Research Article

# CSIS: compressed sensing-based enhanced-embedding capacity image
steganography scheme

Rohit Agrawal1 Kapil Ahuja${}^{1\corr}$<EMAIL_ADDRESS>1Mathematics of Data
Science (MODS) Laboratory, Indian Institute of Technology Indore, Simrol,
Indore, India

###### Abstract

Image steganography plays a vital role in securing secret data by embedding it
in the cover images. Usually, these images are communicated in a compressed
format. Existing techniques achieve this but have low embedding capacity.
Enhancing this capacity causes a deterioration in the visual quality of the
stego-image. Hence, our goal here is to enhance the embedding capacity while
preserving the visual quality of the stego-image. We also intend to ensure
that our scheme is resistant to steganalysis attacks.

This paper proposes a Compressed Sensing Image Steganography (CSIS) scheme to
achieve our goal while embedding binary data in images. The novelty of our
scheme is the combination of three components in attaining the above-listed
goals. First, we use compressed sensing to sparsify cover image block-wise,
obtain its linear measurements, and then uniquely select permissible
measurements. Further, before embedding the secret data, we encrypt it using
the Data Encryption Standard (DES) algorithm, and finally, we embed two bits
of encrypted data into each permissible measurement. This is the first attempt
to rigorously embed more than one bit. Second, we propose a novel data
extraction technique, which is lossless and completely recovers our secret
data. Third, for the reconstruction of the stego-image, we use the least
absolute shrinkage and selection operator (LASSO) for the resultant
optimization problem. This has the advantages of fast convergence and easy
implementation. This component is also new.

We perform experiments on several standard grayscale images and a color image,
and evaluate embedding capacity, Peak Signal-to-Noise Ratio (PSNR) value, mean
Structural Similarity (SSIM) index, Normalized Cross-Correlation (NCC)
coefficients, and entropy. We achieve 1.53 times more embedding capacity as
compared to the most recent scheme. We obtain an average of 37.92 dB PSNR
value, and average values close to $1$ for both the mean SSIM index and the
NCC coefficients, which are considered good. Moreover, the entropy of cover
images and their corresponding stego-images are nearly the same. These
assessment metrics show that CSIS substantially outperforms existing similar
steganography schemes.

##  1 Introduction

The primary concern during the transmission of digital data over communication
media is that anybody can access this data. Hence, to protect this data from
being accessed by illegitimate users, the sender must employ some security
mechanisms. In general, there are two main approaches used to protect secret
data; cryptography Stalling and steganography Abbas . In cryptography, the
encryption process transforms the secret data, known as plain-text, into
cipher-text using an encryption key. This text is in unreadable form, hence,
it attracts the opponents to exploit the content of the cipher-text by
employing some brute-force attacks Stalling . However, steganography avoids
this scenario.

Steganography is derived from the Greek words steganos means “covered or
secret" and graphie means “writing". In steganography, the secret data is
hidden into some other unsuspected cover media so that it is visually
imperceptible. Here, both the secret data as well as the cover media may be
text or multimedia. The media obtain after embedding secret data into cover
media is called stego-media.  Some recent steganography schemes that use text
as cover media are MMT_2020 and Mathematics_2020 . In MMT_2020 , the authors
have proposed an Arabic text steganography scheme, where the secret message is
hidden within the text by using Unicode standard encoding. In Mathematics_2020
, the authors have proposed a character-level text generation-based linguistic
steganography scheme, where the secret message is embedded in the text's
content.

Recently, the steganography schemes that use images as the cover media have
gained a lot of research interest due to their heavy use in Internet-based
applications Typically, these images are transmitted in a compressed format.
So here, we focus on compressed domain-based image steganography. In this, the
challenges are;

  1. 1.

Improving the embedding capacity.

  2. 2.

Maintaining the quality of the stego-image.

  3. 3.

The scheme should be resistant to steganographic attacks.

Although images can be embedded into images, our focus is on embedding binary
data into images.

In the following paragraphs, first we discuss the way in which secret data can
be embedded into cover images, then we summarize some existing schemes and
their limitations, and finally we argue how the scheme presented in this paper
outperforms the existing schemes.

Secret data can be embedded in images by two ways; spatially and by using a
transform. In the spatial domain based image steganography scheme, secret data
is embedded directly into the image by some modification in the values of the
image pixels. Some well-known schemes here are listed in Abbas ; Chan1 ;
Steg_2019_FGT ; Intech_2016 ; Intech2_2016 ; IEEE_2016 ; JIS_2018 ;
Arabian_3D_Mesh_2018 . In the transform domain based image steganography
scheme, first, the image is transformed into frequency components, and then
the secret data is embedded into these components. Some commonly used such
schemes are JSteg Jsteg , F5 F5 , and Outguess Outguess . Some other
techniques, which do not carry specific names are given in references Chang ;
Liu ; Pan ; Steg_ExpertSystem ; Steg_Rohit ; JAIS_2017 ; Elsvier_2018 ;
IEEE_2019 ; Rohit_arXiv .

The spatial domain based image steganography outperforms the transform domain
one in terms of embedding capacity, but the stego-image has a high amount of
redundant data. Digital images transmitted through communication media are
usually of this type. Since transform based schemes reduce the redundancy
present in the image and represent it in a compressed form, they are preferred
for transmission.

Most of the transform domain based scheme follow either Discrete Cosine
Transform (DCT) or Wavelet Transform (WT). The DCT based schemes are also
called the JPEG compression based image steganography techniques. Several
variants of DCT based schemes have been proposed in the literature Jsteg ; F5
; Outguess ; Chang ; Liu ; Pan ; Steg_Rohit ; JAIS_2017 ; Elsvier_2018 ;
IEEE_2019 ; Rohit_arXiv . For the schemes Jsteg ; F5 ; Outguess ; Chang ; Liu
; Pan ; Steg_Rohit ; Elsvier_2018 ; IEEE_2019 , secret data is binary bits,
and for JAIS_2017 ; Rohit_arXiv , secret data is images.

In Jsteg ; Outguess ; Chang , the secret data is embedded by flipping the
least significant bit (LSB) of the quantized DCT coefficients obtained from
the cover image. This process is considered as a direct embedding mechanism.
Alternatively, methods in F5 ; Liu ; Pan ; Steg_Rohit ; Elsvier_2018 ;
IEEE_2019 ; JAIS_2017 ; Rohit_arXiv are considered as indirect steganography
schemes in which the quantized DCT coefficient values are altered according to
certain secret message bits or secret image pixels. By steganalysis, which is
the study of detecting the secret data hidden using steganography, it has been
observed that the indirect steganography mechanism is superior to the direct
one due to its capability in resisting certain statistical attacks. The most
common statistical attacks are the chi-square test, and the shrinkage effect
Westfeld ; Fridrich ; Patsakis . Hence, the schemes Jsteg ; Outguess ; Chang
are not resistant to such attacks, while the schemes F5 ; Liu ; Pan ;
Steg_Rohit ; Elsvier_2018 ; IEEE_2019 ; JAIS_2017 are resistant to them, but
their embedding capacity is limited. If we try to increase the embedding
capacity of the later schemes, then the quality of the stego-images gets
degraded. The scheme Rohit_arXiv has high embedding capacity with resistance
to steganographic attacks, but here, the secret data is the images, which is
different from our goal of embedding binary data in images.

Most recent Wavelet transform based steganography schemes are given in
Steg_ExpertSystem ; DWT_2019 . In Steg_ExpertSystem , the authors have
proposed a steganography scheme based upon edge identification and XOR coding
that uses Wavelet transformation. This scheme is resistant to steganographic
attacks, but here also the embedding capacity is significantly less. As above,
if we try to increase embedding capacity, then the quality of stego-image gets
degraded. The scheme given in DWT_2019 embeds a medical image into a cover
image using Redundant Integer Wavelet Transform (RIWT) and DCT. This scheme's
purpose is again different from ours of embedding binary data in images.

As discussed above, conventional transform domain based image steganography
schemes provide good visual quality stego-image and are resistant to
steganographic attacks, but their embedding capacity is limited. If we try to
increase their embedding capacity, then the stego-image quality degrades. To
overcome this limitation, in this manuscript, we utilize another paradigm, the
compressed sensing, which also fulfills all the requirements of image
steganography. Next, we present literature regarding compressed sensing-based
steganography schemes. These works help to achieve some of the above
objectives of steganography but not all, which we do.

