6 one third probability embedding a new

Published in IET Image Processing
Received on 10th February 2013
Revised on 19th June 2013
Accepted on 14th July 2013
doi: 10.1049/iet-ipr.2013.0109
ISSN 1751-9659
One-third probability embedding: a new ±1
histogram compensating image least significant
bit steganography scheme
Saeed Sarreshtedari, Mohammad Ali Akhaee
Department of Electrical and Computer Engineering, College of Engineering, University of Tehran, Tehran, Iran
E-mail: s.sarreshtedari@ut.ac.ir
Abstract: A new method is introduced for the least significant bit (LSB) image steganography in spatial domain providing the
capacity of one bit per pixel. Compared to the recently proposed image steganography techniques, the new method called one-
third LSB embedding reduces the probability of change per pixel to one-third without sacrificing the embedding capacity. This
improvement results in a better imperceptibility and also higher robustness against well-known LSB detectors. Bits of the message
are carried using a function of three adjacent cover pixels. It is shown that no significant improvement is achieved by increasing
the length of the pixel sequence employed. A closed-form expression for the probability of change per pixel in terms of the
number of pixels used in the pixel groups has been derived. Another advantage of the proposed algorithm is to compensate,
as much as possible, for any changes in the image histogram. It has been demonstrated that one-third probability embedding
outperforms histogram compensating version of the LSB matching in terms of keeping the image histogram unchanged.
1 Introduction
Steganography is a delicate technique in transmitting hidden
data through highly-used media in order not to attract any
unauthorised person’s attention [1]. Since images are
nowadays the most widespread digital media on the
Internet, most of steganographic algorithms are
implemented on images. The least-significant bit (LSB)
steganography algorithms work on LSB of cover images.
These algorithms are widely used because of high
embedding capacity, high quality and low computational
complexity.
The most simple and primitive algorithm of image LSB
steganography is LSB replacement, in which the LSB of
each pixel is replaced with the message bit. In this way,
even pixel values are incremented by one or remain
unchanged. On the other hand, odd values are decremented
by one or kept with no change. As a result, every two
consecutive values in the histogram of the pixel values
converge to the same number and the histogram of the
image will be in a ‘pair-wise’ format. This pair-wise
appearance known as pair of values (PoV) can be detected
by chi-square [2] and RS [3] steganalysis tests.
Referring to this Achilles’ heel, an improvement called
LSB matching (LSBM) was proposed [4]. Here, in order to
overcome the PoVs issue, when the LSB of the cover
image pixel and the message bit are not matched, the pixel
value is decreased or increased by one randomly. More
recent and complicated steganalysis techniques have been
proposed to detect LSBM embedding. An important group
of these analysis techniques are based on the centre of mass
of the histogram of the characteristic function (HCF-COM)
features proposed by Harmsen and Pearlman [5]. Since
HCF-COM method was ineffective for grey-scale images,
Ker proposed adjacency and calibrated HCF-COM [6].
Several steganalysis methods have been proposed since then
for detecting LSB matching [7–9]. Performance of LSBM
was also investigated through universal steganalysis
techniques [10–12].
In LSB matching scheme, it is assumed that the message is
encrypted, the input bit stream is a random sequence and
hence, each bit may match to the LSB of the pixel value
with one-half probability. Therefore, in the LSB matching
method, the probability of change per pixel is 0.5. A
revisited version of LSB matching (LSBMR) was proposed
by Mielikainen, which greatly improved the performance by
lowering the probability of modification per pixel from 0.5
to 0.375 [13]. Our contribution in this regard is to reduce
this probability to one-third. This is why we call our
method one-third probability embedding algorithm. As
expected, the proposed embedding method will exhibit
better performance than LSBM and LSBMR, against
HCF-COM steganalysis.
LSBMR and one-third probability embedding work on
groups of two and three pixels, respectively. There exist
many other steganographic algorithms that work on groups
of pixels [14–16]. Increasing the number of pixels used as
an embedding group to approach infinity, Li showed that
the probability of change per pixel can be reduced to 0.22
[17]. Although the Li’s method is elegant in reducing the
embedding distortion at the expense of finding suitable
groups, it does not have any solution for the histogram
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IET Image Process., 2014, Vol. 8, Iss. 2, pp. 78–89
doi: 10.1049/iet-ipr.2013.0109

preserving which is the main point to several steganalysers.
Several techniques proposed so far are based on LSBM and
LSBMR. Sun et al. [18] exploited the randomness of
changes in LSBM to preserve the histogram of the host
image. The main drawback of this method is its low
embedding efficiency as it is based on LSBM. Human
visual system (HVS) is less sensitive to changes in edges of
the image than smooth areas; thus, Luo et al. [19] adapted
LSBMR to an edge adaptive algorithm. Since LSBM and
LSBMR offered the capacity of only one bit per pixel, LSB
steganographic algorithms with higher capacity were
proposed afterwards [20–25].
Exploiting modification direction (EMD) steganography
method introduced a new class of LSB embedding
technique [26]. Depending on the size of the pixel groups
employed, the capacity of EMD is slightly more than one
bit per pixel. Diamond encoding (DE) was proposed to
improve the capacity of EMD [27]. Adaptive pixel pair
matching steganography method improved the DE in terms
of the mean square error (MSE) between the cover and
stego-image [28]. Sun et al. [29] also proposed another
algorithm to improve the capacity of EMD. Several
schemes have been presented based on EMD to improve
either its payload [30, 31] or imperceptibility [32].
