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where
s
i
is the stego value. Thus, LSB embedding can be operated with a
maximum error
D
= 1. The following examples show what happens in LSB
embedding.
We obtain the expected distortion
D
exp
= (0 + 1 + 1 + 0)/4 = 1/2 and an
embedding rate
R
= 1/1 = 1. Using the rate-distortion pair (
R
,
D
exp
) = (1, 1/2),
we achieve
r
(
D
exp
) = 2
D
exp
. Embedding efficiency is thus
=
(
)
=
rD
D
R
D
exp
E
=
2
.
exp
exp
These results together with an embedding efficiency upper bound are
illustrated in Figure 10.2. To obtain better embedding ability, ternary embed-
ding was devised to improve efficiency.
10.3.2 Ternary embedding
In ±1 steganography, there are three possibilities for each pixel: change it by
±1 or leave it unmodified. Thus, we assume that a pixel's grayscale value
c
i
is
said to be in group
g
if
c
i
modifies 3 =
g
, where
g
= 0, 1, 2, and message symbol
m
i
∈ {0, 1, 2} is to be embedded in each pixel
c
i
. If
c
i
is in group
m
i
, no modifica-
tion is made. Otherwise,
c
i
should be changed to
s
i
in group
m
i
so that |
s
i
-
c
i
|
is minimal. The recipient can determine the message symbol
mi
by simply
looking at the group of
s
i
. The following table describes this ternary embed-
ding more clearly.
We can see that ternary embedding can be operated with maximum error
D
= 1. The expected distortion is
D
exp
= (2 × 3)/9 = 2/3 and the corresponding
embedding rate is
R
= log
2
3/1 ≈ 1.58. Using the resulting rate-distortion pair
(
R
,
D
exp
) = (1og
2
3, 2/3), the embedding efficiency can be computed as:
R
D
log
2
3
2
3
E
=
=
≈
238
.
.
exp
Combining the examples shown in Tables 10.1 and 10.2, and the compari-
son results demonstrated in Table 10.2, we can see that ternary embedding
TaBLe 10.1
Examples of LSB Embedding
c
i
m
i
= 0
m
i
= 1
128 = 10000000
s
i
= 128
D
= 0
s
i
= 129
D
= 1
129 = 10000001
s
i
= 128
D
= 1
s
i
= 129
D
= 0