In Sreedhanya , and Rohit , steganography schemes based on compressed sensing
and Singular Value Decomposition (SVD) have been presented. In these schemes,
secret medical image data is embedded into an image cover media. Both these
approaches use a similar embedding approach, but use compressed sensing
differently. In these, first, encrypted measurements of the secret image are
obtained using the compressed sensing technique, and then these encrypted
measurements are embedded into the cover image using SVD based embedding
algorithm. In Sreedhanya , the PSNR (Peak Signal-to-Noise Ratio, discussed in
Section 4.2.2) value of the stego-image is greater than 30 dB, which shows
that it produces good quality stego-images. But the PSNR value of the
constructed secret image is very low, i.e. the quality of the secret image is
degraded very much. In contrast, in Rohit , both the stego-image as well as
the reconstructed secret image preserved good visual quality. But, the goal in
both these schemes is different from ours. In these schemes, the secret data
is an image. If these techniques are applied on binary data that we want to
embed, the information will be lost. In Pan , the authors have proposed an
image steganography scheme based on sub-sampling and compressed sensing. In
this scheme, the PSNR value of the stego-image is greater than 30 dB, also the
secret data is binary. However, the embedding capacity in this scheme is very
low.

Moreover, some other compressed sensing-based image steganography schemes are
listed in Patsakis , Patsakis1 , and MMT_2019 . In Patsakis , the authors have
presented the application of compressed sensing to detect steganographic
content in the LSB steganography scheme. In Patsakis1 , the authors have
proposed a DCT steganography classifier based on a compressed sensing
technique. Here, the original image is identified from a set of images
containing the original image and some instances of stego images. In MMT_2019
, the authors have proposed an image steganalysis technique for secret signal
recovery. These steganography schemes are not related to our work because the
focus of Patsakis and MMT_2019 is steganalysis, while Patsakis1 focuses on
steganography classifier. Hence, we do not discuss these schemes in detail.

The scheme that we propose satisfies all the goals mentioned in the earlier
paragraphs, i.e. increased embedding capacity without degrading the quality of
stego-images as well as making the scheme resistant to steganalysis attacks.
Our scheme has three components, which we discussed next. The first component
of our scheme consists of three parts; (1) we use compressed sensing to
sparsify cover image block-wise and obtain linear measurements. Here, we
design an adaptive measurement matrix instead of using a random one. Using our
adaptive measurement matrix, we uniquely select a large number of permissible
measurements compared to existing schemes. Hence, we achieve a high embedding
capacity. Moreover, these measurements act as encoded transformed
coefficients, and hence, this adds security to our proposed scheme as well;
(2) we encrypt the secret data using the Data Encryption Standard (DES)
algorithm Stalling . This adds another layer of security to our scheme; (3) we
embed two bits of secret data into each permissible measurement instead of
commonly embedding one bit per measurement. This is a first attemp to
rigorously embed more than one bit. Second, we completely extract secret data
without any loss using our extraction algorithm. Third, we use the alternating
direction method of multipliers (ADMM) solution of the least absolute
shrinkage and selection operator (LASSO) formulation of the underlined
optimization problem in the stego-image construction. The advantages of using
ADMM and LASSO are that they have broad applicability in the domain of image
processing, require a little assumption on the objective function's property,
have fast convergence, and are easy to implement. This is also a completely
new contribution.

For performance evaluation, we perform experiments on standard test images. To
check the quality of stego-image, we reconstruct it from the obtained modified
measurements and then compare it with its corresponding cover image. We
evaluate embedding capacity, Peak Signal-to-Noise Ratio (PSNR) value, mean
Structural Similarity (SSIM) index, Normalized Cross-Correlation (NCC)
coefficient, and entropy. We achieve 1.53 times more embedding capacity when
compared with the most recent scheme of this category. We achieve a maximum of
40.86 dB and an average of 37.92 dB PSNR values, which are considered good.
The average values of mean SSIM index and NCC coefficients are close to 1,
which are again considered good. Moreover, the entropy of cover images and
their corresponding stego-images are nearly the same. In the Experimental
Results section, we also show that our scheme outperforms existing compression
based steganography schemes Jsteg ; F5 ; Outguess ; Liu ; Pan ; Steg_Rohit ;
Steg_2019_FGT ; Steg_ExpertSystem .

The rest of the paper has four more sections. Section 2 describes the
compressed sensing technique. Section 3 explains our proposed steganography
scheme including embedding of the data, extracting it, and stego-image
reconstruction process. Section 4 presents the experimental results. Finally,
Section 5 gives conclusions and future work.

##  2 Compressed Sensing

Compressed sensing is used to acquire and reconstruct the signal efficiently.
Traditionally, the successful reconstruction of the signal from the measured
signal must follow the popular Nyquist/ Shannon sampling theorem, which states
that the sampling rate must be at least twice the signal bandwidth. In many
applications such as image, audio, video, data mining, and wireless
communications & networks, where the signal is sparse or sparsified in some
domain, the Nyquist rate is too high to achieve. There is a fairly new
paradigm, called compressed sensing that can represent the sparse signal by
using a sampling rate significantly lower than the Nyquist sampling rate
Candes ; Donoho . Hence, the application of compressed sensing has gained
popularity in many areas. Some of them are image processing Romberg , radar
system Steeghs , MRI Imaging CS_MRI , and noise separation from data Romberg .

Compressed sensing projects the sparse signal onto a small number of linear
measurements in such a way that the structure of this signal remains the same.
The sparse signal can be reconstructed approximately from these measurements
by an optimization technique. However, the reconstruction of the signal is
possible only when the original signal is sparse, and it satisfies the
Restricted Isometric Property (RIP) Candes2 (discussed in Section 2.2). If the
original signal is not sparse, then it can often be artificially sparsified. A
brief description of signal sparsification, obtaining linear measurements, and
reconstruction of the approximate sparse signal is given next.

###  2.1 Signal Sparsification

Let the original signal be $x\in\mathbb{R}^{N\times 1}$. The signal x is K
sparse when it has maximum K number of non-zeros coefficients, i.e.
$\left\|x\right\|_{0}\leq K$, where $||\cdot||_{0}$ denotes the
$\ell_{0}-norm$ of a vector, and the remaining coefficients are zero or nearly
zero. Let the original signal x not be sparse and be represented in-terms of
$\left\\{\psi_{i}\right\\}_{i=1}^{N}$ basis vectors each of length $N\times
1$, then

$s=\Psi^{T}x,$ (1)

where, $s\in\mathbb{R}^{N\times 1}$ and
$\Psi=\left[\psi_{1},\psi_{2},...,\psi_{N}\right]\in\mathbb{R}^{N\times N}$ is
an orthogonal matrix. If $K\ll N$ then this signal is sparsifiable Candes3 ,
$s$ is the sparse representation of $x$, and $\Psi$ is the corresponding
sparsification matrix.

###  2.2 Sensing Matrix and Linear Measurements

In the compressed sensing framework, we acquire $M(M<N)$ linear measurements
from the inner product between the original signal $x\in\mathbb{R}^{N\times
1}$ and M measurement vectors $\left\\{\phi_{i}\right\\}_{i=1}^{M}$, where
$\phi_{i}\in\mathbb{R}^{N\times 1}$. Considering the measurement/ sensing
matrix as
$\Phi=\left[\phi_{1}^{T};\phi_{2}^{T};...;\phi_{M}^{T}\right]\in\mathbb{R}^{M\times
N}$, the measurements $y\in\mathbb{R}^{M\times 1}$ are given as Candes3

$y=\Phi x.$ (2)

If the input signal is not sparse but sparsifiable, then using the above
theory we get

$y=\Phi\Psi s=\Theta s,$ (3)

where $\Theta=\Phi\Psi$ is again the measurement matrix of size $M\times N$.
Usually, in the compressed sensing framework, the measurement matrix is
nonadaptive. That is, the measurement matrix is fixed and does not depend on
the signal. However, in certain cases, adaptive measurements can lead to
significant performance improvement.

The main concern here is to design the measurement matrix in such a way so
that the most of the information and the structure of the signal is preserved
in the measurements. This would imply that original signal would be recovered
efficiently from these measurements. To achieve this, for all K-sparse signals
$s$, the measurement matrix should hold the following inequality Candes2 .

$\left(1-\delta_{K}\right)\leq\frac{\left\|\Theta
s\right\|_{2}^{2}}{\left\|s\right\|_{2}^{2}}\leq\left(1+\delta_{K}\right),$
(4)

where $\delta_{K}\in(0,1)$ is an isometric constant. The above inequality is
called the RIP that informally says that the $\ell 2-norm$ of the sparse
signal $s$ and the measurement $\Theta s$ should be comparable. Apart from
satisfying the RIP, the minimum number of measurements required, i.e. the
minimum value of $M$, is also a concern in the measurement matrix design.

