CSIS: compressed sensing-based enhanced-embedding capacity image steganography scheme
RResearch Article
CSIS: compressed sensing-basedenhanced-embedding capacity imagesteganography scheme
Rohit Agrawal , Kapil Ahuja ∗ Mathematics of Data Science (MODS) Laboratory, Indian Institute of Technology Indore, Simrol, Indore, India* E-mail: [email protected]
Abstract:
Image steganography plays a vital role in securing secret data by embedding it in the cover images. Usually, theseimages are communicated in a compressed format. Existing techniques achieve this but have low embedding capacity. Enhancingthis capacity causes a deterioration in the visual quality of the stego-image. Hence, our goal here is to enhance the embed-ding capacity while preserving the visual quality of the stego-image. We also intend to ensure that our scheme is resistant tosteganalysis attacks.This paper proposes a Compressed Sensing Image Steganography (CSIS) scheme to achieve our goal while embedding binarydata in images. The novelty of our scheme is the combination of three components in attaining the above-listed goals.
First , weuse compressed sensing to sparsify cover image block-wise, obtain its linear measurements, and then uniquely select permissiblemeasurements. 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 embedmore than one bit.
Second , we propose a novel data extraction technique, which is lossless and completely recovers our secretdata.
Third , for the reconstruction of the stego-image, we use the least absolute shrinkage and selection operator (LASSO) for theresultant optimization problem. This has the advantages of fast convergence and easy implementation. This component is alsonew.We perform experiments on several standard grayscale images and a color image, and evaluate embedding capacity, PeakSignal-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 averageof 37.92 dB PSNR value, and average values close to for both the mean SSIM index and the NCC coefficients, which areconsidered good. Moreover, the entropy of cover images and their corresponding stego-images are nearly the same. Theseassessment metrics show that CSIS substantially outperforms existing similar steganography schemes. The primary concern during the transmission of digital data overcommunication media is that anybody can access this data. Hence,to protect this data from being accessed by illegitimate users, thesender must employ some security mechanisms. In general, thereare two main approaches used to protect secret data; cryptography[1] and steganography [2]. In cryptography, the encryption processtransforms the secret data, known as plain-text, into cipher-text usingan encryption key. This text is in unreadable form, hence, it attractsthe opponents to exploit the content of the cipher-text by employ-ing some brute-force attacks [1]. However, steganography avoids thisscenario.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 mediaso that it is visually imperceptible. Here, both the secret data as wellas the cover media may be text or multimedia. The media obtainafter embedding secret data into cover media is called stego-media.Some recent steganography schemes that use text as cover mediaare [3] and [4]. In [3], the authors have proposed an Arabic textsteganography scheme, where the secret message is hidden withinthe text by using Unicode standard encoding. In [4], the authors haveproposed a character-level text generation-based linguistic steganog-raphy scheme, where the secret message is embedded in the text'scontent.Recently, the steganography schemes that use images as the covermedia have gained a lot of research interest due to their heavy use inInternet-based applications Typically, these images are transmittedin a compressed format. So here, we focus on compressed domain-based image steganography. In this, the challenges are;1. Improving the embedding capacity. 2. Maintaining the quality of the stego-image.3. The scheme should be resistant to steganographic attacks.Although images can be embedded into images, our focus is onembedding binary data into images.In the following paragraphs, first we discuss the way in whichsecret data can be embedded into cover images, then we summarizesome existing schemes and their limitations, and finally we arguehow the scheme presented in this paper outperforms the existingschemes.Secret data can be embedded in images by two ways; spatially andby using a transform. In the spatial domain based image steganogra-phy scheme, secret data is embedded directly into the image by somemodification in the values of the image pixels. Some well-knownschemes here are listed in [2, 5–11]. In the transform domain basedimage steganography scheme, first, the image is transformed into fre-quency components, and then the secret data is embedded into thesecomponents. Some commonly used such schemes are JSteg [12], F5[13], and