Signal Processing 142 (2018) 244–259

Contents lists available at ScienceDirect

Signal Processing
journal homepage: www.elsevier.com/locate/sigpro

Interpolation-based hiding scheme using the modulus function and

re-encoding strategy
Tzu-Chuen Lu1
Department of Information Management, Chaoyang University of Technology, Taichung 41349, Taiwan, ROC

a r t i c l e i n f o a b s t r a c t

Article history: Biswapati et al. proposed a interpolation-based hiding scheme. The scheme directly conceals the infor-
Received 23 February 2017 mation, which records the position of the modified pixel to generate the stego-image. The position value
Revised 20 July 2017
is very large, thus creating a large image distortion. This study reduces the value range of the position
Accepted 22 July 2017
values and re-encodes the values to reduce the distortion.
Available online 24 July 2017
The proposed scheme examines the probabilities for the position values and re-encodes the value ac-
Keywords: cording to its occurrence number. A re-encode function is used to obtain the rank of the position value
Image interpolation in descending order. The most frequent position value is re-encoded to zero. The re-encoded codes are
Reversible information hiding positive numbers, and the values of the codes are still large. To narrow down the value, the re-encoded
Re-encoding codes are ciphered to generate mapping codes with negative and positive numbers. A mapping function is
RS analysis proposed to map the re-encoded code to the mapping code. The mapping code is half of the re-encoded
code such that the image distortion becomes small.
The proposed scheme uses different sizes of embedding blocks to control the hiding rate and image
quality. Compared with other state-of-the-art methods, the proposed scheme is better in terms of hiding
payload and image quality.
© 2017 Elsevier B.V. All rights reserved.

1. Introduction togram shifting, difference expansion, dual images, and image in-
terpolation.
The information hiding technique is used to share secret mes- The difference expansion (DE) technique computes the distance
sages, detect tampered data, verify ownership, track piracy, and between two pixels (or prediction value and pixel) and conceals
augment data. In an information hiding scheme, cover media, such secret bits into any two-time distances. Tian [10], Alattar [1], and Li
as image, video, text, execution file, and audio, could be used to et al. [7] proposed DE-based RDH methods to generate the stego-
carry the secret message. The media that carries the message is image. DE-based RDH schemes can effectively embed secret infor-
called a stego-media. Unauthorized persons cannot detect any dif- mation in the cover image. However, this technique can cause great
ference between the cover media and the stego-media. This study distortion, which diminishes the image quality of the stego-image.
uses an image as the cover media to conceal the secret message Histogram-shifting hiding techniques are proposed to improve
for generating the stego-image [8,10]. the quality of the stego-image using the DE method. The histogram
Information hiding schemes can be categorized into two types, technique computes the probability of pixels to generate a his-
namely, reversible and non-reversible, according to whether the togram and points out the peak pixel in the histogram to embed
stego-image can be reversed or not. The reversible data-hiding the information. The other pixels between the peak pixel and a
(RDH) scheme can recover the original image after the concealed zero pixel are shifted to create a space for hiding the secret mes-
message is extracted. Conversely, the non-reversible hiding scheme sage. For example, Ni, and Lee et al. proposed histogram-shifting-
cannot recover the stego-image to its original state. Research has based hiding schemes [8,6]. The image quality of the histogram-
proposed many related schemes given that the RDH technique shifting-based scheme is high, but the embedding payload is low.
can be used in many applications, such as military use, medical Dual-image-based techniques were proposed in 2014 to en-
purposes, and digital archiving. Recent RDH schemes include his- hance the embedding payload. The dual-image-based RDH scheme
replicates the original image to generate two copy images and con-
ceals information in two images. For example, Qin et al. applied
the modulus function and exploited the modification direction and
E-mail address: tclu@cyut.edu.tw three embedding rules to generate two stego-images [17]. Lu et al.
1
URL: http://www.cyut.edu.tw/∼tclu

http://dx.doi.org/10.1016/j.sigpro.2017.07.025
0165-1684/© 2017 Elsevier B.V. All rights reserved.
 T.-C. Lu / Signal Processing 142 (2018) 244–259 245

utilized the center-folding strategy to fold the secret message be- 2.1. NMI
fore concealing it in the stego-image to enhance the image quality
[13]. Nevertheless, the major drawback of this technique is that it Image interpolation is the process of enlarging the size of an
requires two images to extract the message. original image by inserting virtual pixels between two neighbor-
One technique to solve this problem is image interpolation, ing pixels [15]. Jung and Yoo used these pixels to embed a secret
which extends an extra pixel between two neighboring pixels to message [5]. In their scheme, the virtual pixel is computed by the
embed the secret message instead of generating another image. average value of the neighboring pixels. Fig. 1 shows a diagram of
Many researchers proposed interpolation-based RDH schemes to the scheme. Fig. 1(a) is an original image I. The virtual pixels are
increase the embedding capacity [2,11,12,14,15,16]. For example, computed to generate a cover image using the following equation:
Malik et al. proposed an image interpolation-based RDH scheme
⎧  
using pixel value adjusting feature [14]. Lee et al. proposed a ⎪
⎪ I(i, j−1) + I(i, j+1)
data-hiding method based on reduplicated exploiting modifica- ⎪
⎪ , if i = 2h, j = 2w + 1,
⎪
⎪ 2
tion direction, image interpolation, and canny edge detection [11]. ⎪
⎪
⎪
Lu applied the center-folding strategy and interpolation technique ⎨ I
⎪
+I