This paper introduces another LSB algorithm called
one-third probability embedding, which reduces the
probability of change per pixel. The method offers the
capacity of exactly one bit per pixel, which can be
considered as an improved version of the classic LSBMR.
In this case, the stego image is more similar to the cover
image, and hence embedding is less detectable via
steganalysers like HCF-COM-based techniques. There is
also one degree of freedom in the proposed method, which
has been exploited to change pixels in proper direction to
compensate for mandatory changes and keep the histogram
with less variations. Therefore, the proposed technique has
high embedding efficiency and also preserves the histogram
as much as possible (preliminary results of this work have
been partially reported in IEEE Conference on Multimedia
and Expo (ICME) 2009 [33]).
The paper is organised as follows: In Section 2, we give a
brief introduction to related works. Section 3 details our
proposed algorithm and Section 4 presents the effect of the
extension of our algorithm to pixel groups of more than
three pixels. Numerical results and performance analysis are
presented in Section 5. Finally, Section 6 concludes the paper.
2 Related works
2.1 LSB matching revisited
In the LSB matching revisited method [13], the message
embedding is performed on a group of two cover pixels at
the same time. The grey-level values of those two pixels are
assumed to be xi and xi + 1. After the message embedding,
the value of ith message bit mi equals to the LSB of the
stego image’s ith pixel yi and the value of the i + 1th
message bit mi + 1 is a function of yi and yi + 1 as given below
mi = LSB(yi) (1)
mi+1 = f yi, yi+1 (2)
where
f yi, yi+1 = LSB ⌊yi/2⌋ + yi+1 (3)
It can be easily proved that this function satisfies the
following property (property 1 in [13])
f (l − 1, n) = f (l + 1, n) ∀l, n [ Z (4)
Therefore, when none of mi and mi + 1 are matched to LSB(xi)
and f(xi, xi + 1), by only one change (add or subtract) in xi in a
way to match the defined function to mi + 1, both matching
equations can be satisfied. Another property of this function
is (property 2 in [13])
f (l, n) = f (l, n + 1) ∀l, n [ Z (5)
According to this property, when mi is matched to LSB(xi),
and mi + 1 does not match to the function of two pixels, the
matching can be realised by adding to or subtracting from
xi + 1 one randomly.
To analyse the performance of this method, we define r =
[ri, ri + 1] as
r = ri, ri+1
= xor mi, mi+1 , LSB xi , f xi, xi+1 (6)
which shows the result of matching prior to embedding.
Therefore, there are four possible matching states before
data insertion. The examples of these states are shown in
Table 1. If r = [0, 0], then both matching equations are
satisfied and there exists no need for modification. If r = [0,
1], mi + 1 must be manipulated. According to Property 2,
one change in xi + 1 is sufficient to satisfy the matching
equation. If r = [1, 0], we change xi such that the function
remains unchanged; and if r = [1, 1], we change it in such
a way that the function changes as well, according to
property 1 in (4).
As shown in Table 1, only three output pixels differ from
their corresponding input values through eight. Thus, we
can conclude that the probability of change per pixel as the
result of embedding is 3/8 = 0.375.
2.2 LSB compatible substitution steganography
In LSBM, the pixel value is randomly added or subtracted by
one, when it is not matched to the message bit. LSBMR
exploits this degree of freedom to decrease the probability
of change per pixel to 0.375. Another method called LSB
substitution compatible steganography (LSCS) [18] uses
this randomness to impose less variations into the histogram
of the host image. To this end, every pixel value must be
changed in a direction to compensate for previous pixel
changes if possible. Here, we briefly introduce LSCS
Table 1 Possible states in LSB matching revisited
xi 4 4 4 4
xi + 1 7 7 7 7
mi 0 0 1 1
mi + 1 0 1 0 1
ri 0 0 1 1
ri + 1 1 0 1 0
yi 4 4 3 3
yi + 1 6 or 8 7 7 7
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algorithm with which the histogram preserving property of
our proposed method will be compared.
LSCS embedding is accomplished after two complete
scans of cover images. Comparing the input secret bits with
pixel values at the first scan, the number of pixels with
value u that must be changed is calculated and recorded in
T(u), where u ranges from 0 to 255. We can decide how
many of these changes can be made by subtracting or
adding. These numbers for each value of u are recorded in
two arrays called L(u) and R(u), respectively. L(u) and R(u)
are set to zero for all us at the beginning of the second
scan, and we will have T(u) = R(u) + L(u) at the end. The
LSCS algorithm assigns proper values to L(u) and R(u) in
the second scan. These values are set in an optimum way
such that L(u) will be as close as possible to R(u − 1); that
is the number of changes from u to u − 1, is as close as
possible to the number of changes from u − 1 to u. This
property of the algorithm preserves the histogram of the
cover image in highest extent possible.
3 One-third probability embedding
3.1 Basics and definitions
In the LSBM, if the message bit does not match to the LSB of
the cover pixel, a random change (plus or minus one) in the
pixel value will occur. Therefore, with two pixels, there are
two of these random states. In LSBMR introduced in
Section 2.1, one of these random states is used in order to
obtain better embedding efficiency. In case ri = 1, xi does
not change randomly, but it changes in a way to handle the
following pixel matching and to prevent the next pixel from
being modified, based on the first property of the function.