Outguess [14]. Some other techniques, which do not carryspecific names are given in references [15–23].The spatial domain based image steganography outperforms thetransform domain one in terms of embedding capacity, but thestego-image has a high amount of redundant data. Digital imagestransmitted through communication media are usually of this type.Since transform based schemes reduce the redundancy present in theimage and represent it in a compressed form, they are preferred fortransmission.Most of the transform domain based scheme follow either Dis-crete Cosine Transform (DCT) or Wavelet Transform (WT). TheDCT based schemes are also called the JPEG compression basedimage steganography techniques. Several variants of DCT basedschemes have been proposed in the literature [12–17, 19–23]. For pp. 1–xii i a r X i v : . [ c s . MM ] J a n he schemes [12–17, 19, 21, 22], secret data is binary bits, and for[20, 23], secret data is images.In [12, 14, 15], the secret data is embedded by flipping the leastsignificant bit (LSB) of the quantized DCT coefficients obtainedfrom the cover image. This process is considered as a direct embed-ding mechanism. Alternatively, methods in [13, 16, 17, 19–23] areconsidered as indirect steganography schemes in which the quan-tized DCT coefficient values are altered according to certain secretmessage bits or secret image pixels. By steganalysis, which is thestudy of detecting the secret data hidden using steganography, it hasbeen observed that the indirect steganography mechanism is superiorto the direct one due to its capability in resisting certain statisticalattacks. The most common statistical attacks are the chi-square test,and the shrinkage effect [24–26]. Hence, the schemes [12, 14, 15]are not resistant to such attacks, while the schemes [13, 16, 17, 19–22] are resistant to them, but their embedding capacity is limited. Ifwe try to increase the embedding capacity of the later schemes, thenthe quality of the stego-images gets degraded. The scheme [23] hashigh embedding capacity with resistance to steganographic attacks,but here, the secret data is the images, which is different from ourgoal of embedding binary data in images.Most recent Wavelet transform based steganography schemes aregiven in [18, 27]. In [18], the authors have proposed a steganogra-phy scheme based upon edge identification and XOR coding thatuses Wavelet transformation. This scheme is resistant to stegano-graphic attacks, but here also the embedding capacity is significantlyless. As above, if we try to increase embedding capacity, then thequality of stego-image gets degraded. The scheme given in [27]embeds a medical image into a cover image using Redundant Inte-ger Wavelet Transform (RIWT) and DCT. This scheme's purpose isagain different from ours of embedding binary data in images.As discussed above, conventional transform domain based imagesteganography schemes provide good visual quality stego-imageand are resistant to steganographic attacks, but their embeddingcapacity 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 compressedsensing, which also fulfills all the requirements of image steganogra-phy. Next, we present literature regarding compressed sensing-basedsteganography schemes. These works help to achieve some of theabove objectives of steganography but not all, which we do.In [28], and [29], steganography schemes based on compressedsensing and Singular Value Decomposition (SVD) have been pre-sented. In these schemes, secret medical image data is embedded intoan image cover media. Both these approaches use a similar embed-ding approach, but use compressed sensing differently. In these,first, encrypted measurements of the secret image are obtained usingthe compressed sensing technique, and then these encrypted mea-surements are embedded into the cover image using SVD basedembedding algorithm. In [28], the PSNR (Peak Signal-to-NoiseRatio, discussed in Section 4.2.2) value of the stego-image is greaterthan 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 [29], both the stego-image as well as the reconstructed secretimage preserved good visual quality. But, the goal in both theseschemes is different from ours. In these schemes, the secret data isan image. If these techniques are applied on binary data that we wantto embed, the information will be lost. In [17], the authors have pro-posed an image steganography scheme based on sub-sampling andcompressed 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 steganog-raphy schemes are listed in [26], [30], and [31]. In [26], the authorshave presented the application of compressed sensing to detectsteganographic content in the LSB steganography scheme. In [30],the authors have proposed a DCT steganography classifier basedon a compressed sensing technique. Here, the original image isidentified from a set of images containing the original image andsome