with neighboring pixels (INP) to propose an adaptive interpolation- (i−1, j ) (i+1, j )
C(NMI
i, j ) = , if i = 2h + 1, j = 2w, (1)
based hiding scheme. In this scheme, the secret message is folded ⎪
⎪ 2
⎪
⎪
by the center value to reduce the value range and decrease the ⎪
⎪ 
⎪
⎪
image distortion [12]. Biswapati et al. used a weighted matrix to ⎪ I(i−1, j−1) + I(i−1, j ) + I(i, j−1) , otherwise.
⎪
compute the modulus summation to determine which pixel should ⎩ 2
be modified or not. The position value is added to the interpolated
pixel [2]. In Eq. (1), h and w are the height and width, respectively, of a
In Biswapati et al.’s scheme, an original image is divided into cover image, and (i, j) is the coordinate of the pixel. The interpo-
several parts with a size of 3 × 3. Then, Biswapati et al. used the in- lated image is called cover image C. Jung and Yoo concealed secret
terpolation method to generate a cover block size of 5 × 5 for each bits b in the virtual pixels C(NMI to generate the stego-pixel C(NMI ,
i, j ) i, j )
original part. Each cover block has 12 interpolated pixels, which
as shown in Fig. 1(c). When the receiver receives the stego-image
can be used to conceal 48 secret bits. To enhance the security
C’, the secret bits can be extracted from the stego-pixels C(NMI i, j )
,
of the scheme, Biswapati et al. updated the weighted matrix by
and the original image can be restored by reducing the stego-
using a shared secret key. Only authorized personnel who know
image. Fig. 2 shows how the original image is enlarged and how
the secret key can extract the correct message from the stego-
secret bits are hidden using Jung and Yoo’s scheme. Fig. 2(a) is
image.
the original image I = {84, 86, 88, 81}. Suppose that the secret bit
In Biswapati et al.’s scheme, the secret message is not directly
is b = (101110 )2 . The interpolated pixels are calculated by C(NMI1,2 )
=
concealed in the interpolated pixels. A weighted matrix is used to
compute the modulus sum and compare the sum with the secret  (84+86
2
)
 = 85, C(NMI
2,1 )
=  (84+88
2
)
 = 86, and C(NMI
2,2 )
=  (84+85+86
3
)
=
message to make sure the sum is equal to the secret message. 85. The virtual pixels are shown in Fig. 2(b).
If the sum is not equal to the secret message, then the sum is Next, they divide the cover image into several block sizes of
subtracted from the secret message to obtain a modified position 2 × 2 to embed the secret bits. The first pixel I(1, 1) is the base pixel
value. The position value is then hided into the interpolated pixel used to compute the differences with other interpolated pixels. Let
to generate the stego-pixel. d1NMI , d2NMI , and d3NMI be the differences between I(1, 1) and three
Biswapati et al.’s scheme can hide numerous secret messages in virtual pixels C(NMI , C(NMI , and C(NMI . The equation is expressed as
1,2 ) 2,1 ) 2,2 )
the cover image. However, the image quality of Biswapati et al.’s follows:
scheme can still be improved. ⎧ NMI
The key factor that influences the image quality of Biswapati et ⎪d = C(NMI
1,2 ) − I(1,1 ) ,
⎨ 1
al.’s scheme is the modified position values. The value is usually
d2NMI = C(NMI
2,1 ) − I(1,1 ) , (2)
very large, thus creating a large image distortion between the in- ⎪
⎩
terpolated image and the stego-image. d3NMI = C(NMI
2,2 ) − I(1,1) .
This study reduces the value range of the position values
and re-encodes the values to reduce the distortion. The proposed The differences in Fig. 2(b) are d1NMI = |85 − 84| = 1,
scheme examines the probabilities for the position values and re- d2NMI = |86 − 84| = 2, and d3NMI = |85 − 84| = 1. The difference
encodes the value according to its occurrence number. For the po- dNMI is a key factor for judging the length of the secret bits LNMI ,
sition value with a high occurrence number, the proposed scheme which could be concealed in the interpolated pixel. The length is
encodes it with a small number close to zero. Conversely, the pro- computed by the following:
posed scheme encodes the rare value with a large number. Fre- ⎧ NMI  
⎪L1 = log2 d1NMI ,
quent position value with a small code can effectively reduce the ⎨  
distortion between the interpolated image and the stego-image. LNMI = log2 d2NMI , (3)
Furthermore, the proposed scheme uses different sizes of em- ⎪ 2
⎩ NMI  
bedding blocks to control the hiding rate and image quality. L3 = log2 d3NMI .

In the example in Fig. 2(b), the lengths of the secret

2. Related works
bits are LNMI1
= log2 (1 ) = 0, LNMI
2
= log2 (2 ) = 1, and LNMI
3
=
log2 (1 ) = 0. In this example, LNMI
1
and LNMI
3
are both 0, which in-
Many types of interpolation techniques, such as neighbor mean
dicates that the virtual pixels C(NMI
1,2 )
and C(NMI
2,2 )
are non-embeddable.
interpolation (NMI) and INP, have been proposed. Lu applied INP to
expand an image, and Biswapati et al. used NMI to generate virtual The value of LNMI 2
is 1, which indicates that one secret bit can be
pixels. embedded into the virtual pixel C(NMI 2,1 )
. The scheme transforms the
This study describes NMI (Section 2.1), INP (Section 2.2), Lu’s secret bit of b into a decimal-based symbol β and concealed into
scheme (Section 2.3), and Biswapati et al.’s scheme (Section 2.4) in C(NMI
2,1 )
. The first secret bit of b is (1)2 , and the transformed symbol
Section 2. is β = (1)10 . The secret symbol β is then hidden in C(NMI
2,1 )
to obtain
246 T.-C. Lu / Signal Processing 142 (2018) 244–259

Fig. 1. Diagram of Jung and Yoo’s hiding scheme.

Fig. 2. Example of Jung and Yoo’s scheme.

the stego-pixel C(NMI

2,1 )
= C(NMI
2,1 )
+ β = 86 + 1 = 87. The final results are
illustrated in Fig. 2(c).