However, when ri + 1 = 1, xi + 1 is added or subtracted by
one randomly. Thus, we still have one random state left
which is used in the proposed algorithm to further reduce
the change probability. In order to efficiently use the
existing randomness in the LSB matching, LSBMR utilises
two consecutive pixels where one random state is still left
unexploited. To employ this remaining random state, we
extend LSBMR to three consecutive pixels. Before
introducing the method, we define the minor and major
changes as follows:
Definition 1: A minor change in any x is an addition or a
subtraction by one, such that ⌊x/2⌋ remains unchanged.
Definition 2: A major change in any x is an addition or a
subtraction by one, such that ⌊x/2⌋ changes.
In accordance to these definitions, for any even grey value,
minor and major changes are addition and subtraction by
one, respectively. On the other hand, these changes are vice
versa for odd ones. It can be shown that
minor(x) = ⌊x/2⌋ × 4 + 1 − x (7)
and
major(x) = ⌊(x + 1)/2⌋ × 4 − 1 − x (8)
Now using these new terms, we look at LSBMR with another
point of view. If r = [0, 0], we have no change. In the case of
r = [0, 1], either minor or major changes in xi + 1 satisfy the f
function based on its definition. If mi is not matched but mi + 1
is, then xi should change while f should remain unchanged.
This is done by making a minor change to xi and if r = [1,
1], a major change in xi can satisfy both matching equations
simultaneously. Our algorithm takes three consecutive
pixels at a time. The message bits are embedded in the
functions of these pixels in such a way that after
embedding, we have
mi = f yi, yi+1 = LSB yi + ⌊
yi+1
2
⌋
mi+1 = f yi+1, yi+2 = LSB yi+1 + ⌊
yi+2
2
⌋
mi+2 = f yi+2, yi = LSB yi+2 + ⌊
yi
2
⌋
(9)
where yi, yi + 1 and yi + 2 are pixel values after embedding.
Compared to LSBMR, slight change to the definition of the
function f is observed here. To analyse the performance, we
set a triple result vector r = [ri, ri + 1, ri + 2] as an extension
of (6)
r = xor mi, mi+1, mi+2 ,
f xi, xi+1 , f xi+1, xi+2 , f xi+2, xi
(10)
where xi, xi + 1 and xi + 2 are pixel values before embedding.
As a result, there will be eight possible states for the value
of r. If there is only one 1 in the r vector, one minor
change in the corresponding pixel value will satisfy the
function. If there are two 1s, a major change in the variable
which is common in the two functions will satisfy both
functions at the same time. Finally, in case where r has
triple 1s, a major change in xi and a minor change in xi + 1
will be sufficient to convince all the three functions.
Table 2 shows all these possibilities. The value of the first
column is the decimal description of the vector r. The last
three columns indicate which change needs to be made to
that pixel. It is observed from Table 2 that only eight output
values differ from their corresponding inputs values through
24 states. Thus, the probability of change per pixel is
8/24 = 1/3.
It is interesting to note that there exists still one random
decision case among these eight states, which happens in
the situation corresponding to the last row of Table 2. In
other words, in case when r = [1, 1, 1], that is, none of
three matching equations are satisfied prior to embedding,
all these three decisions can be alternatively taken to satisfy
the three matching equations simultaneously: (major(xi),
minor(xi + 1)), (major(xi + 1), minor(xi + 2)), and (major(xi + 2),
minor(xi)). Now, we explain how this degree of freedom
can be used in the one-third probability embedding method
to compensate for modifications in the histogram of the
image in the best possible form.
Table 2 Different states and their corresponding changes in
one-third probability embedding
R ri ri + 1 ri + 2 xi xi + 1 xi + 2
0 0 0 0 none none none
1 1 0 0 minor none none
2 0 1 0 none minor none
3 1 1 0 none major none
4 0 0 1 none none minor
5 1 0 1 major none none
6 0 1 1 none none major
7 1 1 1 major minor none
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Despite the case 8, in all other cases when a change must
happen – that is the cases corresponding to rows 2–7 of
Table 2 – this change is mandatory and there exists no
randomness left. We define N × N change matrix C, where
N − 1 equals maximum allowed value for pixels and entries
of C are all set to zero at the beginning. Rows and columns
of C are numbered from 0 to N − 1. Embedding is
performed in two distinct phases: mandatory decisions and
random decisions. In mandatory phase, cover image is
scanned and r vectors are calculated. Then decision over all
r vectors with at least one zero – where the changes are
mandatory – are taken according to rows 2–7 of Table 2.
Suppose that we have to change a value from m to m + 1 in
this phase. We increment and decrement entries (m + 1, m)
and (m, m + 1) of the C matrix by one respectively, which
means that later we will need an m + 1 to m change to
compensate this modification and keep the histogram of the
stego image the same as that of the cover one. At the end
of this phase, all mandatory changes in the cover image are
done and the C matrix has non-zero entries only in the
superdiagonal and the subdiagonal. Almost 7/8 of the secret
message is embedded during this mandatory phase. In the
following random phase, we just consider the case r = [1, 1,
1], where three different sets of changes can satisfy all three
matching equations. For each of such cases, we calculate
the ‘score’ for each of these three changing sets. The score
is calculated by adding up the sign of the entries of the C
matrix corresponding to the two possible alterations. For
better illustration, the score calculation is explained through
an example here. Assume pixel intensities take values from
0 to 3, and after the first phase of embedding, the entries of
the change matrix are as below:
C =
0 4 0 0
−4 0 2 0
0 −2 0 1
0 0 −1 0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠ (11)
Now suppose that at the beginning of the second embedding
phase, three message bits equal m1 = 0, m2 = 1 and m3 = 1
have to be embedded in three pixels with values equal x1 =
1, x2 = 1 and x3 = 2, respectively. Therefore
r = xor [0, 1, 1], LSB([(1 + LSB(⌊1/2⌋)),
(1 + LSB(⌊2/2⌋)), (2 + LSB ⌊1/2⌋ )]) = [1, 1, 1]
(12)
All one r-vector implies that matching equation can be
satisfied through three different set of decisions: (i) a major
change in x1 and a minor change in x2, (ii) a major change
in x2 and a minor change in x3 and (iii) a major change in
x3 and a minor change in x1. Now we calculate the scores
for these three sets and pick up the best.
