Embedding Information in Grayscale Images - Signal Processing ...

6 Ternary embedding methods
We start this section with a definition. A grayscale symbol x is said to be in class w iff x
mod 3 = w for w = 0, 1, 2. Now we are ready to discuss uncoded ternary embedding and
after that again two coded embedding methods.
6.1 Uncoded ternary modulation
Suppose that message w ∈ {0, 1, 2} is to be embedded in the grayscale symbol x. The
decoder determines the message w simply by looking at the class of y. If x is in class
w then y = x is chosen, otherwise we change it into the symbol y in class w, such that
D(x, y) = (y − x) 2 is minimal, see the tables below.
x w = 0 1 2
· · ·
9 y = 9 10 8
10 9 10 11
11 12 10 11
· · ·
x w = 0 1 2
· · ·
9 D = 0 1 1
10 1 0 1
11 1 1 0
· · ·
The obtained embedding rate R = log 2 3 ≈ 1.585. The corresponding average distortion
¯D = 2/3 · 1 = 2/3. This results in the ratio R/ ¯D = 3/2 log 2 3 ≈ 2.378 which is quite good!
6.2 Embedding with ternary Hamming codes
We can also design codes for the modulating the class, based on ternary Hamming codes. For
a given value m, i.e. the number of parity check symbols, the codeword length is (3 m − 1)/2.
Therefore
R = 2m log 2 3
3 m − 1 and ¯D = 2
3 m .
Hence
see figure 5.
R/ ¯D = m3m log 2 3
3 m − 1 ,
6.3 Embedding using the ternary Golay code
If we use the (11, 6, 5) ternary Golay code we get the following rate and distortion:
R = 5 log 2 3
11
≈ 0.7204 and ¯D =
( 11
) (
1 · 1 · 2 + 11
)
2 · 2 · 4
243 · 11
= 42
243 ≈ 0.1728.
Hence R/ ¯D ≈ 4.1682 (see figure 5).