2.2. INP

Lee and Huang proposed an interpolation using the INP scheme

[6]. In Lee and Huang’s scheme, the virtual pixel C(INP
i, j )
is the aver-
age of two neighboring pixels, and the weight of the pixel on the Fig. 3. Example of the INP scheme.
right-hand side and the upper side is higher than that of the pixel
on the left side. The interpolated pixel is computed by The differences are calculated by
⎧ INP
⎧⎢ ⎛  ⎞⎥ ⎪d = C(INP
1,2 ) − Max ,
⎪⎢ ⎥ ⎨ 1
⎪⎢ ⎝I + I(i, j−1) + I(i, j+1) ⎠ ⎥
⎪
⎪
⎪⎢ (i, j−1 ) ⎥ d2INP = C(INP
2,1 ) − Max , (6)
⎪
⎪⎢ 2 ⎥ ⎪
⎩
⎪
⎪⎢ ⎥, if i = 2h, j = 2w + 1, = − Max .
⎪
⎪⎣ 2
⎦ d3INP C(INP
2,2 )
⎪
⎪
⎨ The differences are then used to compute the lengths of the
C(INP
i, j ) = ⎢ ⎥ secret bits. The equation is expressed as
⎪
⎪⎢ I + (I(i−1, j) +I(i+1, j) ) ⎥ ⎧  
⎪
⎪⎢ ( ) ⎥
= log2 d1INP ,
i −1 , j
⎣ ⎦, if i = 2h + 1, j = 2w, ⎨LINP
2
⎪
⎪ 1  
⎪
⎪ 2
LINP = log2 d2INP ,
⎪
⎪ ⎩ INP
2   (7)
⎪ C(INP
⎪
⎪

L3 = log2 d3INP .
⎩ i−1, j) +2 C(i, j−1) , otherwise.
INP

The maximum value of I(1, 1) , I(1, 2) , I(2, 1) , and I(2, 2) in Fig. 3(a) is
(4) 88. Therefore, the differences are d1INP = |84 − 88| = 4, d2INP = |85 −
88| = 3, d3INP = |84 − 88| = 4. The lengths of the differences are
Using the sample example, the original image
LINP = log2 (d1INP ) = log2 (4 ) = 2, LINP = log2 (3 ) = 1, and LINP =
is I = {84, 86, 88, 81}. The virtual pixels are C(INP = 1 2 2
1,2 ) log2 (4 ) = 2. Two secret bits b = (10)2 can be embedded in the
( I( 1 , 1 ) + I( 1 , 2 ) )
(I(1,1) + 2 )/2 = (84 + (84+86
2
)
)/2 = 84, C(INP
2,1 )
= first interpolated pixel C(INP 1,2 )
, as LINP
1
= 2. The transformed decimal
(I(1,1) +
( I( 1 , 1 ) + I( 2 , 1 ) )
= (84 + (84+88 )
)/2
)/2 = 85, and C(INP = secret symbol of b is β =(2)10 , and it is added to the interpolated
2,2 )
(84+85 )
2 2
pixel C(INP
1,2 )
to obtain the stego-pixel C(INP 1,2 )
= C(INP
1,2 )
+ β = 84 + 2 =
 2  = 84. Fig. 3(a) shows the INP results. 86. The final stego-pixels are shown in Fig. 3.
Lee and Huang then computed the differences between the
maximum value of the block and the interpolated pixels to deter- 2.3 Lu’s (t, F1 ) scheme
mine the lengths of the secret bits. The maximum is calculated by

  Lu [12] proposed an adaptive interpolation-based hiding

Max = max I(1,1) , I(1,2) , I(2,1) , I(2,2) . (5) scheme to improve Lee and Huang’s scheme. In the scheme, a pa-
 T.-C. Lu / Signal Processing 142 (2018) 244–259 247

