Embedding Information in Grayscale Images - Signal Processing ...

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0.9
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0.7
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0.5
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0.1
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Figure 5: Rate-distortion curve, time-sharing R = 1-LSB modulation (...), binary Hamming
codes (o), ternary Hamming codes (*), and both Golay codes (x).
7 Two-dimensional codes
Finally we consider a method that modifies pairs of grayscale-symbols (x a , x b ). Therefore
we ”color” the two-dimensional rectangular grid with five colors. A point and its nearest
neighbors all have a different color, see table below.
x a →
1 2 3 4 0 1 2 3 4 0 1 2 3 4 0
x b 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3
↓ 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
3 4 0 1 2 3 4 0 1 2 3 4 0 1 2
We want to embed in x a and x b a message w ∈ {0, 1, 2, 3, 4}. The decoder just looks at the
color of pair (y a , y b ). If (x a , x b ) does not have color w change it into the nearest neighbor
(y a , y b ) with color w. Now
¯D = 1 · 0 + 4 · 1
5 · 2
= 2/5 and R = log 2 5
2
= 1.16096.
and R/ ¯D = 2.90241. This basic code can be used together with 5-ary Hamming codes.
For a given m, i.e. the number of parity checks, we get a code length of (5 m − 1)/4 5-ary
symbols. This code processes (5 m − 1)/2 grayscale-symbols and the embedding rate is
R = 2m log 2 (5)
5 m − 1
and the distortion ¯D = 2
5 m .
The resulting distortion-rate pairs are shown in figure 6.