1: Major(x1):1 2, Minor(x2):1 0
⇒ Score(1) = sign(C(1, 2)) + sign(C(1, 0)) = 0
2:Major(x2):1 2, Minor x3 :2 3
⇒ Score(2) = sign(C(1, 2)) + sign(C(2, 3)) = 2
3:Major(x3):2 1, Minor x1 :1 0
⇒ Score(3) = sign(C(2, 1)) + sign(C(1; 0)) = −2
(13)
Consequently, the best decision is
y1 = x1 = 1, y2 = Major x2 = 2, y3 = Minor x3 = 3
(14)
Note that score ‘2’ for this set of variations indicates that these
two changes can compensate for two other changes already
happened in the mandatory phase in the cover image
histogram. In fact, this is a better pair of actions than the
other two sets which cannot compensate (set 1) or even
deteriorate (set 3) the histogram variations caused by the
mandatory phase. It is important to consider the effect of
this compensation in the change matrix by refreshing the
entries corresponding to a decision already taken and
updating the matrix C as follows
C =
0 4 0 0
−4 0 1 0
0 −1 0 0
0 0 0 0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠ (15)
As is seen, every change influences two entries of C, same as
the mandatory phase. Now suppose there exists more than one
change set with maximum score. In such situations, set of
changes with maximum sum value of the corresponding
entries in C has priority. For example, consider the C
matrix as below
C =
0 −4 0 0
4 0 2 0
0 −2 0 1
0 0 −1 0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠ (16)
Calculating scores of all three changing sets results in 2, 2 and
0 this time. But considering the second condition, we
calculate the secondary scores for the first two changing
sets as 2 + 4 = 6, and 2 + 1 = 3, respectively. Therefore in
this case, we choose the first action set. Proper changes in
C are applied accordingly. If the change sets were the same
in the second condition as well, the decision is taken
randomly.
3.2 Embedding algorithm
Now considering the above explanation, the procedure of our
embedding algorithm can be stated as follows.
Step 1 (Initialisation): The cover image is copied into the
stego one. Pixels at extremities of N − 1 and 0 are
decreased and increased by one, respectively. The N × N
matrix C where N − 1 is the maximum value allowed for
pixel values is defined and all of its entries is set to zero.
Step 2 (First scan): The cover image is divided into groups of
three pixels (xi, xi + 1, xi + 2) which are called embedding units.
((xi, xi + 1, xi + 2) are the same as (yi, yi + 1, yi + 2) at the
beginning). The order of selecting pixels is determined by a
secret key shared between the transmitter and receiver;
therefore, no one else can extract the secret message
without having the secret key and the security of the
algorithm is guaranteed. According to the details of the
method explained above, three secret bits can be embedded
into each embedding unit. Thus, three secret bits (mi, mi + 1,
mi + 2) are fetched from the input for each embedding unit.
(yi, yi + 1, yi + 2) will contain values for corresponding pixels
in the stego image after data embedding. For each
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embedding unit (xi, xi + 1, xi + 2) and its corresponding secret
bits (mi, mi + 1, mi + 2), the result of matching vector is
calculated as (10). This calculation is repeated until the
result vector is derived for all embedding units. Based on
the corresponding result vector, the embedding units are
divided into three groups. If result vector is all zero, the
embedding unit is added to unchanged list. The pixels in
this list will remain unchanged. Those embedding units
leading to a result vector of one or two 1s along with
corresponding secret bits will be added to mandatory
change list. One certain pixel among three must endure a
certain minor or major change. Finally, the all-one result
vector leads its corresponding embedding unit to the
random change list. Three possibilities are available for
each member of this list to satisfy the matching conditions.
Step 3 (Mandatory embedding phase): For each embedding
unit in this list, pixel values and corresponding secret bits
are fetched. The pixel values are changed according to the
following six cases:
Case#1: r = [1, 0, 0]:
yi = Minor xi ,
C yi, xi = C yi, xi + 1,
C xi, yi = C xi, yi − 1.
Case#2: r = [0, 1, 0]:
yi+1 = Minor xi+1 ,
C yi+1, xi+1 = C yi+1, xi+1 + 1,
C xi+1, yi+1 = C xi+1, yi+1 − 1.
Case#3: r = [1, 1, 0]:
yi+1 = Major xi+1 ,
C yi+1, xi+1 = C yi+1, xi+1 + 1,
C xi+1, yi+1 = C xi+1, yi+1 − 1.
Case#4: r = [0, 0, 1]:
yi+2 = Minor xi+2 ,
C yi+2, xi+2 = C yi+2, xi+2 + 1,
C xi+2, yi+2 = C xi+2, yi+2 − 1.