The value pos is then added to the interpolated pixel to gen-

erate the stego-pixel. For example, Fig. 7(a) shows the origi-
nal image. The interpolated pixel C(BIS
1,2 )
is computed by C(BIS
1,2 )
=
 84+86 
2 = 85. The cover image is shown in Fig. 7(b). The origi-
Fig. 4. Diagram of Lu’s scheme with (t, F1 ) = (2, 3). nal image block is used to compute the modulus value with the
weighted matrix, as shown in Fig. 6(c). The modulus value is Val =
[(84 × 1 + 86 × 2 + · · · + 86 × 8)] mod 16 = 9. Assume that the se-
cret bit is b = (1011 )2 and that the transformed secret symbol is
β =(11)10 . The symbol is not equal to the modulus value. The mod-
ified position is pos = β − V al = 11 − 9 = 2. The first stego-pixel
C(BIS
1,2 )
is then computed by C(BIS
1,2 )
= C(BIS
1,2 )
+ pos = 85 + 2 = 87. The
final stego-image is shown in Fig. 7(c).
In Biswapati et al.’s scheme, each block size of 5 × 5 can conceal
48 secret bits. The hiding capacity of the scheme is large. How-
Fig. 5. Example of Lu’s scheme. ever, the scheme directly adds the position value to the interpo-
lated pixel, thereby causing a large distortion.
To improve the image quality of Biswapati et al.’s scheme, this
rameter pair (t, F1 ) is used to control the image quality and hiding study re-encodes these position values by using a pre-processing
capacity. The parameter t is the total number of thresholds, and procedure. The frequent position values are mapped to a small
F1 is the length of the secret message for the smoothest area. For value close to zero to reduce the distortion between the interpo-
example, the parameter pair (t, F1 ) is set to (2, 3), which means lated pixel and the stego-pixel.
that two thresholds are used to determine the embedding length,
and the first length of the smoothest area is 3. The diagram of (t,
3. Proposed scheme
F1 ) = (2, 3) is shown in Fig. 4. Suppose that the first threshold is
T1 = 0 and that the second threshold is T2 = 5. If the variance of an
The diagram of the proposed scheme is illustrated in Fig. 8. In
embedding block is located in [5, ∞), then the length of the se-
the diagram, Fig. 8(a) is the original image. The proposed scheme
cret bits is equal to F2 , where L = 4. The scheme extracts four se-
applies Lee and Huang’s INP interpolation method to enlarge the
cret bits from b = (1011 )2 and transforms it to a decimal-based se-
original image for generating the cover image. Fig. 8(b) presents
cret symbol β =(11)10 . To reduce the image distortion, Lu folds the
the cover image, which is divided into several blocks. A weighted
secret symbol using the center-folding strategy. The center value
matrix shown in Fig. 8(c) is used to compute the modulus value.
2L−1 = 24−1 = 8 is subtracted from the secret symbol β to obtain
The scheme computes the modulus value for each block and con-
the folded value β̄ = 11 − 8 = 3. The folded value is then added to
ceals the secret message in the interpolated pixels. Each interpo-
the interpolated pixel to generate the stego-pixel. Lu’s scheme ap-
lated pixel in the block can carry L secret bits. The scheme extracts
plies the INP method to create the interpolated image. Therefore,
L secret bits from the secret message b and transforms it into a
the interpolated pixels are C(Lu = 84, C(Lu = 85, and C(Lu = 84.
1,2 ) 2,1 ) 2,2 ) decimal-based secret symbol β . The differences between the mod-
The first folded value β̄ = 3 that is added to the interpolated pixel ulus value and the symbol β are counted to generate a histogram.
C(Lu
1,2 )
can obtain the first stego-pixel C(Lu
1,2 )
= 84 + 3 = 87. The fi- The differences are re-encoded according to their occurrence num-
nal stego-pixels are shown in Fig. 5. ber. To further decrease the image distortion, the proposed scheme
narrows down the re-coded code to generate the mapping code.
2.4. Biswapati et al.’s weighted matrix scheme The mapping code is half of the re-coded code. The mapping codes
are then added to the interpolated pixels to generate the stego-
In Biswapati et al.’s scheme, a weighted matrix is used to com- pixels. The detailed embedding process is illustrated in Fig. 8.
pute the modulus function value to indicate the embedding in-
formation [2]. An original image is divided into several blocks 3.1. Embedding process
with a size of 3 × 3, and the blocks are enlarged into cover blocks
with a size of 5 × 5. The interpolated pixel is the average value After generating the cover image using Eq. (4), the scheme
of the neighboring original pixels. For example, the interpolated divides the image into several blocks with two different sizes
I( 1 , 1 ) + I ( 1 , 2 )
pixel C(BIS is computed by C(BIS = , C(BIS is com- n × n, n ∈ [3, 4]. Fig. 9 shows the blocks with different sizes. The
1,2 ) 1,2 ) 2 2,1 )
I  proposed scheme uses block size to control the hiding capacity. For
( 1 , 1 ) + I( 2 , 1 )
puted by C(BIS
2,1 )
= 2 , and C(BIS
2,2 )
is computed by C(BIS
2,2 )
= a block size of 3 × 3, five interpolated pixels can be used to con-
I  ceal messages. For a block size of 4 × 4, 12 interpolated pixels can
( 1 , 1 ) + I( 1 , 2 ) + I( 2 , 1 ) + I( 2 , 2 )
4 . The diagram of Biswapati et al.’s scheme is be used to embed secret bits.
illustrated in Fig. 6. The interpolated image is called a cover image, Each block has four original pixels I = {I(1,1) , I(1,2) , I(2,1) , I(2,2) }
as shown in Fig. 6(b). A weighted matrix (wm) shown in Fig. 6(c) is that are used to compute a modulus value with a weighted matrix.
used to compute the weighted modulus value, which is computed Let wm = {wm(1,1) , wm(1,2) , wm(2,1) , wwm(2,2) | wm(i, j ) ∈ [1, 4]}
by be the weighted matrix. The matrix can be generated using a se-
   cret key to increase the security of the scheme. The modulus value
V al = I(i, j ) × wm(i, j ) mod 16. (8) is computed by
  
Each modulus value can carry four secret bits. Therefore, the se- V al = I(i, j ) × wm(i, j ) mod (2 × k + 1 ), (10)
cret bits are transformed to a decimal secret symbol and compared
with the modulus value. If the modulus value is not equal to the where k is a constant used to control the length of the secret bit
secret symbol, then the modulus value will be subtracted from the concealed in each interpolated pixel. The length of the secret bit is
secret symbol to obtain the modified position calculated by

pos = β − V al. (9) L = log2 (2 × k + 1 ). (11)

248 T.-C. Lu / Signal Processing 142 (2018) 244–259

Fig. 6. Diagram of Biswapati et al.’s hiding scheme.

Fig. 7. Example of Biswapati’s scheme.

Fig. 8. Diagram of the proposed scheme.