###  2.3 Reconstruction of the Approximate Signal

As discussed in the previous subsection, size of the measurement $y=\Phi
x=\Phi\Psi s=\Theta s$ is less than the size of the original signal $s$.
Hence, the reconstruction of the signal from measurements becomes an ill-posed
problem. That is, the solution of an under-determined linear system of
equations is to be found. If the matrix $\Theta$ satisfies the RIP, then the
sparse signal $s$ can be reconstructed approximately by solving the following
optimization problem Baraniuk :

$\displaystyle\min_{s}\left\\{\text{number of i such
that}\>s\left(i\right)\neq 0\right\\}$ (5) $\displaystyle\text{Subject
to}\hskip 2.84544pt\Phi\Psi s=y.$

In the above equation, the function to be minimized is simply the number of
nonzero coefficients in the vector $s$. This equation is referred to as
$\ell_{0}-norm$ minimization problem. It is combinatorial and an NP-hard
problem Baraniuk . The other approach is to substitute the $\ell_{0}-norm$ by
the closest convex norm, i.e. the $\ell_{1}-norm$, or

$\displaystyle\min_{s}\left\|s\right\|_{1}$ (6) $\displaystyle\text{Subject
to}\hskip 2.84544pt\Phi\Psi s=y,$

where $||\cdot||_{1}$ denotes the $\ell_{1}-norm$ of a vector. The approach to
reconstruct the sparse signal $s$ by solving the above equation is termed as a
convex optimization method.

Other approaches such as Greedy based (OMP OMP , CoSaMP Needell ), sparse
reconstruction by separable approximation Stephen , Bayesian strategy David ,
and ADMM solution of the LASSO formulation of the above optimization problem
can also be used to reconstruct the sparse signal from the measurements
ADMMSBoyd ; Lasso .

Next, we give a brief idea of LASSO and ADMM, which we use. The general LASSO
problem is given as Lasso

$\min_{z}\left\\{\frac{1}{2}\|Az-b\|_{2}^{2}+\lambda\|z\|_{1}\right\\},$ (7)

where $z\in\mathbb{R}^{n}$, $A\in\mathbb{R}^{p\times n}$,
$b\in\mathbb{R}^{p}$, $\|\cdot\|_{2}$ is the $\ell_{2}$ norm and $\lambda>0$
is a scalar regularization parameter also called Lagrangian parameter
ParaLarPD . Further, (7) is transformed into a form solvable by ADMM ADMMSBoyd
. That is

$\displaystyle\begin{split}&\min\limits_{z,z_{1}}\left\\{\frac{1}{2}\|Az-b\|_{2}^{2}+\lambda\|z_{1}\|_{1}\right\\}\\\
&\text{ Subject to }z-z_{1}=0.\end{split}$ (8)

Finally, ADMM solve the above optimization problem.

Now, we discuss how to solve our signal reconstruction problem, i.e.
(LABEL:eq:_CS_reconstruction_l1) by LASSO and ADMM. For our case,
$\Theta=\Phi\Psi$ is the measurement matrix, and $\Theta\in\mathbb{R}^{M\times
N}$. In the compressed sensing framework, matrix $\Theta$ is underdetermined,
i.e. $M<N$. Hence, there is equivalent solution of
(LABEL:eq:_CS_reconstruction_l1), which is given as l1_Lasso

$\displaystyle\min_{s}\left\\{\frac{1}{2}\left\|\Theta
s-y\right\|_{2}^{2}+\lambda\left\|s\right\|_{1}\right\\}$ (9)

Here, we observe that (LABEL:eq:_CS_reconstruction_Lasso) is equivalent to (7)
with $\Theta=A$, $s=z$ and $y=b$.

Finaly, we briefly mention a theoretical result related to reconstruction. In
Candes4 , it is shown that for sufficiently small constant C ($C>0$), the
K-sparse signal $s$ of size N can be approximately reconstructed from M
measurements $y$ if $M\geq CK\left(\log N\right)$. After recovering the sparse
signal $s$, the original signal $x$ can be obtained as $x=\Psi s$. For us,
this property holds.

##  3 Proposed Method

Our proposed compressed sensing-based image steganography scheme consists of
the following components; data embedding, data extraction, and stego-image
construction, which are discussed in the respective sections below.

Algorithm 1 Embedding Rule

1:

  * •

$y$: Sequence of transform coefficients.

  * •

S: Encrypted secret bit sequences which is to be embedded.

2:

  * •

$z$: The modified version of transform coefficients.

3:if ($length(S)<2\times length(y)$) then

4: for $j=1$ to $length(y)$ do

5: if (${y}\left(j\right)=-1$ or ${y}\left(j\right)=0$ or
${y}\left(j\right)=+1$) then

6: ${z}={y}$ (Do not embed in these measurements)

7: else

8: if (${y}\left(j\right)\%2=0$) then

9: if (${y}\left(j\right)\%4=0$) then

10: if ($S\left(j\right)=00$) then

11: ${z}={y}+1$

12: else if ($S\left(j\right)=01$) then

13: ${z}={y}$

14: else if ($S\left(j\right)=10$) then

15: ${z}={y}-1$

16: else if ($S\left(j\right)=11$) then

17: ${z}={y}+2$ or ${z}={y}-2$

18: end if

19: else

20: if ($S\left(j\right)=00$) then

21: if (${y}\neq 2$) then

22: ${z}={y}-1$

23: else

24: ${z}={y}+3$

25: end if

26: else if ($S\left(j\right)=01$) then

27: if (${y}\neq-2$) then

28: ${z}={y}+2$

29: else

30: ${z}={y}-2$

31: end if

32: else if ($S\left(j\right)=10$) then

33: if (${y}\neq-2$) then

34: ${z}={y}+1$

35: else

36: ${z}={y}-3$

37: end if

38: else if ($S\left(j\right)=11$) then

39: ${z}={y}$

40: end if

41: end if

42: else

43: if ((${y}\left(j\right)-1)\%4=0$) then

44: if ($S\left(j\right)=00$) then

45: ${z}={y}$

46: else if ($S\left(j\right)=01$) then

47: ${z}={y}-1$

48: else if ($S\left(j\right)=10$) then

49: ${z}={y}-2$

50: else if ($S\left(j\right)=11$) then

51: ${z}={y}+1$

52: end if

53: else

54: if ($S\left(j\right)=00$) then

55: ${z}={y}+2$

56: else if ($S\left(j\right)=01$) then

57: ${z}={y}+1$

58: else if ($S\left(j\right)=10$) then

59: ${z}={y}$

60: else if ($S\left(j\right)=11$) then

61: ${z}={y}-1$

62: end if

63: end if

64: end if

65: end if

66: end for

67:else

68: Whole secret data cannot be embedded. Try short length secret data.

69:end if

70:return $z$

###  3.1 Data Embedding

Figure 1: The Embedding Process

The first step in any compressed sensing-based image steganography scheme is
the input image's sparsification if it is not sparse at the start. This step
is equivalent to the signal sparsification of Section 2.1. Methods such as
K-SVD, DCT, Discrete Walsh Transform, Stationary Wavelet Transform, and
Discrete Rajan Transform provide good sparsification. Since the distortion due
to DCT is less, we use it as our sparsifying agent. To further reduce the
distortion, instead of sparsifying the whole image at once, first, we
decompose the cover image into non-overlapping blocks of the same size, and
then each block is sparsified.

Let the image I's size be $r1\times r2$ and each block size be $B\times B$,
then we have $(r1\times r2)/B^{2}$ number of blocks. In our case, $r1=r2$ and
$B$ completely divides $r1$. The block-wise sparsification is now done as

$s_{i}=DCT\left(x_{i}\right),$ (10)

where $i=1,2,\cdots,(r1\times r2)/B^{2}$, $x_{i}$ and $s_{i}$ are the $i^{th}$
original and sparse blocks of the same size, i.e. $B\times B$, respectively.
Next, we convert each block into their vector representation by stacking them
column-wise. Thus, $s_{i}$ becomes a vector of size $B^{2}\times 1$. Because
of sparsification, each sparse vector has few coefficients of large values and
the remaining coefficients of very small values or zero. Hence, we categories
each vector into two groups. Let $p_{1}$ be the number of coefficients having
large values and $p_{2}$ be the number of coefficients having small values or
zero values. Note that here, $p_{1}<p_{2}$ as each of these vectors are sparse
in nature and $p_{1}+p_{2}=B^{2}$. We represent each vector in two groups
based upon these coefficients, i.e. $s_{i,u}\in\mathbb{R}^{p_{1}}$ and
$s_{i,v}\in\mathbb{R}^{p_{2}}$. Now, we project each sparse vector onto linear
measurements using a measurement matrix, which is equivalent to Section 2.2.