Case#5: r = [1, 0, 1]:
yi = Major xi ,
C yi, xi = C yi, xi + 1,
C xi, yi = C xi, yi − 1.
Case#6: r = [0, 1, 1]:
yi+2 = Major xi+2 ,
C yi+2, xi+2 = C yi+2, xi+2 + 1,
C xi+2, yi+2 = C xi+2, yi+2 − 1.
(17)
The score matrix (C) is passed to the next phase.
Step 4 (Random embedding phase): For all embedding units
in this list, we have r = [1, 1, 1]. Therefore, for each
embedding unit in this list, the scores of these three
possible changes are calculated; the action set with the
highest score is taken and the matrix C is updated as
explained before
Action set #1: yi = Major xi , yi+1 = Minor xi+1
Action set #2: yi+1 = Major xi+1 , yi+2 = Minor xi+2
Action set#3: yi+2 = Major xi+2 , yi = Minor xi (18)
Step 5 (Stego image generation): The processed pixels from
all three lists are integrated and located in their original
coordinates to generate the stego image. A block diagram
of the embedding phase is sketched in Fig. 1.
3.3 Extraction algorithm
The stego image is ﬁrst divided into groups of three pixels (yi,
yi + 1, yi + 2) which we call embedded units. The order of
selecting pixels is determined using the secret key. Three
secret bits are extracted from each embedded unit according
to (9). The process is continued until all embedded bits are
detected. The simple block diagram of the extraction phase
is shown in Fig. 2.
4 Extension of the proposed method to the
groups of more than three
Assuming the one-third probability embedding algorithm as a
level-three extension of a general LSB technique, its
Fig. 1 Block diagram of the embedding phase
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level-four extension can be shown as
yi+1
2
⌋
mi+1 = f yi+1, yi+2 = LSB yi+1 + ⌊
yi+2
2
⌋
mi+2 = f yi+2, yi+3 = LSB yi+2 + ⌊
yi+3
2
⌋
mi+3 = f yi+3, yi = LSB yi+3 + ⌊
yi
2
⌋ (19)
In general form, the level-k extension of this algorithm is
given as
yi+1
2
⌋
mi+1 = f yi+1, yi+2 = LSB yi+1 + ⌊
yi+2
2
⌋
..
.
mi+k−2 = f yi+k−2, yi+k−1 = LSB yi+k−2 + ⌊
yi+k−1
2
⌋
mi+k−1 = f yi+k−1, yi = LSB yi+k−1 + ⌊
yi
2
⌋ (20)
Here, we still have the r vector with its new definition similar
to its level-three form. Assuming that there exist two
consecutive ‘1’s in this vector, only one major change in
the common pixel can satisfy both corresponding matching
equations due to the extended functions defined in (20).
Therefore, we should find these consecutive ‘1’s and
remove them by a series of major changes in the most
suitable form. Afterwards, every ‘1’ which is left isolated
(‘1’ with two ‘0’ at both sides) can be handled via a minor
change in the corresponding pixel. Note that in this
analysis, the change matrix and histogram compensating
ability are ignored for the sake of simplicity. The
generalised level-k embedding procedure can be
summarised as below:
Step 1: Copy the cover image into stego image. All of the
mentioned changes in the next steps are done on this stego
image. Construct the k function of k consecutive pixels
according to (20). Continue with other steps until the last
message bit is embedded.
Step 2: Get k input bits and k corresponding stego image
pixels and calculate the appropriate result vector r with k
binary elements describing the matching results.
Step 3: Find the first pair of consecutive ‘1’s in r. Consider r
as a circular vector. It means r(k) and r(1) can be a pair too.
Go to step 5 if there is no such a pair.
Step 4: Make a major change to the common pixel of the pair
of equations corresponding to this pair of 1s in the r vector.
Then flip both bits in the r vector to zero and go to step 3.
Step 5: Look for the first ‘1’ left isolated in r. Stego image
is produced and embedding is over, if there exists no
isolated ‘1’.
Step 6: Make a minor change to the first pixel in equation
corresponding to this isolated ‘1’ in the r vector. Then flip
it to zero and go to step 5.
As an example, consider the level-7 embedding algorithm
and its embedding function. Assume that we are in step
3. Stego image has been initialised with corresponding
values of the cover image after copying, and the r vector
equals to [1011101]. We begin with r(1). As seen, there
exists no ‘1’ after. The next ‘1’ is found in r(3) forming a
pair of consecutive 1s with r(4). Hence, we have a major
change in yi + 3, which is a common pixel to the third and
fourth equations of (20). The new vector is r = [1000101]
and we are again in step 3. The next ‘1’ is isolated but
there is a ‘1’ in r(7) with immediately following ‘1’ in r(1).
Therefore, we have a major change in yi which is a
common pixel to the first and last equations in (20). The
resulted r is [0000100]. There will be nothing left to do in
step 3. There exists an isolated ‘1’ in r(5) found in step 5
which imposes a minor change in yi + 4. Now the matching
equations are satisfied.
To see the performance of the proposed technique, we have
calculated the probability of change per pixel for up to 20
levels. The results are shown in Fig. 3. The level-2
algorithm is another interpretation of the LSB matching
Fig. 2 Block diagram of the extraction phase
Fig. 3 Performance of one-third probability embedding against
embedding level
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83

revisited using these two functions
yi+1
2
⌋
mi+1 = f yi+1, yi = LSB yi+1 + ⌊
yi
2
⌋ (21)
where level-1 extension results in the LSB matching.