instances of stego images. In [31], the authors have proposedan image steganalysis technique for secret signal recovery. These steganography schemes are not related to our work because the focusof [26] and [31] is steganalysis, while [30] focuses on steganographyclassifier. Hence, we do not discuss these schemes in detail.The scheme that we propose satisfies all the goals mentioned inthe earlier paragraphs, i.e. increased embedding capacity withoutdegrading the quality of stego-images as well as making the schemeresistant to steganalysis attacks. Our scheme has three components,which we discussed next. The first component of our scheme con-sists of three parts; (i) we use compressed sensing to sparsify coverimage block-wise and obtain linear measurements. Here, we designan adaptive measurement matrix instead of using a random one.Using our adaptive measurement matrix, we uniquely select a largenumber of permissible measurements compared to existing schemes.Hence, we achieve a high embedding capacity. Moreover, these mea-surements act as encoded transformed coefficients, and hence, thisadds security to our proposed scheme as well; (ii) we encrypt thesecret data using the Data Encryption Standard (DES) algorithm[1]. This adds another layer of security to our scheme; (iii) weembed two bits of secret data into each permissible measurementinstead of commonly embedding one bit per measurement. This isa first attemp to rigorously embed more than one bit. Second , wecompletely extract secret data without any loss using our extrac-tion algorithm.
Third , we use the alternating direction method ofmultipliers (ADMM) solution of the least absolute shrinkage andselection operator (LASSO) formulation of the underlined optimiza-tion problem in the stego-image construction. The advantages ofusing ADMM and LASSO are that they have broad applicability inthe domain of image processing, require a little assumption on theobjective function's property, have fast convergence, and are easy toimplement. This is also a completely new contribution.For performance evaluation, we perform experiments on standardtest images. To check the quality of stego-image, we reconstruct itfrom the obtained modified measurements and then compare it withits corresponding cover image. We evaluate embedding capacity,Peak Signal-to-Noise Ratio (PSNR) value, mean Structural Similar-ity (SSIM) index, Normalized Cross-Correlation (NCC) coefficient,and entropy. We achieve 1.53 times more embedding capacity whencompared with the most recent scheme of this category. We achievea maximum of 40.86 dB and an average of 37.92 dB PSNR val-ues, which are considered good. The average values of mean SSIMindex and NCC coefficients are close to 1, which are again consid-ered good. Moreover, the entropy of cover images and their cor-responding stego-images are nearly the same. In the ExperimentalResults section, we also show that our scheme outperforms existingcompression based steganography schemes [6, 12–14, 16–19].The rest of the paper has four more sections. Section 2 describesthe compressed sensing technique. Section 3 explains our proposedsteganography scheme including embedding of the data, extractingit, and stego-image reconstruction process. Section 4 presents theexperimental results. Finally, Section 5 gives conclusions and futurework.
Compressed sensing is used to acquire and reconstruct the signalefficiently. Traditionally, the successful reconstruction of the sig-nal from the measured signal must follow the popular Nyquist/Shannon sampling theorem, which states that the sampling ratemust be at least twice the signal bandwidth. In many applicationssuch as image, audio, video, data mining, and wireless communica-tions & networks, where the signal is sparse or sparsified in somedomain, the Nyquist rate is too high to achieve. There is a fairly newparadigm, called compressed sensing that can represent the sparsesignal by using a sampling rate significantly lower than the Nyquistsampling rate [32, 33]. Hence, the application of compressed sens-ing has gained popularity in many areas. Some of them are imageprocessing [34], radar system [35], MRI Imaging [36], and noiseseparation from data [34].Compressed sensing projects the sparse signal onto a small num-ber of linear measurements in such a way that the structure of thissignal remains the same. The sparse signal can be reconstructed pp. 1–xiiii pproximately from these measurements by an optimization tech-nique. However, the reconstruction of the signal is possible onlywhen the original signal is sparse, and it satisfies the Restricted Iso-metric Property (RIP) [37] (discussed in Section 2.2). If the originalsignal is not sparse, then it can often be artificially sparsified. A briefdescription of signal sparsification, obtaining linear measurements,and reconstruction of the approximate sparse signal is given next.