 T.-C. Lu / Signal Processing 142 (2018) 244–259 249

The re-encoding map among diff, H, H , and di f f is shown in

Fig. 13.
The mapping code di f f instead of the difference diff is con-
cealed in the interpolated pixel to generate the stego-pixel. The
equation is
C(reE reE
i, j ) = C(i, j ) + di f f . (14)
Following the same example, the
 re-encoded
 code Idx(di f f ) =
Fig. 9. Difference block sizes of the proposed scheme. 1 is computed by diff = sign(1 ) × 1+1
2 = 1 . The first stego-pixel is
calculated by C(reE
1,2 )
= C(reE
1,2 )
+ di f f = 84 + 1 = 85. The final stego-
When the constant k is set to four, the length of the secret image is shown in Fig. 10(c).
bits is L = log2 (2 × 4 + 1) = 3. Each interpolated pixel can con-
ceal three secret bits. When k is set to eight, then the length is 3.2. Overflow/Underflow problem
L = log2 (2 × 8 + 1) = 4. Four secret bits can be embedded into
each interpolated pixel. The proposed scheme may suffer from an overflow or under-
The scheme extracts L secret bits from the secret message and flow problem when the interpolated pixel is smaller than k − 1 (or
transforms it into a decimal-based secret symbol β . Then, the higher than 255- k) and the re-encoded code is smaller than 0 (or
modulus value is subtracted from the secret symbol to obtain the higher than k). For example, assume that the interpolated pixel is
difference C(reE
i, j )
= 254 and the re-encoded code is di f f = 4. The stego-pixel
is C(reE = C(reE + di f f = 254 + 4 = 258. The pixel has an overflow
di f f = (β − V al )% (2 × k + 1 ). (12) i, j ) i, j )
problem. To solve this problem, the scheme modifies Eq. (14) as
The scheme counts the occurrence number of each diff to gen- follows:
erate a histogram. Let H = {h0 , h1 , · · · , h2×k } be the occurrence his-
togram, where hτ is the occurrence number of diffτ . The differ- C(reE
i, j )
= C(reE
i, j ) + di f f , and (15)
ences are then ordered by the occurrence number in decreasing ⎧
order to obtain H  = {h0 , h1 , · · · , h 2×k |hτ ≥ hτ +1 }. Let Idx(.) be a ⎪
⎨C(i, j ) , i f 0 ≤ C(i, j ) ≤ 255,
reE reE
function that returns the location of diff in H and Idx(diff) be the 
C(i, j ) = C(i, j ) + 2 × (2 × k + 1 ), i f C(reE
reE reE
i, j )
< 0, (16)
re-encoded code of diff. ⎪
⎩C reE − 2 × (2 × k + 1), i f C reE > 255.
Fig. 10 shows an example. Fig. 10(a) is the cover block (i, j ) (i, j )
size of 3 × 3. The four original pixels are I = {84, 86, 88, 81},
and the weighted matrix is wm = {2, 3, 4, 1}. In this exam- Following the same example, the temporary pixel C(reE i, j )
= C(reE
i, j )
+
ple, we want to hide four bits for each interpolated pixel; di f f = 258 is greater than 255. The stego-pixel is modified as
thus, the constant k is set to 8. The length of the secret
C(reE
i, j )
= C(reE
i, j )
− 2 × (2 × k + 1) = 258 − 2 × (2 × 8 + 1) = 224.
bit is L = log2 (2 × 8 + 1) = 4. The modulus value is V al =
The proposed scheme contains several procedures, which in-
[(84 × 2 ) + (86 × 3 ) + (88 × 4 ) + (81 × 1 )] mod (2 × 8 + 1)= 9. As-
clude dividing blocks, computing the modulus value, transforming
sume that the secret bits is b = (0 0 0 0)2 and the trans-
the secret message, calculating the difference diff to generate the
formed secret symbol is β = (0 )10 The difference is di f f =
histogram H, sorting the histogram to obtain H , re-encoding diff to
( 0 − 9 )% ( 2 × 8 + 1 ) = 8 .
obtain Idx(diff), mapping Idx(diff) to obtain di f f , and adding di f f
Suppose that the occurrence number of each diff is shown in
to the interpolated pixel to generate the stego-pixel. The pseu-
H of Fig. 11. The occurrence number of di f f = 8 is h8 = 7. The
docode of the proposed scheme is shown below.
difference after ordering is denoted as H . The occurrence number
h8 = 7 is ranked in the second position, and the index of di f f = 8 Input: an interpolated image of size h × w
is Idx(di f f ) = 1. Output: the stego-image
The re-encoding processing is to set the most frequent position Pre-process:
(1) Divide the image into server blocks with size n × n.
value to zero. However, the re-encoded codes Idx(diff) are positive
(2) For each block
numbers. The values of the other codes are still very large. To nar- (3) Compute the modulus value by using Eq. 10.
row down the value range, the re-encoded codes are ciphered to (4) For each pixel in the block
generate mapping codes with negative and positive numbers. The (5) Extract the secret message and transform it to a secret symbol β .
closest codes to zero are − 1 and 1; thus, the re-encoded codes 1 (6) Compute the difference di f f by using Eq. 12.
(7) Generate a histogram H to record the occurrence number of each
and 2 are mapped to − 1 and 1, respectively. The next close codes
di f f .
are − 2 and 2; hence, the re-encoded codes 3 and 4 are mapped to (8) End for
− 2 and 2, respectively, and so on. Fig. 12 shows the diagram of the (9) End for
mapping process. The largest re-encoded codes 2 × k and 2 × k + 1 (10) Sort H in descending order to obtain H  .
are mapped to −k and k, respectively.
The formal equation used to cipher the re-encoded code to gen- Embedding process:
erate a mapping code is
  (1) Divide the image into server blocks with size n × n.
Idx(diff ) + 1 (2) For each block
diff = sign(Idx(diff ) ) × , and (13) (3) Compute the modulus value by using Eq. 10.
2 (4) For each pixel in the block
(5) Extract the secret message and transform it to a secret symbol β .
−1, i f x % 2 = 0, (6) Compute the difference di f f by using Eq. 12.
sign(x ) = Re-encode di f f to obtain Idx(di f f ).
1, otherwise. (7)
(8) Compute the mapping code di f f by using Eq. 13.
In the equation, di f f is the mapping code, and sign(x) returns (9) Generate the stego-pixel by using Eq. 16.
(10) End for
the sign of x. If x is an even number, then sign(x ) = −1. Otherwise, (11) End for
sign(x ) = 1.
250 T.-C. Lu / Signal Processing 142 (2018) 244–259

Fig. 10. Example of the proposed scheme.

Fig. 11. Re-encoding process of the example.

Fig. 12. Diagram of the mapping code.

Fig. 13. Re-encoding map of the proposed scheme.

 T.-C. Lu / Signal Processing 142 (2018) 244–259 251

Fig. 14. Test images.

Fig. 15. Experimental results of the proposed scheme and the original scheme with Fig. 16. Experimental results of the proposed scheme and the original scheme with
different n and k for Lena. different n and k for Airplane.

The algorithm complexity of the proposed scheme includes two

ceiver. The receiver divides the stego-image into several blocks
parts, namely, pre-process and the embedding process. Steps (2)–
with sizes of n × n. Each block has four original pixels I =
(9) of the pre-process include two ‘for’ loops, but it processes each
pixel only once. Hence, the algorithm complexity is O(h × w). Step
{I(1,1) , I(1,2) , I(2,1) , I(2,2)} that are the same as the original image
(10) sorts the histogram. Only 2 × k elements are present in the I = {I(1,1) , I(1,2) , I(2,1) , I(2,2) }. The original pixels are used to com-
2
histogram. Thus, the sorting complexity is O( (2 × k ) ) = O(k2 ). The pute the interpolated pixels C(reE i, j )
using Eq. (4) and the modulus
value of k is no more than 8. Hence, the sorting complexity can be value Val with the weighted matrix wm using Eq. (10).
ignored. Then, the interpolated pixel is subtracted from the stego-pixel
The complexity of each step in the embedding process is O(1). to obtain the concealed message. The equation is as follows:
Thus, the algorithm complexity of the embedding process is the
di f f  = C(reE reE
i, j ) − C(i, j ) . (17)
same as that of the pre-process. The total algorithm complexity of
the proposed scheme is O(2 × (h × w)). If diff is larger than 2 × k + 1, then the pixel will have an un-
derflow problem and will have added 2 × (2 × k + 1) previously.
3.3. Extraction and recovery Therefore, the value 2 × (2 × k + 1) needs to be subtracted from
the value diff to obtain the original di f f . Another condition is that
The stego-image I’, the numbers of n and k, the re- if diff is smaller than zero, then the pixel will have an overflow
encoding map, and the weighted matrix wm are sent to the re- problem and will have subtracted 2 × (2 × k + 1) previously. There-
252 T.-C. Lu / Signal Processing 142 (2018) 244–259