There are two ways to choose the measurement matrix: either randomly or
deterministically. Randomly generated matrices such as the Independent and
Identically Distributed (i.i.d.) Gaussian matrix, the Bernoulli matrix or
other matrices generated by probabilistic methods are nonadaptive, although
they satisfy the RIP. Deterministically generated matrices are the ones that
are designed such that specific properties are satisfied, e.g., adaptiveness
and the RIP. We design a deterministic matrix that is adaptive to our sparse
vector since this improves the efficiency of compressed sensing. To achieve
RIP here, the projected linear measurements are enforced to have almost the
same $\ell_{2}-norm$ as that of the sparse vector.

One way to design the measurement matrix is to first analyze the distribution
of all $B^{2}$ coefficients in each sparse vector, and then find the $m$
indices out of these that give maximum $\ell_{2}-norm$ Xiaorong . That is ,

$E_{max}^{|m|}=\max_{i\in m\subset B^{2}}\left\|s_{i}\right\|_{2}^{2},$ (11)

where $|m|$ is the number of entries in set $m$ and $E_{max}^{|m|}$ is a
variable that stores the maximum value of square of $\ell_{2}$-norm of vector
$s_{i}$ for $i\in m\subset B^{2}$. However, in this paper, we use the property
of DCT to design the measurement matrix. This property states that DCT
coefficients can be divided into three sets; low frequency, middle frequency,
and high frequency components. Low frequency corresponds to the overall image
information, middle frequency corresponds to the structure of the image, and
high frequency corresponds to the noise or small variance. For image
reconstruction, only lower and middle frequency components are useful. Hence,
we select $m$ indices out of all $B^{2}$ indices that correspond to these two
sets of frequency Chang . Here, $|m|$ is a user-defined parameter such that
$p_{1}<|m|<p_{1}+p_{2}$, and is discussed in Experimental Results section. As
discussed earlier, in this subsection we have two groups of sparse vectors
$s_{i,u}$ and $s_{i,v}$. Hence, we design two different measurement matrices
$\Phi_{u}$ and $\Phi_{v}$ corresponding to $s_{i,u}$ and $s_{i,v}$,
respectively.

Since $\left\|s_{i,u}\right\|_{2}$ is close to $\left\|s_{i}\right\|_{2}$
because $s_{i,u}$ contains large value coefficients of $s_{i}$, we project
$s_{i,u}$ onto the same number of linear measurements. Thus, we have
$\Phi_{u}=\alpha I_{p_{1}}$, where $I_{p_{1}}$ is the identity matrix of size
$p_{1}\times p_{1}$, and $\alpha$ is a small constant.

As mentioned in Section 2.2, the main purpose of measurement matrix is to
project the sparse vector onto less number of linear measurements. Hence, we
project $s_{i,v}$ onto $|m|-p_{1}$ measurements or the size of $\Phi_{v}$ is
$(|m|-p_{1})\times p_{2}$. To construct $\Phi_{v}$, we first take a random
Hadamard matrix of size $p_{2}\times p_{2}$, which is a standard procedure in
compressed sensing literature Hadamard , and then we choose $|m|-p_{1}$ rows
from the available $p_{2}$ rows. These rows map to the last of $|m|-p_{1}$
indices from the index set $m$. This is because the first $p_{1}$ indices have
the overall image information, and hence, map to construction of $\Phi_{u}$.

We use the same measurement matrices for all blocks. This is because, for all
blocks of an image, the distribution of coefficients of the generated sparse
vectors is almost the same. Thus, for each block $i=1,2,\dots,(r1\times
r2)/B^{2}$, the block-wise linear measurements vector
$y_{i}\in\mathbb{R}^{|m|}$ is given as

$y_{i}=\begin{bmatrix}y_{i,u}\\\
y_{i,v}\end{bmatrix}=\begin{bmatrix}\Phi_{u}s_{i,u}\\\
\Phi_{v}s_{i,v}\end{bmatrix}.$ (12)

Using the standard terminology Candes ; Donoho , the measurements $y_{i,u}$
are called the ordinary samples or non-compressed samples, and the
measurements $y_{i,v}$ are called the compressed sensing samples.

Next, we discuss the encryption process of the secret data $D$ that is to be
embedded. This data is a sequence of $0s$ and $1s$. As mentioned in the
Introduction, this provides an extra layer of security to the embedded data.
For this, we first encrypt this data by using DES algorithm to obtain the
encrypted secret data $S$ (which is also a sequence of $0s$ and $1s$) Stalling
. DES is a fairly standard algorithm used for data encryption Stalling . Then,
we represent $S$ as a set of two-two bits, i.e.
$S=\left\\{S_{1},S_{2},\dots,S_{n}\right\\}$, where each $S_{L}$ consists of
two bits.

Next, we embed the secret data in our linear measurements $y_{i}$. The
embedding rule is summarized in Algorithm 1, and helps to embed two bits into
the transform coefficients. The rule is designed in such a way so that the
secret data could be extracted without any loss, discussed in Data Extraction
and Experimental Result sections. We embed the data in $y_{i,v}$ and not
$y_{i,u}$. This is because $y_{i,u}$ corresponds to sparse vector coefficients
of large values, and embedding in it leads to degradation of image quality.
Further, in $y_{i,v}$, the secret data is embedded selectively. We do not
embed in $y_{i,v}$ with measurement value of $-1$, $0$ and $1$. This is
because our embedding algorithm concatenates the measurement values with
integers from $-3$ to $+3$, and if these values are $-1$, $0$ or $1$, then we
may end up getting many $0s$ after concatenation, which leads to difficulty in
the extraction process. After embedding in other measurement values of
$y_{i,v}$, we obtain the modified $y_{i,v}$, which is termed as $z_{i,v}$.
That is,

$z_{i,v}=\begin{cases}y_{i,v}&\text{if }y_{i,v}=-1,\>0\>or\>1\\\
y_{i,v}+c&\text{otherwise},\end{cases}$ (13)

where $c\in\\{-3,-2,-1,0,1,2,3\\}$. We obtain our stego-data by concatenating
the measurements $y_{i,u}$ and $z_{i,v}$ as $\begin{bmatrix}y_{i,u}\\\
z_{i,v}\end{bmatrix}$. The block diagram for this complete data embedding
process is given in Fig. 1.

###  3.2 Data Extraction

In this section, we explain the process of extracting embedded secret data
from our stego-data. The steps of this extraction process are given below,
which are exactly reverse to our data embedding process.

  1. 1.

Separate the measurements $z_{i,v}$ from the stego-data, i.e.
$\begin{bmatrix}y_{i,u}\\\ z_{i,v}\end{bmatrix}$, where $i=1,2,\dots,(r1\times
r2)/B^{2}$ is the block number, and $u,v$ are indices available from the
previous subsection.

  2. 2.

Extract only those measurements from $z_{i,v}$ whose values are not equal to
$-1$, $0$ or $1$. The embedding rule ensures that the embedded data could be
extracted without loss. In other words, Algorithm 1 ensures that no secret
data is embedded in measurements with values $-1$, $0$ and $1$.

  3. 3.

Extract the encrypted message ${S}^{\prime}$ from the measurements obtained in
the above step by applying Algorithm 2.

  4. 4.

Decrypt this ${S}^{\prime}$ by DES algorithm, and obtain the extracted secret
data $D^{\prime}$.

Figure 2: The Extraction Process

Now, we check the correctness of this extracted secret data $D^{\prime}$ by
comparing it with original secret data $D$. For this, we use the Bit Error
Rate (BER), which is given as Pan

$\displaystyle\text{Error Bits (EB)}=D\bigoplus{D}^{\prime},$ (14)
$\displaystyle BER=\frac{\text{Number of ones in EB}}{\text{Size of D}}\times
100\%,$ (15)

where $\bigoplus$ denotes the bitwise XOR/ Exclusive OR operation. The BER
value for our steganography scheme is $0\%$, i.e. we successfully extract
complete secret data without any error. This is the property of our embedding
rule. The above extraction process is represented via a block diagram in Fig.
2.

Algorithm 2 Extraction Rule

1:

  * •

$z$: Sequence of modified linear measurements. These are $z_{i,v}$ that are
not having value equal to $0$, $1$ or $-1$. See extraction process in Section
3.2.

2:

  * •

${S}^{\prime}$: Encrypted secret bit sequences.

