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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
 
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