.5,
z
= 2.5,
z
= 4.5,
…