3:for $j=1$ to $length(z)$ do

4: if (${y}\left(j\right)=-1$ or ${y}\left(j\right)=0$ or
${y}\left(j\right)=+1$) then

5: Continue

6: else

7: if (${z}\left(j\right)\%2=0$) then

8: if (${z}\left(j\right)\%4=0$) then

9: ${S}^{\prime}\left(j\right)=01$

10: else

11: ${S}^{\prime}\left(j\right)=11$

12: end if

13: else

14: if ((${z}\left(j\right)-1)\%4=0$) then

15: ${S}^{\prime}\left(j\right)=00$

16: else

17: ${S}^{\prime}\left(j\right)=10$

18: end if

19: end if

20: end if

21:end for

22:return ${S}^{\prime}$

###  3.3 Stego-Image Construction

When the stego-data is transferred over a communication media, the intruder
can access this data from the public channel and can try to construct the
stego-image. If the intruder obtains a high visual quality image, then the
goal of steganography is fulfilled. This is because he/ she will not be able
to judge whether some data is hidden in the image or not. Therefore, in this
subsection, we give the steps to construct the stego-image from the stego-
data, which is equivalent to Section 2.3. We refer this process as
construction rather than reconstruction.

  1. 1.

Obtain the approximate sparse vector ${s}^{\prime}$ from the stego-data and
measurement matrices $\Phi_{u}$ and $\Phi_{v}$ as (recall (12))

$\displaystyle{s}^{\prime}_{i,u}=\Phi_{u}^{-1}y_{i,u},\>and$ (16)
$\displaystyle{s}^{\prime}_{i,v}=ADMM\\_LASSO\left(z_{i,v},\Phi_{v}\right).$

Here, as discussed in Section 2.3, we use ADMM and LASSO to construct
${s}^{\prime}_{i,v}$. The sparse vector ${s}^{\prime}$ is obtained by
concatenating ${s}^{\prime}_{i,u}$ and ${s}^{\prime}_{i,v}$. Here, the size of
${s}^{\prime}_{i,u}$, ${s}^{\prime}_{i,v}$, and ${s}^{\prime}$ is the same as
that of $s_{i,u}$, $s_{i,v}$, and $s$, respectively.

  2. 2.

Convert each vector ${s}^{\prime}_{i}$ into a block of size $B\times B$.

  3. 3.

Apply two-dimensional Inverse DCT (IDCT) to each of these blocks to generate
blocks ${x}^{\prime}_{i}$ of image. That is, recall (10),

${x}^{\prime}_{i}=IDCT\left({s}^{\prime}_{i}\right).$ (17)

  4. 4.

Construct the stego-image of size $r1\times r2$ by arranging all these blocks
${x}^{\prime}_{i}$.

The block representation of these steps is given in Fig. 3. We show in the
Experimental Results section that image obtained from this stego-data
preserves the quality of the original image.

Figure 3: Stego-Image Construction

As earlier, we term our proposed steganography scheme as Compressed-Sensing-
Image-Steganography (CSIS) because we use compressed sensing to enhance the
embedding capacity of the image steganography scheme.

##  4 Experimental Results

Experiments are carried out in MATLAB on a machine with an Intel Core i3
processor @2.30 GHz and 4GB RAM. We use a set of standard grayscale images to
test our CSIS. Sample test images are shown in Fig. 4 and Fig. 5. These images
have the varying texture property and are taken from the miscellaneous
category of USC-SIPI image database SIPI and two other public domain databases
ImageDatabase1 ; ImageDatabase2 .

The miscellaneous category of USC-SIPI database consists of 24 grayscale
images. Some images, such as Lena, and Tiffany are no longer available in this
database. These images have played a significant role in image processing, and
literature. Thus, we use other public-domain test images databases
ImageDatabase1 ; ImageDatabase2 for them. A total of seven such images are
chosen. Hence, we have a total of 31 grayscale images. Our CSIS is also
applicable to color images, and we pick one of them from USC-SIPI database.

In this manuscript, we report average values of all the 31 images with
detailed results for 10 images due to space limitations. This is further
justified by the fact that the image processing literature has used these 10
images or a subset of them.

The size of each of test images is $512\times 512$, i.e. $r1\times r2$. We
take blocks of size $8\times 8$, i.e. $B\times B$. As earlier, the size of
measurement matrix $\Phi_{u}$ is $p_{1}\times p_{1}$. Recall from Section 3.1,
$p_{1}$ is the number of coefficients with large values/ low frequency in the
input sparse vector. For commonly used images, this value is between $10$ and
$14$ Chang ; NC . Since the measurement matrix cannot be different for every
input matrix, we do experiments with three different values of $p_{1}$ ($10$,
$12$ and $14$) to find the optimal one here. Again from Section 3.1, the size
of measurement matrix $\Phi_{v}$ is $(|m|-p_{1})\times p_{2}$. We take $|m|$
from the following range Chang ; NC : $\\{32,35,36,37,39,40,42,47\\}$, and as
before, $p_{2}=B\times B-p_{1}$ (i.e. $p_{2}=64-p_{1}$). For secret data, we
use randomly generated data, which is sequence of $0$ and $1$ bits.

First, we check the embedding capacity of our proposed scheme. Second, we do
the similarity analysis between the cover images and the constructed stego-
images by assessing . Third, in the remainder of this section, we do security
analysis, perform five comparisons with existing steganography schemes, and
also experiment with a color image.

(a) Lena

(b) Peppers

(c) Boat

(d) Goldhill

(e) Zelda

Figure 4: Test images used in our experiments

(a) Tiffany

(b) Living room

(c) Tank

(d) Airplane

(e) Camera man

Figure 5: Continued from Fig. 4; test images used in our experiments

###  4.1 Embedding Capacity Analysis

Embedding capacity is defined as the maximum number of bits embedded in the
cover media, which is the image here. The embedding capacity of our proposed
steganography scheme depends on the sampling rate (SR), which is given as

$\displaystyle SR=\frac{\text{Total Linear Measurements}}{\text{Total Pixels
in Cover Image}}.$ (18)

We have $r_{1}\times r_{2}$ total pixels in the cover image and $|m|$ linear
measurements for each block with $\frac{r1\times r2}{B\times B}$ number of
blocks. Therefore, our sampling rate is

$\displaystyle SR=\left(\frac{|m|}{r1\times
r2}\right)\times\left(\frac{r1\times r2}{B\times
B}\right)=\frac{\left|m\right|}{B\times B}.$ (19)

From this definition, it is evident that embedding capacity mainly depends
upon $|m|$, however, the compressed image quality depends upon both $p_{1}$
and $|m|$. Therefore, to maintain the quality of stego-image while enhancing
embedding capacity, the combination of these parameters is critical.

For different combinations of $p_{1}$ and $|m|$, in Table 4.1, we give the
embedding capacity in bits of our proposed CSIS for the 10 test images of Fig.
4 and Fig. 5 and the average capacity for all the 31 images. We analyze the
data of this table by comparing $p_{1}$ and $|m|-p_{1}$ instead of $p_{1}$ and
$|m|$ because the former set directly maps to the number of ordinary samples
and compressed sensing samples, respectively. When $p_{1}$ is constant, and
$|m|-p_{1}$ is increased, the number of compressed sensing samples increases,
where the secret data bits are embedded, leading to increased capacity. For
example, consider columns $2$ and $3$ of Table 4.1, we can observe that the
embedding capacity increases when $p_{1}$ is constant, i.e. $10$ and
$|m|-p_{1}$ is increased from $22$ to $25$. When $|m|-p_{1}$ is constant and
$p_{1}$ is increased, the number of compressed sensing samples decrease
leading to decreased embedding capacity. For example, consider columns $3$ and
$4$, we observe that embedding capacity decreases when $|m|-p_{1}$ is
constant, i.e. $25$ and $p_{1}$ is increased from $10$ to $12$.

Embedding capacity (in bits) obtain by proposed CSIS for different parameters
and for different test images  Test image Parameters $p_{1}=10$ $|m|=32$
$p_{1}=10$ $|m|=35$ $p_{1}=12$ $|m|=37$ $p_{1}=12$ $|m|=40$ $p_{1}=12$
$|m|=42$ $p_{1}=12$ $|m|=47$ $p_{1}=14$ $|m|=36$ $p_{1}=14$ $|m|=39$ Lena
171087 194519 194265 217491 232924 272130 170361 193679 Peppers 173091 196725
196357 219641 235265 274890 172304 196193 Boat 171563 194819 194559 217665
233430 272162 170738 194167 Goldhill 174359 198019 197477 221155 236888 276297
173674 197031 Zelda 170447 193811 193635 216639 232441 270830 170080 192951
Tiffany 170457 193717 193291 216419 231924 270386 169747 192739 Living room
174534 198336 198216 222186 238076 277904 174402 198336 Tank 174961 198933
198395 222165 238276 277972 174564 198223 Airplane 167255 189865 189195 212003
227341 265207 165822 188313 Camera man 161201 183181 180375 202601 215917
251596 157618 177801 Avg. of 10 images 170895 194192 193576 216796 232248
270937 169931 192943 Avg. of 31 images 152786 176645 174678 198080 214135
251989 150023 173564

###  4.2 Stego-image Quality Assessment

In general, when the embedding capacity increases, the visual quality of
stego-image degrades. Hence, with increased embedding capacity, preserving the
visual quality of stego-image is also essential. There is no universal metric
to judge the quality of stego-image. However, we check the quality of stego-
image by examining the similarity between cover images and their corresponding
stego-images.

