Embedding Information in Grayscale Images - Signal Processing ...

which is closest to x1 7 in Hamming-sense. To obtain the composite sequence just replace the
LSB vector x1 7 by y7 1 .
Next we determine the average distortion ¯D of this method. The Hamming code is perfect
with d H = 3, thus we will find a word y1 7 ∈ C i at distance 1 from x1 7 with probability 7/8 and
a word at distance 0 with probability 1/8 (some smoothness of P(x) is assumed here). Hence
¯D = (7/8)/7 = 1/8. The decoder can detect the coset to which the composite sequence y1
7
belongs, hence reliable transmission is possible with rate R = (log 2 8)/7 = 3/7 bit/symbol.
Thus we achieve ( ¯D, R) = (1/8, 3/7). The R/ ¯D-ratio = 24/7 ≈ 3.428 which is a factor
1.714 higher than LSB modulation. A Hamming code of length 2 m − 1 for m = 2, 3, 4, · · ·
gives
R =
m
2 m − 1 and ¯D = 1
2 m .
Hence the ratio
R/ ¯D = m2m
2 m − 1 ,
which is approximately equal to m for large m (see figure 4).
5.2 Embedding based on the binary Golay code
Instead of the binary Hamming code we can use the (23, 12, 7) binary Golay code. This
leads to
R = 11
( 23
) (
23 ≈ 0.478 and 1 · 1 + 23
) (
2 · 2 + 23
)
3 · 3
¯D =
≈ 0.124.
2048 · 23
Now the ratio R/ ¯D ≈ 3.85, which is almost a factor two better what can be achieved with
LSB modulation (see figure 4).
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Figure 4: Rate-distortion curve, time-sharing R = 1-LSB modulation (...), binary Hamming
codes (o), and the binary Golay code (x).