Signal Sparsification
Let the original signal be x ∈ R N × . The signal x is K sparse whenit has maximum K number of non-zeros coefficients, i.e. (cid:107) x (cid:107) ≤ K ,where || · || denotes the (cid:96) − norm of a vector, and the remainingcoefficients are zero or nearly zero. Let the original signal x not besparse and be represented in-terms of { ψ i } Ni =1 basis vectors each oflength N × , then s = Ψ T x, (1)where, s ∈ R N × and Ψ = [ ψ , ψ , ..., ψ N ] ∈ R N × N is anorthogonal matrix. If K (cid:28) N then this signal is sparsifiable [38], s is the sparse representation of x , and Ψ is the correspondingsparsification matrix. Sensing Matrix and Linear Measurements
In the compressed sensing framework, we acquire M ( M < N ) linear measurements from the inner product between the originalsignal x ∈ R N × and M measurement vectors { φ i } Mi =1 , where φ i ∈ R N × . Considering the measurement/ sensing matrix as Φ = (cid:104) φ T ; φ T ; ... ; φ TM (cid:105) ∈ R M × N , the measurements y ∈ R M × aregiven as [38] y = Φ x. (2)If the input signal is not sparse but sparsifiable, then using the abovetheory we get y = ΦΨ s = Θ s, (3)where Θ = ΦΨ is again the measurement matrix of size M × N .Usually, in the compressed sensing framework, the measurementmatrix is nonadaptive. That is, the measurement matrix is fixed anddoes not depend on the signal. However, in certain cases, adaptivemeasurements can lead to significant performance improvement.The main concern here is to design the measurement matrix insuch a way so that the most of the information and the structure ofthe signal is preserved in the measurements. This would imply thatoriginal signal would be recovered efficiently from these measure-ments. To achieve this, for all K-sparse signals s , the measurementmatrix should hold the following inequality [37]. (1 − δ K ) ≤ (cid:107) Θ s (cid:107) (cid:107) s (cid:107) ≤ (1 + δ K ) , (4)where δ K ∈ (0 , is an isometric constant. The above inequality iscalled the RIP that informally says that the (cid:96) − norm of the sparsesignal s and the measurement Θ s should be comparable. Apart fromsatisfying the RIP, the minimum number of measurements required,i.e. the minimum value of M , is also a concern in the measurementmatrix design. Reconstruction of the Approximate Signal
As discussed in the previous subsection, size of the measurement y = Φ x = ΦΨ s = Θ s is less than the size of the original signal s .Hence, the reconstruction of the signal from measurements becomesan ill-posed problem. That is, the solution of an under-determinedlinear system of equations is to be found. If the matrix Θ satisfiesthe RIP, then the sparse signal s can be reconstructed approximately by solving the following optimization problem [39]: min s { number of i such that s ( i ) (cid:54) = 0 } Subject to ΦΨ s = y. (5)In the above equation, the function to be minimized is simplythe number of nonzero coefficients in the vector s . This equation isreferred to as (cid:96) − norm minimization problem. It is combinatorialand an NP-hard problem [39]. The other approach is to substitute the (cid:96) − norm by the closest convex norm, i.e. the (cid:96) − norm , or min s (cid:107) s (cid:107) Subject to ΦΨ s = y, (6)where || · || denotes the (cid:96) − norm of a vector. The approach toreconstruct the sparse signal s by solving the above equation istermed as a convex optimization method.Other approaches such as Greedy based (OMP [40], CoSaMP[41]), sparse reconstruction by separable approximation [42],Bayesian strategy [43], and ADMM solution of the LASSO for-mulation of the above optimization problem can also be used toreconstruct the sparse signal from the measurements [44, 45].Next, we give a brief idea of LASSO and ADMM, which we use.The general LASSO problem is given as [45] min z (cid:26) (cid:107) Az − b (cid:107) + λ (cid:107) z (cid:107) (cid:27) , (7)where z ∈ R n , A ∈ R p × n , b ∈ R p , (cid:107) · (cid:107) is the (cid:96) norm and λ > is a