This check is done in two ways. Initially we perform a visual or subjective
check. The subjective measure is a good way to assess the quality of stego-
image, but it depends on many factors like viewing distance, the display
device, the lighting condition, viewer's vision ability, and viewer's mood.
Therefore, it is necessary to design mathematical models to assess the quality
of stego-images, which we discuss next.

####  4.2.1 Subjective or Visual Measure

Human observers are the final arbiter of image quality. Therefore, the
subjective measure is a perfect way of assessing the quality of the images.
Here, we construct stego-images corresponding to different test images used in
our experiment for different combinations of $p_{1}$ and $|m|$. This result
shows that the stego-images are almost similar to their corresponding cover
images. The same is true for their corresponding histograms also. As an
example, we present the visual comparison for ‘Pepper’ cover image for one set
of parameters; $p_{1}=12$ and $|m|=37$. Fig. 6 shows the (a) ‘Pepper’ cover
image (b) ‘Pepper’ cover image histogram (c) ‘Pepper’ stego-image (d) ‘Pepper’
stego-image histogram. From these figures, we observe that the stego-image is
almost similar to its corresponding cover image and their corresponding
histograms are also very similar.

We also construct the edge map diagrams for both the cover image and its
corresponding stego-image for this same example. These edge maps are shown in
Fig. 7(a) and Fig. 7(b), respectively. We can see from these figures that both
the edge maps are almost the same. Hence, the visual quality of the cover
image and its corresponding stego-image is almost similar.

(a) ‘Pepper’ cover image

(b) Cover image histogram

(c) ‘Pepper’ stego-image

(d) Stego-image histogram

Figure 6: ‘Pepper’ cover image, its stego-image, and their corresponding
histogram using parameter $p_{1}$=12 and $|m|$=37.

(a) Cover image edge map

(b) Stego-image edge map

Figure 7: Edge maps of ‘Pepper’ cover image and its stego-image using
parameter $p_{1}$=12 and $|m|$=37.

####  4.2.2 Objective or Numerical Measures

These measures compare the cover images and their corresponding stego-images
based on some numerical criteria that do not require extensive subjective
studies. Hence, in recent times, these measures are more commonly used for
image quality assessment. These include; Peak Signal-to-Noise Ratio (PSNR),
mean Structural Similarity (SSIM) index, Normalized Cross-Correlation (NNC)
coefficient, and entropy. We discuss all of them below.

##### PSNR

We compute the PSNR value to evaluate the imperceptibility of stego-images.
That is,

$PSNR=10\log_{10}\frac{R^{2}}{MSE}\>dB,$ (20)

where $MSE$ represents the mean square error between the cover image $I$ and
the stego-image $SI$, $R$ is the maximum intensity of pixel, which is $255$
for grayscale images, and dB refer to decibel. The $MSE$ is calculated as

$MSE=\frac{\sum_{i=1}^{r1}\sum_{j=1}^{r2}\left(I\left(i,\,j\right)-SI\left(i,\,j\right)\right)^{2}}{r1\times
r2},$ (21)

where $r1$ and $r2$ represent the row and column numbers of the digital image,
respectively, and $I(i,j)$ and $SI(i,j)$ represent the pixel value of the
cover image and the constructed stego-image, respectively.

A higher PSNR value indicates the higher imperceptibility of the stego-image.
In general, a value higher than 30 dB is considered to be good since human
eyes can hardly distinguish the distortion in the stego-image Liu ; Zhang2013
. The PSNR values of the stego-images corresponding to 10 test images of Fig.
4 and 5, and average for all 31 images for different combination of $p_{1}$
and $|m|$ are given in Table 4.2.2. From this table, we can easily observe
that this value is higher than 30 dB for all combinations of parameters and
for all images.

Value of PSNR (in dB) obtain by proposed CSIS for different parameters and for
different test images  Test image Parameters $p_{1}=10$ $|m|=32$ $p_{1}=10$
$|m|=35$ $p_{1}=12$ $|m|=37$ $p_{1}=12$ $|m|=40$ $p_{1}=12$ $|m|=42$
$p_{1}=12$ $|m|=47$ $p_{1}=14$ $|m|=36$ $p_{1}=14$ $|m|=39$ Lena 34.34 35.11
35.62 36.15 36.71 37.31 36.33 36.91 Peppers 34.05 34.35 35.23 35.76 36.21
36.98 35.44 35.81 Boat 32.67 33.07 33.84 34.25 34.72 36.84 34.37 34.81
Goldhill 32.69 33.61 34.06 34.54 35.12 35.32 34.33 34.93 Zelda 39.31 39.61
40.10 41.32 40.02 42.67 40.73 42.46 Tiffany 33.64 33.96 34.69 35.88 36.49
37.23 35.73 36.37 Living room 30.94 31.29 32.07 32.98 33.31 33.78 33.48 33.64
Tank 34.27 34.32 35.13 35.62 35.98 36.98 35.36 35.87 Airplane 32.89 34.15
34.88 35.78 36.39 37.91 34.43 35.42 Camera man 35.71 36.89 37.52 38.86 39.38
40.86 40.04 40.65 Avg. of 10 images 34.051 34.636 35.314 36.114 36.433 37.588
36.024 36.687 Avg. of 31 images 34.245 34.883 35.593 36.379 36.668 37.921
36.282 36.901

##### Means SSIM Index

It is an image quality assessment metric used to measure the structural
similarity between two images SSIM . This measure is based on the assumption
that the human visual system (HVS) is more adapted to the image's structural
information. The mean SSIM (MSSIM) index is given as

$\displaystyle
SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})},$
(22) $\displaystyle MSSIM(I,SI)=\frac{1}{M}\sum_{j=1}^{M}SSIM(i_{j},si_{j}),$
(23)

where $SSIM(x,y)$ calculates the SSIM index for vectors $x$ and $y$, and
$MSSIM$ $(I,SI)$ calculates the mean SSIM between cover image $I$ and stego-
image $SI$, i.e. for the overall image quality. Here, $\mu_{x}$ is the
weighted mean of $x$, $\mu_{y}$ is the weighted mean of $y$, $\sigma_{x}$ is
the weighted standard deviation of $x$, $\sigma_{y}$ is the weighted standard
deviation of $y$, $\sigma_{xy}$ is the weighted covariance between $x$ and
$y$, $C_{1}$ & $C_{2}$ are arbitrary constants, $i_{j}$ & $si_{j}$ are the
content of the cover image and stego-image, respectively, at the $j^{th}$
local window, and $M$ is the number of local windows. We took the values of
all these parameters according to SSIM . The value of the mean SSIM index lies
between $0$ and $1$, where the value $0$ indicates that there is no similarity
between the two images, and the value $1$ indicates that the images are
exactly similar.

The mean SSIM index values between the stego-images and their corresponding
cover images for different combination of $p_{1}$ and $|m|$ are given in Table
4.2.2. As earlier, 10 images from 4 and 5 are extensively analyze and average
of 31 images is reported. From this table, we observe that all these values
are close to $1$, which represents that the stego-images are very much similar
in structure to their corresponding cover images.

