Embedding Information in Grayscale Images - Signal Processing ...

Embedding Information in Grayscale Images
Marten van Dijk and Frans Willems
Philips Research Laboratories, Eindhoven
Abstract
Grayscale images can be represented as sequences of integer-valued symbols. If
such a symbol has alphabet {0, 1, · · · , 2 B − 1} it could be represented by B binary
digits. To embed information in these symbols, we are allowed to distort them. The
distortion measure that we consider here is mean squared error. In this setup there is
a so-called “rate-distortion function” that tells us what the largest embedding rate is,
given a certain distortion level. First we determine this rate-distortion function. Next
we compare the performance of LSB modulation to this rate-distortion function. Then
several embedding codes are proposed: (i) codes that only affect the LSB of a symbol,
(ii) so-called ternary codes in which a symbol can be changed by at most one, and (iii)
codes that process a pair of symbols each time.
1 System description
discrete
memoryless
source
X N 1
✲
encoder
Y N
1
✲
destination
✻
W
✲
decoder
✲
Ŵ
Figure 1: A model of an information embedding system.
The embedding system that we study is shown in figure 1. A discrete memoryless source
emits the host sequence x1
N = (x 1, x 2 , · · · , x N ). The symbols x n for n = 1, N assume values
from the alphabet X . Their distribution is {P(x), x ∈ X }. A message source produces the
message index w ∈ {1, 2, · · · , M} with probability 1/M, independent of x1 N . Based on the
message index w and the host sequence x1
N the encoder (embedder) produces the composite
sequence
= f (x 1 N , w).
y N 1
1

The symbols from y1
N = (y 1, y 2 , · · · , y N ) assume values in the alphabet Y. We require that
from the composite sequence y1
N the embedded message can be reconstructed, i.e. there is a
decoder producing the estimate Ŵ = g(y1 N ) such that
Pr{Ŵ ̸= W } = 0.
Moreover the sequences y N 1
must be “close” to x N 1
, i.e. the expected distortion
¯D = E[d(X1 N , Y 1 N )] = ∑ Pr{X1 N = x 1 N } Pr{W = w}d(x 1 N , f (x 1 N , w)) ≤ ,
x1 N ,w
where is the maximum allowable distortion and
d(x1 N , y 1 N ) = 1 ∑
N
n=1,N
D(x n , y n )
is the distortion between x1
N and y 1 N
define the embedding-rate R as
, for some specified distortion matrix D(·, ·). Finally we
R = 1 N log 2 (M).
2 The ”rate-distortion” function
We are obviously interested in finding codes that combine a large embedding rate R with a
small distortion ¯D. Only recently several authors, e.g. [3] and [1], realized that this setup
is strongly related to the “defect channel”-problem which was first investigated in a general
form by Gelfand and Pinsker [2]. In [4] the so-called ”rate-distortion function”, that
corresponds to a slightly more general embedding model than the one we study here, was
determined. This function gives the maximum embedding rate R for a given distortion level
. For our model (Z ≡ Y ) this function turns out to be
R() =
max
{P(y|x): ∑ H(Y |X).
x,y P(x)P(y|x)D(x,y)≤}
Note that this is not a rate-distortion function in the usual sense.
3 Grayscale symbols, mean squared error distortion
0
1
2
x y
2 B − 1
❅ ❅
D(x, y) = (x − y) 2
Figure 2: The grayscale alphabet G and mean squared error distortion.
A grayscale symbol assumes a value from an integer alphabet G. The cardinality of G
should no be too small. E.g. let G = {0, 1, · · · , 2 B − 1} for some positive integer B, then