scalar regularization parameter also called Lagrangian parameter[46]. Further, (7) is transformed into a form solvable by ADMM[44]. That is min z,z (cid:26) (cid:107) Az − b (cid:107) + λ (cid:107) z (cid:107) (cid:27) Subject to z − z = 0 . (8)Finally, ADMM solve the above optimization problem.Now, we discuss how to solve our signal reconstruction prob-lem, i.e. (6) by LASSO and ADMM. For our case, Θ = ΦΨ is themeasurement matrix, and Θ ∈ R M × N . In the compressed sensingframework, matrix Θ is underdetermined, i.e. M < N . Hence, thereis equivalent solution of (6), which is given as [47] min s (cid:26) (cid:107) Θ s − y (cid:107) + λ (cid:107) s (cid:107) (cid:27) (9)Here, we observe that (9) is equivalent to (7) with Θ = A , s = z and y = b .Finaly, we briefly mention a theoretical result related to recon-struction. In [48], it is shown that for sufficiently small constant C (
C > ) , the K-sparse signal s of size N can be approximatelyreconstructed from M measurements y if M ≥ CK (log N ) . Afterrecovering the sparse signal s , the original signal x can be obtainedas x = Ψ s . For us, this property holds. Our proposed compressed sensing-based image steganographyscheme consists of the following components; data embedding, dataextraction, and stego-image construction, which are discussed in therespective sections below.
Data Embedding
The first step in any compressed sensing-based image steganographyscheme is the input image's sparsification if it is not sparse at the pp. 1–xii iii over Image I (Size r1 X r2 ) Non-overlappingblocks x i (each of size B X B ) Block Decomposition
Sparse Blocks s i (in vector form) Blocks havinglarge value s i,u Blocks havinglow value si,v
MeasurementMatrix Φ u MeasurementMatrix Φ v X XLinearMeasurement y i,v LinearMeasurement y i,u Secret Data(0 or 1sequence)EncryptedSecret Data(0 or 1sequence)
Encryption usingDES Algorithm
Embeddingusing
Algorithm 1
Modified LinearMeasurement z i,v Concatenation MeasurementStream [y i,u z i,v] Fig. 1 : The Embedding Processstart. This step is equivalent to the signal sparsification of Section2.1. Methods such as K-SVD, DCT, Discrete Walsh Transform, Sta-tionary Wavelet Transform, and Discrete Rajan Transform providegood sparsification. Since the distortion due to DCT is less, we useit as our sparsifying agent. To further reduce the distortion, instead ofsparsifying the whole image at once, first, we decompose the coverimage into non-overlapping blocks of the same size, and then eachblock is sparsified.Let the image I's size be r × r and each block size be B × B ,then we have ( r × r /B number of blocks. In our case, r r and B completely divides r . The block-wise sparsification is nowdone as s i = DCT ( x i ) , (10)where i = 1 , , · · · , ( r × r /B , x i and s i are the i th originaland sparse blocks of the same size, i.e. B × B , respectively. Next,we convert each block into their vector representation by stack-ing them column-wise. Thus, s i becomes a vector of size B × .Because of sparsification, each sparse vector has few coefficients oflarge values and the remaining coefficients of very small values orzero. Hence, we categories each vector into two groups. Let p bethe number of coefficients having large values and p be the num-ber of coefficients having small values or zero values. Note thathere, p < p as each of these vectors are sparse in nature and p + p = B . We represent each vector in two groups based uponthese coefficients, i.e. s i,u ∈ R p and s i,v ∈ R p . Now, we projecteach sparse vector onto linear measurements using a measurementmatrix, which is equivalent to Section 2.2.There are two ways to choose the measurement matrix: either ran-domly or deterministically. Randomly generated matrices such as theIndependent and Identically Distributed (i.i.d.) Gaussian matrix, theBernoulli matrix or other matrices generated by probabilistic meth-ods are nonadaptive, although they satisfy the RIP. Deterministicallygenerated matrices are the ones that