As observed, we have a little improvement (about 0.2%)
for level-5, compared to the level-3 algorithm described in
Section 3. As illustrated in Fig. 3, the total trend of
performance characteristic (average changes per pixel)
against the level increasing is a damping oscillation around
1/3. To avoid inessential complexity, we neglect this 0.2%
improvement and implement the original one-third
probability embedding described in Section 3. A
closed-form expression for the probability of change per
pixel in terms of the number of embedding unit pixels is
derived in Appendix 1.
5 Experimental result
5.1 Embedding distortion
Steganographic algorithms are required to be designed in a
way to leave the least possible distortion on the cover
image quality, that is, no trace of manipulation must be
perceivable in the stego image compared to the original
one. There are plenty of works measuring the similarity
between the cover and stego images. In this experiment, we
compare the distortion of different algorithms by mean
square error (MSE) and peak signal-to-noise ratio (PSNR),
defined as below
MSE =
1
MN
M
i=1
N
j=1
C(i, j) − S(i, j)
2
(22)
PSNR = 10 log10
Max2
MSE
(23)
where M and N stands for the height and width of the images,
C(i, j) and S(i, j) represent the corresponding cover and stego
image pixels, and the Max is 255 for 8-bit images. Assuming
p as the probability of change per pixel for any LSB
algorithm, we have
MSE =
1
MN
(p × MN) = p (24)
⇒ PSNR = 10 log10
255 × 255
p
It was shown that the probability of change per pixel equals to
0.5, 0.5, 0.375 and 0.333 for LSBM, LSCS, LSBMR and
one-third probability embedding, which results in the
average PSNR of 51.1411, 51.1411, 52.3905 and 52.9064,
respectively. From the four proposed sample images listed
in Table 3, we see that the resulting PSNR are well agreed
with the expected value calculated theoretically. Since the
PSNR of approximately 35 dB is assumed as the threshold
where the HVS notices the distortion, we can be sure that
all of the proposed methods are completely invisible as
shown in Fig. 4. In this figure, the original sample images
are shown in the left column, while a secret random
message of the rate of one bit per pixel is embedded into
them using the proposed method and the resulting stego
images are shown in the right column.
5.2 Reducing the probability of change per pixel
Here, the efficiency of our proposed method is compared with
the LSBM and LSBMR [13] methods. Four thousand colour
JPEG images of size 400 × 300 are used and converted into
8-bit grey-scale images. Two of the best known detectors
for the LSBM technique are calibrated and adjacency
HCF-COM introduced by Ker [6]. These features are
improvements of Harmsen’s HCF-COM detector [5], which
does not work neither on grey-scale images nor coloured
ones. Receiver operation characteristics (ROC) of these two
detectors for three embedding methods are derived and
compared in Figs. 5 and 6. These figures are generated
using different threshold values for discrimination between
the clean and stego images. Each threshold value results in
a certain pair of probability of detection-false alarm on the
ROC curve. From Figs. 5 and 6, it is perceived that the
ROC curve of our proposed method stands lower than
the other two methods, which results in less probability of
detection by steganalysers of this kind. This result was
expected since the HCF-COM detectors work on the
histogram of pixel values which is left more intact in our
algorithm with 1/3 average change per pixel as compared to
3/8 and 1/2 in the LSBMR and LSBM embedding,
respectively. Table 4 supports the above facts. In our
experiment, there exist four groups of images: The cover
images and the stego images of LSBM, LSBMR, and
one-third probability embedding algorithms. There are three
features for each group. Each row of this table shows what
ratio of each group is more similar to corresponding covers
out of whole images and according to feature specified in
that row. For example, suppose that we have 100 cover
images and 100 stego images from each one of three
embedding methods. Now assume that we calculate normal
HCF-COM feature for all 100 cover images and 300 stego
images. For each cover image, we find corresponding stego
image with the most similar normal HCF-COM feature, and
record the embedding method resulted in the stego image.
The first row and first column entry of Table 4 illustrates
that nearly 9% of stego images generated via LSBM has
normal HCF-COM feature which is more similar to the
corresponding cover image than stego images generated
from the same cover image using LSBMR and one-third
probability embedding methods. This number is called
similarity percent. From this table, it is seen that the
features of one-third embedded images are closer to that of
the original ones. Using our proposed algorithm, the normal
HCF-COM looks more natural than that of two other
features. This is the reason why one-third probability
method shows better resistance against the calibrated HCF
detector which uses normal HCF-COM of the image than
adjacency HCF-COM [6].
Table 3 PSNR values resulting from different algorithms and
images
Lena Peppers Baboon Cameraman
LSBM 51.1444 51.1439 51.1440 51.1357
LSCS 51.1444 51.1439 51.1440 51.1357
LSBMR 52.3887 52.3919 52.3858 52.3831
one-third 52.9024 52.8968 52.8963 52.9310
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5.3 Histogram compensating characteristic of
one-third probability embedding
In this section, the histogram preserving characteristic of the
one-third probability embedding is evaluated. Here, this
characteristic is compared to LSCS algorithm [18] that was
introduced in Section 2.2. Secret data are embedded in 50
grey-scale 512 × 512 Jpeg images via four different
methods. The criteria has been the number of
uncompensated changes in the histogram of image after
embedding with full capacity, which equals to off-centre
diameter sum value of C matrix, that can be stated as
uncompensated changes =
255
i=0
hc(i) − hs(i)
2
(25)
where hc and hs are histograms of the cover and stego images,
respectively. Each uncompensated change makes two
histograms to differ in two bins. This is why a factor of two
is seen in the denominator.