Fig. 17. Stego-images of the proposed scheme and the original scheme.
 T.-C. Lu / Signal Processing 142 (2018) 244–259 253

Fig. 18. Image quality comparisons among the proposed scheme and other methods.
254 T.-C. Lu / Signal Processing 142 (2018) 244–259

Fig. 18. Continued

fore, the value 2 × (2 × k + 1) needs to be added to the value diff Continuing the example above, in Fig. 10(c), the stego-
to obtain the original di f f . Otherwise, di f f is equal to diff . The pixels are I = {84, 86, 88, 81}. The interpolated pixel is
equation is expressed as follows: C(reE
1,2 )
= 84 by using Eq. (4). The modulus value is V al =
! [(84 × 2 ) + (86 × 3 ) + (88 × 4 ) + (81 × 1 )] mod (2 × 8 + 1)= 9 by
di f f  − 2 × (2 × k + 1 ), i f di f f  > 2 × (2 × k + 1 ), using Eq. (10). The extracted difference is di f f  = |C(reE − C(reE |=
1,2 ) 1,2 )
di f f = di f f  + 2 × (2 × k + 1 ), i f di f f  < 0,
di f f  , otherwise. 85 − 84 = 1. As 0 < di f f  < 2 × (2 × k + 1), the value di f f is
equal to diff , where di f f = 1. According to the re-encoding map
(18) shown in Fig. 11, the corresponding value of di f f = 1 is di f f = 8.
The re-encoding map is then used to find the corresponding The secret symbol is computed by β = (8 + 9 )% (2 × 8 + 1) = 0.
value diff of di f f . The proposed scheme computes the secret sym- The length of the secret bits is L = log2 (2 × 8 + 1) = 4. There-
bol using the following equation: fore, the secret symbol is transformed to the binary bit string
(0 0 0 0)2 with a length of 4 to obtain the original secret
β = (di f f + V al )% (2 × k + 1). (19) bits.
Finally, the proposed scheme transforms the secret symbol β
into a binary string to obtain the original secret bits.
 T.-C. Lu / Signal Processing 142 (2018) 244–259 255

Fig. 19. Re-encoding map among diff, H, H , and di f f of Lena with n = 3 and k = 4.

Fig. 20. Comparison results of NMI, INP, CRS, Lu, and the proposed method.
256 T.-C. Lu / Signal Processing 142 (2018) 244–259

Fig. 21. Histogram steganalysis of Lena and Mandrill.

Fig. 22. RS diagram steganalysis of Lena.

4. Experimental results and discussion

where TSB is the total number of secret bits embedded in the cover
image, and bpp means bits per pixel.
This study compares the proposed scheme with five state-of-
the-art methods, namely, NMI (Jung and Yoo [5]), INP (Lee and
Huang [6]), CRS (Tang et al. [9]), Lu [12], and Biswapati [2]. Six 4.1. Determining the number of n and k
grayscale images shown in Fig. 14 are used for testing the perfor-
mance of the proposed scheme. The system is developed by MAT- In the proposed scheme, the constants n and k are used to con-
LAB R2012a. trol the image quality and hiding payload. The first experiment
The image quality is measured by the peak signal-to-noise ratio aims to test the performance of the proposed scheme with differ-
(PSNR) computed by ent n and k. To examine the efficiency of the re-encoding strat-
  egy, the study implements the proposed method without the re-
2552 encoding strategy to generate another scheme called the original
PSNR(I ) = 10 × log10 (dB ), and (20)
MSE scheme. The proposed scheme is called reEncode. In the origi-
nal scheme, the difference diff is added directly to the interpo-
1  2
h−1 w−1
lated pixel without re-encoding. The stego-pixel is computed by
MSE = I(i, j ) − I(i, j ) , (21)
h×w C(reE
i, j )
= C(reE
i, j )
+ di f f .
i=0 j=0
Table 1 compares the original scheme and the reEncode scheme
where h and w are the image height and width, respectively; I is with difference parameters n and k. The image qualities of the
the cover image I = {I(1,1) , I(1,2) , · · · , I(h,w) }; and MSE is the mean original scheme and the reEncode scheme are 37.23 and 43.31 dbs,
square errors. A high PSNR value means a low difference between respectively, when the block size is set to n × n = 3 × 3. The hid-
the cover image and the stego-image. Conversely, a low PSNR value ing capacity is 433,500 bits when the hiding bit is set to k = 4.
means a high difference. When the block size is set to n × n = 4 × 4, the image quality is
The hiding payload is calculated by decreased to 35.90 and 41.91 dbs by using the original scheme and
T SB the reEncode scheme, respectively. However, the hiding capacity
bpp = (bpp ), (22) can increase to 589,824 bits with k = 4. However, this condition
h×w
 T.-C. Lu / Signal Processing 142 (2018) 244–259 257