Value of Mean SSIM index obtain by proposed CSIS for different parameters and
for different test images  Test image Parameter $p_{1}=10$ $|m|=32$ $p_{1}=10$
$|m|=35$ $p_{1}=12$ $|m|=37$ $p_{1}=12$ $|m|=40$ $p_{1}=12$ $|m|=42$
$p_{1}=12$ $|m|=47$ $p_{1}=14$ $|m|=36$ $p_{1}=14$ $|m|=39$ Lena 0.9308 0.9394
0.9475 0.9518 0.9562 0.9672 0.9512 0.9558 Peppers 0.9203 0.9225 0.9291 0.9463
0.9424 0.9547 0.9333 0.9383 Boat 0.9211 0.9356 0.9444 0.9517 0.9575 0.9663
0.9484 0.9545 Goldhill 0.9011 0.9122 0.9236 0.9343 0.9421 0.9532 0.9227 0.9359
Zelda 0.9512 0.9563 0.9613 0.9657 0.9694 0.9768 0.9628 0.9678 Tiffany 0.9239
0.9315 0.9357 0.9434 0.9501 0.9596 0.9369 0.9437 Living room 0.9012 0.9092
0.9211 0.9332 0.9384 0.9460 0.9341 0.9382 Tank 0.8835 0.8857 0.9006 0.9094
0.9197 0.9388 0.9086 0.9131 Airplane 0.9463 0.9525 0.9605 0.9646 0.9673 0.9755
0.9607 0.9692 Camera man 0.9677 0.9752 0.9839 0.9864 0.9871 0.9907 0.9838
0.9868 Avg. of 10 images 0.9198 0.9272 0.9360 0.9445 0.9492 0.9598 0.9398
0.9463 Avg. of 31 Images 0.9206 0.9276 0.9365 0.9449 0.9498 0.9601 0.9412
0.9471

##### NCC Coefficient

Normalized correlation (NC) metric measures the degree of similarity between
two images, and when the two images are independent, this correlation is
called normalized cross-correlation (NCC) NC . The NCC coefficient is given as

$\displaystyle
NCC=\frac{\sum_{i=1}^{r1}\sum_{j=1}^{r2}I(i,j)SI(i,j)}{\sum_{i=1}^{r1}\sum_{j=1}^{r2}I^{2}(i,j)},$
(24)

where $r1$ and $r2$ represent the row and column numbers of the digital image,
respectively. $I(i,j)$ and $SI(i,j)$ represent the pixel value of the cover
image and the constructed stego-image, respectively. The value equal to 1
indicates that both the images are exactly similar. For our experiments, the
values of NCC are given in Table 4.2.2. The set of images used are same as for
PSNR and SSIM. We observe that all these values are close to 1, which means
that the stego-images are almost identical to their corresponding cover
images.

Value of normalized cross-correlation obtain by proposed CSIS for different
parameters and for different test images  Test image Parameter $p_{1}=10$
$|m|=32$ $p_{1}=10$ $|m|=35$ $p_{1}=12$ $|m|=37$ $p_{1}=12$ $|m|=40$
$p_{1}=12$ $|m|=42$ $p_{1}=12$ $|m|=47$ $p_{1}=14$ $|m|=36$ $p_{1}=14$
$|m|=39$ Lena 0.9985 0.9988 0.9989 0.9991 0.9991 0.9992 0.9991 0.9993 Peppers
0.9982 0.9983 0.9985 0.9987 0.9988 0.9989 0.9985 0.9988 Boat 0.9979 0.9983
0.9985 0.9987 0.9988 0.9989 0.9987 0.9989 Goldhill 0.9976 0.9981 0.9983 0.9986
0.9987 0.9988 0.9982 0.9986 Zelda 0.9991 0.9993 0.9994 0.9995 0.9995 0.9997
0.9994 0.9995 Tiffany 0.9992 0.9993 0.9994 0.9995 0.9995 0.9996 0.9994 0.9995
Living room 0.9962 0.9961 0.9962 0.9971 0.9972 0.9964 0.9970 0.9982 Tank
0.9987 0.9988 0.9992 0.9991 0.9992 0.9993 0.9993 0.9991 Airplane 0.9989 0.9991
0.9993 0.9994 0.9994 0.9995 0.9993 0.9995 Cameraman 0.9989 0.9991 0.9994
0.9994 0.9994 0.9995 0.9995 0.9996 Avg. of 10 images 0.9983 0.9985 0.9989
0.9989 0.9989 0.9989 0.9988 0.9991 Avg. of 31 images 0.9983 0.9985 0.9990
0.9989 0.9989 0.9989 0.9990 0.9991

##### Entropy

In general, entropy is defined as the measure of average uncertainty of a
random variable, which here is the average number of bits required to describe
the random variable. In the context of an image, it is a statistical measure
of randomness that can be used to characterize the texture of the image
Gonzalez . For a grayscale image, entropy is given as

$\displaystyle Entropy=-\sum_{i=0}^{255}(p_{i}\log_{2}p_{i}),$ (25)

where $p_{i}$ is the probability of value $i$ pixel of the image. Table 4.2.2
gives the entropy values for the cover images and their corresponding stego-
images for different combinations of $p_{1}$ and $|m|$. The set of images used
are same as for PSNR, SSIM, and NCC. From this table, we observe that for all
these combinations of $p_{1}$ and $|m|$, the entropy of the cover images and
their corresponding stego-images are almost similar.

Entropy comparison of cover images and their corresponding stego-images obtain
by proposed CSIS using different parameters  Test image Cover image Stego-
image using different parameters $p_{1}=10$ $|m|=32$ $p_{1}=10$ $|m|=35$
$p_{1}=12$ $|m|=37$ $p_{1}=12$ $|m|=40$ $p_{1}=12$ $|m|=42$ $p_{1}=12$
$|m|=47$ $p_{1}=14$ $|m|=36$ $p_{1}=14$ $|m|=39$ Lena 7.4456 7.4552 7.4581
7..4569 7.456 7.4545 7.4534 7.4551 7.4536 Peppers 7.5715 7.5924 7.5924 7.5908
7.5911 7.5901 7.5889 7.5897 7.5898 Boat 7.1238 7.1323 7.1339 7.1322 7.1334
7.1337 7.1331 7.1277 7.1304 Goldhill 7.4778 7.4653 7.4686 7.4704 7.4723 7.4719
7.4731 7.469 7.4717 Zelda 7.2668 7.2625 7.2635 7.2638 7.2643 7.2649 7.2652
7.2633 7.2642 Tiffany 6.6015 6.6076 6.6063 6.6046 6.606 6.6074 6.607 6.6096
6.6076 Living room 7.2950 7.4200 7.4200 7.4253 7.4260 7.4261 7.4262 7.4267
7.4278 Tank 5.4957 6.3614 6.3728 6.3771 6.3829 6.3846 6.3871 6.3709 6.3815
Airplane 6.7025 6.773 6.7637 6.7535 6.7501 6.7468 6.7396 6.7614 6.7454 Camera
man 7.0482 7.0743 7.0763 7.0738 7.0703 7.0683 7.0664 7.0726 7.0661 Avg. of 10
images 7.0028 7.1144 7.0691 7.07678 7.1152 7.1148 7.1140 7.1145 7.1138 Avg. of
31 images 6.9985 6.6451 7.7132 6.7124 6.6476 6.6462 6.6447 6.6469 6.6448

###  4.3 Security Analysis

Since the proposed CSIS is a transform domain based technique and it employs
indirect embedding strategy, i.e. it does not follow the LSB flipping
method,and hence, it is immune to statistical attacks Westfeld ;
PM1_steganography . Also, CSIS does not lead to the shrinkage effect. That
means, after embedding, the nonzero coefficients do not modify to zero value,
and hence attacks against F5 Fridrich ; PM1_steganography are not considered.

Moreover, in CSIS, the measurement matrix $\Phi$ is considered as the secret-
key, which is shared between the sender and the legitimate receiver. If the
eavesdropper intercepts the stego-image by a randomly generated measurement
matrix, he cannot not enter the embedding domain without the original secret-
key. Hence, we achieve increased security in our proposed system. To justify
this, we extract the secret data in two ways, i.e. by using the correct
measurement matrix and by using a measurement matrix that is very close to the
original one, and obtain the BER (discussed in Section 3.2) between the
original secret data the extracted one.

In Fig. 8, we present this BER for earlier discussed 10 cover images, and for
the parameter $p_{1}$=12 and $|m|$=37. In this figure, we see that for the
correct secret-key, the BER is 0, and for a tiny difference in the measurement
matrix, i.e. wrong secret-key, the BER is very high, which is 35% to 40%. That
is, even a small change in the secret-key will lead to an extreme shift in
accuracy between the original secret data and the extracted one.

Figure 8: BER with the correct and with a wrong secret-key (i.e. measurement
matrix)

In addition to the above security analysis, we also measure the security by
analyzing the distribution of the measurements and their corresponding
modified measurements, i.e. after embedding the secret data. For ‘Pepper’
image with parameter $p_{1}$=12 and $|m|$=37, this distribution of the
original measurements and the modified measurements is shown in Fig. 9(a) and
Fig. 9(b), respectively. The green and blue colors are automatically added by
Matlab and do not have any significance here. From these figures, we see that
the distribution for both cases is almost the same. We also check these
distributions for all the images and obtain the same results. We do not
include these in this manuscript due to space limitations.