The ﬁrst technique is the one-third probability embedding.
The second is LSBMR, the third is LSCS and the last one is
LSBM, that is actually LSCS without histogram
compensating. The improvement in LSCS method
compared to its non-compensating version is clear in Fig. 7.
This ﬁgure presents the number of uncompensated changes
for 50 images in which a secret message is embedded at the
rate of one bit per pixel using four different methods.
Although LSCS has a higher probability of change per
pixel compared to LSBMR, but it outperforms LSBMR inFig. 5 ROC curves for the calibrated HCF COM
Fig. 6 ROC curves for the adjacency HCF COM
Table 4 Similarity percent of three stego-image groups and
original image for each feature
LSB
matching, %
LSB
revisited, %
LSB
one-third, %
normal
HCF-COM
8.74 24.55 66.71
calibrated
HCF-COM
17.74 38.67 43.59
adjacency
HCF-COM
8.17 40.95 50.88
Fig. 4 Cover images (left) and corresponding stego images (right)
generated through one-third probability embedding algorithm with
full capacity (1bpp)
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85

terms of histogram preserving. In the following, we just keep
the curves of LSCS and our proposed methods for more
accurate comparison.
From Fig. 8, it can be seen that the proposed method
outperforms LSCS in terms of histogram preserving. Thus,
one-third probability embedding outperforms LSCS and
LSBMR, in both reducing probability of change per pixel
and histogram compensating. Averaging over whole set of
50 images, the average uncompensated histogram changes
are derived as 2210, 5062, 3273 and 6374, for one-third
probability embedding, LSBMR, LSCS and LSBM,
respectively. From these average values, it can be deduced
that both one-third and LSCS methods act better in having
less manipulation at the histogram of the cover image than
other two methods not compensating the histogram.
Besides, one-third probability embedding outperforms
LSCS in terms of histogram preserving.
6 Conclusion
In this paper, we introduced a novel ± 1 image LSB
steganography algorithm with capacity of one bit per pixel.
As the result of reducing the probability of change per
pixel, in the same capacity, the stego image pixels are more
similar to the corresponding cover pixels. This feature
makes the proposed method more imperceptible and also
less detectable against well-known steganalysers such as
HCF-COM-based methods compared to LSBM and
LSBMR. This algorithm also exploits one degree of
freedom, which is used to keep the histogram of the stego
image unchanged more possibly. This advantage will help
the proposed method to show a better resistance against
histogram-based detectors. Since the extension of pixel
groups from two pixels in LSBMR to three pixels in
one-third probability embedding reduces the probability of
change per pixel, and it is demanding to investigate if there
is much more improvement to extend the one-third
probability algorithm for more than three pixels. Therefore,
we found a closed-form expression for the probability of
change per pixel for any level-k extension of our
generalised algorithm. It is proved that three is optimum
value for the number of pixels in the groups.
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8 Appendix 1
8.1 Probability of change per pixel for different
number of pixels in embedding units
A closed-form expression for the probability of change per
pixel at the kth level of the proposed steganography
algorithm is deduced in this section. In all definitions and
proofs, the k-ary vector v is assumed to be circular. An
r-length run in v consists of r consecutive one enclosed by
two zeros. For any k-ary vector v, there are 2k
combinations
where each vector can carry runs with the lengths of 1, 2
and k. Now we consider a run with the length of r in v. A
major change happens if two consecutive 1s are converted
to 0, and if there were a ‘1’ left, it will be converted to 0 as
a minor change. Therefore, for the sake of matching, the
runs with the length of 1 and 2 will be changed in one
position; the runs with the length of 3 and 4 will change in
two positions and so forth. As a matter of fact, an r-length
run, will cause ⌈r/2⌉ changes where ⌈r⌉ is the smallest
integer greater than or equal to r. To derive the probability
of change per pixel at the kth level of the algorithm, the
total number of changes in different combinations of v is
divided by k × 2k
. Assuming nr as the number of r-length
runs, the total number of changes at the level k equals to
N =
k
r=1
nr⌈r/2⌉ (26)
Lemma 1: For an integer number, ⌈r/2⌉ equals to
⌈
r
2
⌉ =
r
2
+
1 + (−1)r+1
4
(27)
Proof: Case 1: Even numbers: In this case, (1 + ( − 1)r + 1
)/
4 = 0, thus, (27) reduces to r/2 and is correct.