Table 1 Lena image. Furthermore, the color of the image is the closest to
Image quality and hiding capacity comparisons with dif-
the original one.
ferent n and k.
Fig. 19 shows the re-encoding map example among diff, H,
n k Original reEncode Capacity bpp H , and di f f of Lena with n = 3 and k = 4. The total numbers of
3 4 37.23 43.31 433,500 1.65 the most frequent occurrence differences di f f = 1 and di f f = 4
4 4 35.90 41.97 589,824 2.25 are 18,444 and 18,009, respectively. The original scheme directly
5 4 35.44 41.51 156,060 0.60 adds the values to the interpolated pixels. The distortions made
6 4 35.19 41.26 108,375 0.41
by the differences will be di f f × di f f × H (di f f ) = 1 × 1 × 18444
7 4 35.03 41.10 79,935 0.30
8 4 34.93 41.00 61,440 0.23 of di f f = 1, and 4 × 4 × 18009 = 288144 of di f f = 4. The total dis-
9 4 35.00 41.07 47,040 0.18 tortion of the original scheme is 3224467, and the MSE is com-
10 4 34.86 40.93 39,015 0.15 puted as (3224467
512×512 )
= 12.3.
3 8 31.82 37.38 578,0 0 0 2.20
4 8 30.48 36.03 786,432 3.00
Considering the same image, the re-Encode scheme sorts H
5 8 30.03 35.58 208,080 0.79 in descending order to obtain H and the re-encoded code di f f .
6 8 29.78 35.34 144,500 0.55 Hence, di f f = 1 and di f f = 4 are re-encoded to 0 and 1. Further-
7 8 29.62 35.18 106,580 0.41 more, the codes are mapped to obtain the mapping codes 0 and
8 8 29.51 35.08 81,920 0.31
− 1. The distortions made by the differences are 0 and 18,090. The
9 8 29.59 35.15 62,720 0.24
10 8 29.45 35.01 52,020 0.20 total distortion of the re-Encoded scheme is 790,893, and MSE is
3 16 25.20 31.38 722,500 2.76 3.017. We can see that the distortion is greatly decreased such that
4 16 23.87 30.04 983,040 3.75 the image quality of the stego-image is increased effectively.

4.2. Comparison Results

does not mean that a large block size can obtain better results. Table 2 shows the comparison between the proposed method
From Table 1, we can see that the hiding capacity is decreased to with n = 4 and k = 8 and the four state-of-the-art methods in terms
156,060 bits, and image quality is reduced to 35.44 and 41.51 dbs. of PSNR and bpp. The PSNR of the proposed scheme is 4 db better
These results are the same as when the hiding bit is set to k = 8. than those of the other three methods on the other images and
Therefore, the parameter n set to 3 or 4 can obtain better results. 5 db better than that of Lu’s scheme. However, the PSNR is 2.75 db
Furthermore, the study increased the hiding bit to 16, k = 16. less than that of NMI on Tiffany.
Experimental results show that the hiding capacity can increase The comparison figures with different hiding payloads are
to 722,500 and 983,040 bits of n = 3 and n = 4, respectively. How- shown in Fig. 20. The image quality and the hiding payload of the
ever, the image qualities of the stego-images are lower than 35 db, proposed scheme with n = 4 and k = 8 are better than those of the
which may be perceived by the human eye. Hence, the suggested other methods.
hiding bit is set to k = 4 or k = 8. Table 3 shows the execution time comparisons of NMI, INP, CRS,
Figs. 15 and 16 show more results with different n and k. The Lu, and the proposed method. The proposed scheme statistics the
proposed scheme (called reEncode) with n = 3 and k = 4, can ob- histogram of the differences and ranking the values. Hence, the ex-
tain the highest image quality. However, the hiding payload of the ecution time (approximately 1.14 s) is worse than that of the other
proposed scheme with n = 3 and k = 4 is less than that of the pro- methods. However, the total execution time is still acceptable.
posed scheme with n = 4 and k = 4. The parameters set to n = 4 Table 4 shows the comparison between Biswapati’s scheme and
and k = 8 can obtain the highest hiding payload. Therefore, the pro- the proposed scheme with n = 4 and k = 8, and n = 4 and k = 4, re-
posed scheme can obtain improved image quality when the pa- spectively. In the high hiding capacity, the PSNR value of the pro-
rameters are set to n = 3 and k = 4. The parameters can be set to posed scheme is higher than that of Biswapati’s scheme by approx-
n = 4 and k = 8 to obtain a high hiding payload. imately 0.23 db. In the low hiding capacity, the PSNR value of the
The image quality and the hiding payload of the proposed proposed scheme is higher than that of Biswapati’s scheme by ap-
scheme are higher than those of the original scheme, thus indi- proximately 4.73 db.
cating that the re-encoding strategy is workable.
The cover images and the stego-image using the original 4.3. Steganalysis
scheme and the proposed scheme are shown in Fig. 17. The figures
indicate that the stego-images are similar to the cover images. To prove the security of the proposed scheme, steganalysis tests
Fig. 18 shows the image quality comparisons among the pro- such as histogram steganalysis, RS steganalysis, primary sets, Chi
posed scheme and the other methods. Fig. 18(a) is the original im- square, sample pairs, RS analysis, and fusion detection are con-
age Lena. The columns Face and Eye are partial magnification of ducted.
Lena’s face and eye. Fig. 18(b) shows the stego-image of CRS. The Histogram steganalysis is used to compare the shapes of the
eye part of Fig. 18(b) has a large amount of noise, and edges have histograms of the cover image and the stego-image to determine
be severely modified. A considerable amount of salt-and-pepper if a message has been concealed in the image. Figs. 21(a) and (b)
appeared in the edges. show the histogram comparison between Lena and Mandrill, re-
Fig. 18(c) shows the stego-image of NMI. The image quality of spectively. The curve starting with the symbol “∗ ” is the histogram
Fig. 18(c) is worse than that of Fig. 18(b). The image contains nu- of the stego-image. In the figure, the shape of the stego-image is
merous redundant white lines, and the eye becomes very fuzzy. almost the same as that of the cover image.
The same situation is observed in Fig. 18(d). RS steganalysis is also conducted. Fridrich et al. [4] proposed RS
Figs. 18(e) and (f) show the stego-images obtained by using Lu’s steganalysis in 2001. In their scheme, pixels are categorized into
scheme and the original scheme without re-encoding. Both images different groups by using a judgment function and a flipping func-
do not have the salt-and-pepper noise and redundant white lines. tion. The judgment function determines the smoothness or regu-
However, the color of the images is lighter than that of the original larity of each group. The flipping function defines the groups into
image. three different categories: regular (R), singular (S), and unusable
Fig. 18(g) shows the stego-image obtained by using the pro- (U). The group percentages of regular, singular, and unusable with
posed scheme. The image is the most similar image to the original mask M = [1 0 0 1] and -M = [− 1 0 0 − 1] are indicated by R_M_G,
258 T.-C. Lu / Signal Processing 142 (2018) 244–259

Table 2
PSNR and bpp comparison of NMI, INP, CRS, Lu, and the proposed method for stego-image and cover image.