(a) Original measurements

(b) Modified measurements

Figure 9: Distribution of measurements for ‘Peppers’ image

The preservation of distribution of measurements in the earlier two histograms
can also be justified by the probability of addition and subtraction operation
decided by our algorithm. In Fig. 10, we plot this probability. From this
figure, we see that the lines of probabilities of addition and subtraction
operation oscillate around 0.5. Here, the minimum and maximum deviation to 0.5
are 0.02 and 0.07, respectively, i.e. for proposed CSIS, the probabilities of
both the addition and the subtraction are nearly the same. The distribution of
measurements and the probability of addition & subtraction operation as
discussed have justified that for our proposed CSIS, the likelihood of
detecting data embedding by an eavesdropper is significantly low.

Figure 10: Probability of addition and subtraction operation

###  4.4 Performance Comparison

In this subsection, we compare the performance of the proposed CSIS with the
existing steganography schemes. This result is given in Table 4.4. In this
table, the first column represents the comparison metrics, and the remaining
columns give the metric data for different steganography schemes.

Performance comparison between proposed CSIS and various other steganography
schemes  Metrics Steganography Schemes CSIS Ref. Steg_2019_FGT Ref. Jsteg Ref.
F5 Ref. Outguess Ref. Liu Ref. Pan Ref. Steg_ExpertSystem Ref. Steg_Rohit
Capacity (in bits) 174678 262144 41267 41451 20644 55001 950 57568 113960 PSNR
(in dB) 30.94 to 40.86 36.51 34.39 35.00 34.54 32.54 35.52 49.89 36.64
Compression Based Yes No Yes Yes Yes Yes Yes No Yes Resistant to Chi-square
Yes No No Yes Yes Yes Yes Yes Yes Resistant to Shrinkage Effect Yes NA Yes No
Yes Yes Yes Yes Yes Secret Key Yes No No No No No Yes No No

In the first row of Table 4.4, we compare the average embedding capacity over
all the 31 images. We report these embedding capacity for the parameter
$p_{1}=12$ & $|m|=37$. In this table, we do not compare these results for all
the images because the existing schemes' data are not available for all the
images. From the first row of this table, we observe that on an average our
steganography scheme has approximately $0.67$, $4.23$, $4.21$, $8.46$, $3.18$,
$183.87$, $3.03$ and $1.53$ times embedding capacity as compared to references
Steg_2019_FGT , Jsteg , F5 , Outguess , Liu , Pan , Steg_ExpertSystem , and
Steg_Rohit , respectively. Here, we can see that our proposed scheme has a
higher embedding capacity compared to all schemes except the one, which is
Steg_2019_FGT . The reason for this is that this scheme is based on embedding
secret data in the spatial domain. As discussed in the Introduction, spatial
domain based embedding techniques have a higher embedding capacity, but they
are prone to security issues. Also, these techniques are not based on
compression, which is the main motivation of this manuscript. Further, as
evident from Table 4.1, for a set of parameters $p_{1}=12$ and $|m|=47$, CSIS
has 270937, and 251989 bits embedding capacity for the average of 10 and 31
images, respectively. Hence, for this set of parameters, CSIS has
approximately the same embedding capacity as that of Steg_2019_FGT .

In the second row of this table, for our scheme we report the range of PSNR
values when considering all sets of parameters and again all 31 images. From
the second row of this table, we observe that similar to existing
steganography schemes, our CSIS also has PSNR values greater than 30 dB, which
is considered good Liu ; Zhang2013 .

The purpose of the proposed CSIS is to embed secret data in the compressed
domain. Hence, in the third row of Table 4.4, we check which schemes are based
on compression and which are not. From this row, we observe that except
Steg_2019_FGT ; Steg_ExpertSystem , our CSIS and all other schemes are based
on compression. Finally, from the fourth row to the sixth row of Table 4.4, we
compare the security of these schemes by checking whether they are resistant
to chi-square attack or not, resistant to shrinkage effect or not, and use any
secret-key or not. We observe that only our proposed CSIS and Pan schemes pass
all the three security tests. Hence, we can conclude that out of all these
schemes, only CSIS fulfills all the goals of steganography with higher
embedding capacity.

###  4.5 Experiments on Color Image

All the above experiments were performed on the grayscale images. However, we
also show the applicability of our proposed CSIS on a color image. For this we
only use ‘Pepper’ color image of resolution $512\times 512$, and perform
experiments for $p_{1}=12$ and $|m|=37$ as well as $p_{1}=14$ and $|m|=36$.

The performance analysis of our proposed scheme on color cover image
($512\times 512$ Pepper color image) using different parameters.  Parameters
Embedding capacity PSNR Mean SSIM Normalized cross-correlation (NCC) (for
different color component) Entropy Red Green Blue Average Cover image Stego
image $p_{1}=12$, $|m|=37$ 503863 33.89 0.9913 0.9990 0.9981 0.9963 0.9978
7.669 7.723 $p_{1}=14$, $|m|=36$ 573657 33.71 0.9843 0.9989 0.9979 0.9960
0.9976 7.669 7.724 Average 538760 33.80 0.9878 0.9990 0.9980 0.9962 0.9978
7.669 7.723

(a) Color cover image

(b) Color stego image

Figure 11: $512\times 512$ ‘Pepper’ color cover image and stego-image using
parameter $p_{1}$=12, $|m|$=37.

Fig. 11 shows the subjective/ visual measure for ‘Pepper’ color image for
$p_{1}=12$, $|m|=37$. From this figure, we observe that the cover image and
its corresponding stego-image are almost similar. Table 4.5 gives the results
for other measures like embedding capacity, PSNR values, mean SSIM index, NCC
coefficients for the different color components, and entropy for both cover
image and stego-image. We can observe from this table that the embedding
capacity of our color image is approximately three times the embedding
capacity of ‘Pepper’ grayscale image for the same set of parameters. Please
see columns $4$ and $8$ of Table 4.1. This is because of the presence of three
color components in the color image. Also, the PSNR values here are greater
than 30 dB, and mean SSIM index & NCC coefficients are all close to $1$, which
shows that the stego-image is almost similar to its corresponding cover
images. Finally, we compare the entropy of the cover image and the stego-
image. We see that entropy for both these images is almost the same.

##  5 Conclusions and Future Work

We present an enhanced-embedding capacity image steganography scheme based on
compressed sensing technique. Here, we combine three components to achieve
increased embedding capacity without degrading the quality of stego-images, as
well as making it resistant to steganalysis attacks. First, we use compressed
sensing to sparsify cover image block-wise and obtain its linear measurements
using a matrix. We uniquely select a large number of permissible measurements.
Hence, we achieve a high embedding capacity. Since the measurement matrix is a
secret-key that is shared between the sender and the legitimate receiver, this
adds extra security to our scheme. Also, we encrypt the secret data using the
DES algorithm and then embed two bits of secret data into each permissible
measurement instead of embedding one bit per measurement. Second, we propose a
technique of data extraction that is lossless and recovers our secret data
entirely. Third, we use ADMM solution of the LASSO formulation of the obtained
optimization problem in the stego-image construction. The reason for selecting
them is that they have broad applicability in the field of image processing,
require less assumptions on the property of the objective function, have fast
convergence, and are easy to implement.

We initially perform experiments on several standard grayscale images that
vary in texture, and with different sets of parameters and randomly generated
binary data as our secret data. For performance evaluation, we calculate
embedding capacity, PSNR value, mean SSIM index, NCC coefficient, and entropy.
Experiments show that our proposed CSIS achieves higher embedding capacity
than existing steganography schemes that follow compression. We achieve 1.53
times more embedding capacity as compared to the most recent scheme of the
similar category. PSNR values coming out of our scheme are more than 30 dB,
which is considered good. Both mean SSIM index and NCC coefficients values are
close to one, which shows that the cover images and their corresponding stego-
images are almost similar. This similarity is further supported by the fact
that we obtain approximately the same entropy value for both the cover images
and their corresponding stego-images. Further, we also show the applicability
of CSIS on a color image. Again, the results obtained are almost the same as
that of grayscale images. However, we get approximately three times higher
embedding capacity for the color image because of the presence of the three
component in color images.

In future, we plan to embed the secret data in text, audio, and video. Other
future works include extending this work for a real-time application such as
hiding fingerprint data, iris data, medical information of patients, and
personal signature. As mentioned in the Introduction, another line of work is
embedding images inside images. Since a lot of work has been done in embedding
a single image, we will focus on hiding multiple secret images and multilevel
image steganography scheme.

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