each symbol can be represented by a vector of B binary digits. Now let X = Y = G. If a
grayscale symbol x ∈ G is changed into a grayscale symbol y ∈ G then the resulting squared
error distortion is
D(x, y) = (y − x) 2 ,
see figure 2. To determine R() for this case, let p i =
∑y−x=i P(x)P(y|x) for all integer
i, then
H(Y |X) = H(Y − X|X) ≤ H(Y − X) = ∑ 1
p i log 2
p
i
i
and
∑
P(x)P(y|x)(y − x) 2 = ∑
x,y
i
p i i 2 .
The corresponding bound for R() can be achieved by taking i = y − x independent of x,
which is possible if we assume that |G| is large and we neglect boundary-effects. Now we
have to maximize ∑ i p 1
i log 2 p i
under the constraint ∑ i p ii 2 ≤ . Therefore consider for
some α > 0 the “Gaussian” distribution {pi ∗} where
p ∗ i
= exp(−αi 2 )/β,
with β = ∑ i exp(−αi 2 ) and having variance ∑ i p∗ i i 2 = . Then for any distribution {p i }
with variance ∑ i p ii 2 ≤ we can write
∑
p i ln 1 pi
∗
i
= ∑ i
p i ln(β exp(αi 2 )) = ln β + α ∑ i
p i i 2 ≤ ln β + α,
and therefore its entropy (in nats)
∑
p i ln 1 = ∑ p
i i
i
p i ln p∗ i
p i
+ ∑ i
p i ln 1 p ∗ i
≤ ln β + α.
Equality is achieved only for {p i } equal to the Gaussian distribution {pi ∗ }. Therefore to
compute the rate-distortion function we only need to vary α and compute the entropy and
variance of {pi ∗ }. We can now make a plot of R(), see figure 3. This plot shows that to
achieve a rate of 1 bit/symb. we need an average distortion of at least ≈ 0.22.
4 Least-significant bit(s) modulation
Traditional embedding methods assume that a grayscale symbol x ∈ G is represented as a
binary vector b B−1 , · · · , b 1 , b 0 such that x = ∑ i=0,B−1 b i2 i . Messages are now embedded
only in the least-significant bits (LSB’s) in this binary representation. Therefore this form of
embedding is called least-significant bits modulation. Assume that message w is embedded
in the R LSB’s (R is a positive integer). The following tables now show what happens for
R = 1. Observe that for R = 1 the distortion ¯D = 1/4 · 2 = 1/2.

3
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3
Figure 3: R() in bits per symbol as a function of the distortion per symbol (horizontally),
together with two LSB distortion-rate pairs (o) and time-sharing the R = 1-LSB method (...).
x w = 0 1
· · ·
8=1000 y = 8 9
9=1001 8 9
· · ·
x w = 0 1
· · ·
8=1000 D = 0 1
9=1001 1 0
· · ·
Similarly for R = 2 the average distortion is ¯D = 1/16(6 · 1 + 4 · 4 + 2 · 9) = 5/2. The
distortion-rate pairs ( ¯D, R) = (1/2, 1) and (5/2, 2) are plotted in figure 3 denoted by o’-s.
Using the LSB modulation scheme ( ¯D, R) = (1/2, 1) only for a fraction of the symbols, we
achieve
R( ¯D) = 2 ¯D, for 0 ≤ ¯D ≤ 1/2.
The resulting distortion-rate pairs are plotted in the figure with a dotted line. The problem
we want to address next is: ”How can we do better than R( ¯D)/ ¯D = 2?”
5 Coded LSB modulation
In this section we consider coding methods that affect only the LSB of each grayscale symbol.
The first method is based on Hamming codes, after that we consider a code based on the
binary Golay code.
5.1 Embedding based on binary Hamming codes
To describe the embedding code consider the (7, 4, 3) Hamming code. Fix a certain coset C i ,
for some i = 0, 1, · · · , 7. Consider only the vector of LSB’s of seven host symbols. Denote
this vector by x 7 1 = x 1, x 2 , · · · , x 7 . Determine a binary vector y 7 1 = y 1, y 2 , · · · , y 7 ∈ C i

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).

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).

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 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.

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Figure 6: Rate-distortion curve, time-sharing R = 1 LSB modulation (...), binary Hamming
codes (o), ternary Hamming codes (*), the Golay codes (x), and the 5-ary codes (+).
8 Conclusion
We have only looked at small distortions here. Moreover we have concentrated on perfect
codes since this was so easy. It can be shown that Hamming codes are quite good for D → 0.
For a moderate rate we have seen that Golay codes are rather good, but what about even better
codes for even higher rates? How do we generalize the coloring results to larger number of
colors and more dimensions?
References
[1] J. Chou, S. Pradhan, L. El Ghaoui and K. Ramchandran, “A Robust Optimization Solution
to the Data Hiding Problem Using Distributed Source Coding Principles,” preprint,
2000.
[2] S. Gelfand and M. Pinsker, “Coding for a Channel with Random Parameters”, Problems
of Control and Information Theory, vol. 9, pp. 19-31, 1980.
[3] P. Moulin and J. O’Sullivan, “Information-theoretic Analysis of Information Hiding,”
preprint, 1999.
[4] F.M.J. Willems, ”An Informationtheoretical Approach to Information Embedding,”
Proc. 21st Symp. Inform. Theory in the Benelux, pp. 255-260, Wassenaar, May 25-26,
2000.