Method NMI INP CRS Lu Original reEncode

File Name PSNR bpp PSNR bpp PSNR bpp PSNR bpp PSNR bpp PSNR bpp

Airplane 33.05 1.05 32.64 1.19 31.54 1.51 30.74 2.83 30.48 3.00 36.03 3.00
Tiffany 37.77 0.93 37.15 1.09 36.00 1.37 30.93 2.83 30.75 3.00 35.02 3.00
Lake 32.48 1.10 31.76 1.25 30.75 1.60 30.74 2.83 30.48 3.00 36.03 3.00
Lena 34.89 1.10 34.32 1.25 33.19 1.57 30.74 2.83 30.48 3.00 36.03 3.00
Mandrill 32.40 1.10 31.65 1.25 31.36 1.60 30.74 2.83 30.48 3.00 35.99 3.00
Pepper 34.27 1.10 33.72 1.25 33.39 1.53 30.31 2.83 30.43 3.00 34.85 3.00

Table 3 U_FM_G have the same situations. Therefore, the proposed scheme
Execution time comparisons of NMI, INP,
cannot be detected by the RS steganalysis.
CRS, Lu, and the proposed method.
The experimental results of the primary sets, chi square, sample
Method Execution time (s) pairs, RS analysis, and fusion detection [3] are shown in Table 5. All
Proposed scheme 3.592 experimental numbers are small, thus indicating that the proposed
Original 2.349 scheme is secure and robust.
Lu 3.025
CRS 2.854
INP 2.100
NMI 1.900 5. Conclusion

This study proposes an interpolation-based reversible hiding

Table 4
scheme by using the difference in the re-encoding strategy. The
PSNR and bpp comparison between Biswapati’s scheme
and the proposed method.
scheme applies the INP interpolation method to enlarge the orig-
inal image for generating the cover image. The secret message is
Method Biswapati reEncode
then concealed in the interpolated pixel. The proposed scheme
File name Capacity PSNR Capacity PSNR
uses a modulus function and a weighted matrix to compute a
Lena 776,224 35.80 786,732 36.03 modulus value for hiding secret bits. The difference between the
560,0 0 0 37.24 589,824 41.97
modulus value and the secret message is embedded in the virtual
Airplane 776,224 35.81 786,732 36.03
560,0 0 0 37.24 589,824 41.97 pixel. Before embedding, the differences are re-encoded according
to their occurrence number.
The proposed scheme re-encodes the frequency difference with
a small value. Conversely, the rare difference is re-encoded with a
R_FM_G, S_M_G, S_FM_G, U_M_G, and U_FM_G, respectively. The large value. Furthermore, the proposed scheme uses two parame-
hypotheses are R_M_G ∼ = R_FM_G, S_M_G ∼ = S_FM_G, and U_M_G ters n and k to control the image quality and hiding payload, where
∼
= U_FM_G. Figs. 22(a) and (b) show the RS steganalysis results of n × n is the size of a block, and k is the constant used to decide the
Lena using the original scheme and the proposed scheme, respec- length of the secret bits. To obtain a high image quality, small n
tively. and k are recommended, for example, n = 3 and k = 4. By contrast,
The steganalysis results reveal that the curve of R_M_G is simi- to obtain a high hiding payload, large n and k are suggested, for
lar to that of R_FM_G. The curves of S_M_G, S_FM_G, U_M_G, and example, n = 4 and k = 8.

Table 5
Steganalysis results of StegExpose.

File name n k Primary sets Chi square Sample pairs RS analysis Fusion (mean)

Airplane 3 4 0.00640 0.00329 0.00775 0.00378 0.00530

3 8 0.00561 0.00581 0.00214 0.00670 0.00506
4 4 0.00792 0.00305 0.01047 0.01122 0.00816
4 8 0.00865 0.00388 0.00482 0.00844 0.00645
Tiffany 3 4 0.00764 0.00381 0.01028 0.01341 0.00878
3 8 0.02325 0.00847 0.02814 0.02569 0.02139
4 4 0.00176 0.00561 0.0 0 083 0.00890 0.00428
4 8 0.02910 0.00757 0.02454 0.03490 0.02403
Lake 3 4 0.00113 0.00183 0.00669 0.00404 0.00342
3 8 0.03037 0.00163 0.01820 0.01912 0.01733
4 4 0.00135 0.00117 0.00188 0.0 0 058 0.00124
4 8 0.03081 0.00115 0.01497 0.01514 0.01552
Lena 3 4 0.01325 0.0 0 0 09 0.00192 0.0 0 081 0.00402
3 8 0.0 020 0 0.0 0 034 0.00231 0.00623 0.00272
4 4 0.01833 0.0 0 065 0.01556 0.00345 0.00950
4 8 0.01163 0.00127 0.01363 0.00272 0.00731
Mandrill 3 4 0.01126 0.00137 0.00780 0.02601 0.01161
3 8 0.00970 0.0 0 093 0.03324 0.01517 0.01476
4 4 0.02282 0.00121 0.01492 0.03324 0.01805
4 8 0.0 0 097 0.00247 0.00489 0.03562 0.01099
Pepper 3 4 0.03750 0.00264 0.02881 0.01831 0.02182
3 8 0.10726 0.00190 0.08165 0.07413 0.06623
4 4 0.00407 0.00263 0.00264 0.01292 0.00556
4 8 0.08059 0.00332 0.08103 0.08464 0.06239
 T.-C. Lu / Signal Processing 142 (2018) 244–259 259

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