Case 2: Odd numbers: The proof is the same. □
Lemma 2: The number of r-length runs for a k-ary circular
vector where r ≤ k equals to
nr =
k × 2k−r−2
, r , k − 1
k, r = k − 1
1, r = k
⎧
⎨
⎩
(28)
Proof: With the assumption of r ≤ k, for r = k and r = k − 1,
the proofs are straightforward. For r < k − 1, there will be r
consecutive 1s enclosed by two 0s. Consequently, there
exists 2k − r−2
consequent 1s starting from anywhere in
vector v which in total, we will have k × 2k − r−2
runs. □
Now by considering Lemma 1, (26) can be rewritten as
N = ⌈
k
2
⌉ + k⌈
k − 1
2
⌉
+
k−2
r=1
k × 2k−r−2
× ⌈
r
2
⌉ = A + B
(29)
where
A = ⌈
k
2
⌉ + k⌈
k − 1
2
⌉ (30)
and
B =
k−2
r=1
k × 2k−r−2
× ⌈
r
2
⌉ (31)
The variables A and B are calculated separately and eventually
will be substituted in (29). We use Lemma 1, in order to
calculate the values of A and B
A = ⌈
k
2
⌉ + k⌈
k − 1
2
⌉
=
k
2
+
1 + ( − 1)k+1
4
+ k
k − 1
2
+
1 + ( − 1)k
4
=
k
2
+
1
4
+
( − 1)k+1
4
+
k2
2
−
k
2
+
k
4
+
k( − 1)k
4
=
1
4
−
1
4
+
k2
2
+
k
4
+
k
4
=
k2
+ k
2
k even
1
4
+
1
4
+
k2
2
+
k
4
−
k
4
=
k2
+ 1
2
k odd
⎧
⎪⎪⎨
⎪⎪⎩
(32)
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and
B =
1
4
× k × 2k
×
k−2
r=1
1
2
r
r
2
+
1
4
+
( − 1)r+1
4
=
1
4
× k × 2k
×
k−2
r=1
r
1
2
r+1
+
k−2
r=1
1
2
r+2
−
k−2
r=1
−1
2
r+2
=
1
4
× k × 2k
× (C + D − E)
(33)
To calculate C, we use the following lemma:
Lemma 3:
k−3
r=0
(r + 1)
1
2
r
= 4 1 − k
1
2
k−1
(34)
Proof:
k−3
r=0
xr+1
= x
k−3
r=0
xr
=
x − xk−1
1 − x
(35)
Finding the derivative of (35) yields
k−3
r=0
(r + 1)xr
=
1 − (k − 1)xk−2
(1 − x) + x − xk−1
(1 − x)2
(36)
Now, by substituting x with 1/2, we have
k−3
r=0
(r + 1)
1
2
r
=
(1/2) − (k − 1)(1/2)k−1
+ (1/2) − (1/2)k−1
(1/4)
= 4 1 − k
1
2
k−1
(37)
Proof is complete. □
Now using Lemma 3
C =
k−2
r=1
r
1
2
r+1
=
1
4
k−2
r=1
r
1
2
r−1
=
1
4
k−3
r=0
(r + 1)
1
2
r
= 1 − k
1
2
k−1
(38)
The calculations of D and E are more straightforward
D =
k−2
r=1
1
2
r+2
=
1
8
k−2
r=1
1
2
r−1
=
1
8
k−3
r=0
1
2
r
=
1
8
1 − (1/2)k−2
(1/2)
=
1
4
1 −
1
2
k−2
(39)
E =
k−2
r=1
−1
2
r+2
=
−1
8
k−2
r=1
−1
2
r−1
=
−1
8
k−3
r=0
−1
2
r
=
−1
8
1 − ( − 1/2)k−2
(3/2)
=
−1
12
1 −
−1
2
k−2
(40)
Therefore, the value of B equals to
B =
1
4
× k × 2k
× 1 − k
1
2
k−1
+
1
4
−
1
2
k
+
1
12
−
1
3
−1
2
k
=
1
4
× k × 2k
×
4
3
− (2 k + 1)
1
2
k
−
1
3
−1
2
k
=
k
3
× 2k
−
k2
2
−
k
4
−
k
12
( − 1)k
=
k
3
× 2k
−
k2
2
−
k
3
, k even
k
3
× 2k
−
k2
2
−
k
6
, k odd
⎧
⎪⎪⎨
⎪⎪⎩
(41)
Finally
N
=
k
3
× 2k
−
k2
2
−
k
3
+
k2
2
+
k
2
=
k
3
× 2k
+
k
6
, k even
k
3
× 2k
−
k2
2
−
k
6
+
k2
2
+
1
2
=
k
3
× 2k
+
3 − k
6
, k odd
⎧
⎪⎪⎨
⎪⎪⎩
(42)
Considering all, the probability of change per pixel for the kth
level of one-third probability steganography which is N/(k ×
2k
) equals to
P(k) =
1
3
+
1
6 × 2k
, k even
1
3
+
3 − k
6 × k × 2k
, k odd
⎧
⎪⎨
⎪⎩
(43)
This probability asymptotically equals to 1/3 which was
observed in Fig. 3 of Section 4 as well. The even function
of P(k) is a monotonic descending function in that it ﬁnds
its minimum asymptotically. For the odd function of P(k),
by equating numerator of its derivative (Nom) to zero we
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obtain
Nom = −6 k × 2k
− 6 × 2k
+ 6 k ln (2) × 2k
(3 − k) = 0
⇒ −k − (1 + k ln (2))(3 − k) = 0
⇒ ln (2)k2
− 3 ln (2)k − 3 = 0 (44)
The valid (positive) root of (44) is 4.06. Since for this case,
we are interested only in odd integer numbers, the desired
numbers that can minimise the odd part of P(k) are 3 and
5. Therefore, the minimum will occur among these cases
P(3) = 0.33333, P(5) = 0.33125,
P(1) = 0.33333
(45)
Equation (45) shows that the minimum probability of change
per pixel is achieved working on groups of five pixels as
embedding units. However, since the advantage of working
on five pixels instead of three pixels is not significant, for
the sake of simplicity, we implement the algorithm just on
groups of three pixels, as explained in Section 3.
www.ietdl.org
doi: 10.1049/iet-ipr.2013.0109
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