are designed such that specificproperties are satisfied, e.g., adaptiveness and the RIP. We design adeterministic matrix that is adaptive to our sparse vector since thisimproves the efficiency of compressed sensing. To achieve RIP here,the projected linear measurements are enforced to have almost thesame (cid:96) − norm as that of the sparse vector.One way to design the measurement matrix is to first analyze thedistribution of all B coefficients in each sparse vector, and then findthe m indices out of these that give maximum (cid:96) − norm [49]. Thatis , E | m | max = max i ∈ m ⊂ B (cid:107) s i (cid:107) , (11) where | m | is the number of entries in set m and E | m | max is a vari-able that stores the maximum value of square of (cid:96) -norm of vector s i for i ∈ m ⊂ B . However, in this paper, we use the propertyof DCT to design the measurement matrix. This property statesthat DCT coefficients can be divided into three sets; low frequency,middle frequency, and high frequency components. Low frequencycorresponds to the overall image information, middle frequencycorresponds to the structure of the image, and high frequency cor-responds 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 indices that correspond to thesetwo sets of frequency [15]. Here, | m | is a user-defined parametersuch that p < | m | < p + p , and is discussed in ExperimentalResults section. As discussed earlier, in this subsection we have twogroups of sparse vectors s i,u and s i,v . Hence, we design two dif-ferent measurement matrices Φ u and Φ v corresponding to s i,u and s i,v , respectively.Since (cid:13)(cid:13) s i,u (cid:13)(cid:13) is close to (cid:107) s i (cid:107) because s i,u contains large valuecoefficients of s i , we project s i,u onto the same number of linearmeasurements. Thus, we have Φ u = αI p , where I p is the identitymatrix of size p × p , and α is a small constant.As mentioned in Section 2.2, the main purpose of measurementmatrix is to project the sparse vector onto less number of linear mea-surements. Hence, we project s i,v onto | m | − p measurements orthe size of Φ v is ( | m | − p ) × p . To construct Φ v , we first take arandom Hadamard matrix of size p × p , which is a standard pro-cedure in compressed sensing literature [50], and then we choose | m | − p rows from the available p rows. These rows map to thelast of | m | − p indices from the index set m . This is because thefirst p indices have the overall image information, and hence, mapto construction of Φ u .We use the same measurement matrices for all blocks. This isbecause, for all blocks of an image, the distribution of coefficients ofthe generated sparse vectors is almost the same. Thus, for each block i = 1 , , . . . , ( r × r /B , the block-wise linear measurementsvector y i ∈ R | m | is given as y i = (cid:20) y i,u y i,v (cid:21) = (cid:20) Φ u s i,u Φ v s i,v (cid:21) . (12)Using the standard terminology [32, 33], the measurements y i,u arecalled the ordinary samples or non-compressed samples, and themeasurements y i,v are called the compressed sensing samples.Next, we discuss the encryption process of the secret data D thatis to be embedded. This data is a sequence of s and s . As men-tioned in the Introduction, this provides an extra layer of security pp. 1–xiiiv lgorithm 1 Embedding Rule
Input: • y : Sequence of transform coefficients. • S: Encrypted secret bit sequences which is to be embedded.
Output: • z : The modified version of transform coefficients. if ( length ( S ) < × length ( y ) ) then for j = 1 to length ( y ) do if ( y ( j ) = − or y ( j ) = 0 or y ( j ) = +1 ) then z = y (Do not embed in these measurements) else if ( y ( j ) %2 = 0 ) then if ( y ( j ) %4 = 0 ) then if ( S ( j ) = 00 ) then z = y + 1 else if ( S ( j ) = 01 ) then z = y else if ( S ( j ) = 10 ) then z = y − else if ( S ( j ) = 11 ) then z = y + 2 or z = y − end if else if ( S ( j ) = 00 ) then if ( y (cid:54) = 2 ) then z = y − else z = y + 3 end if else if ( S ( j ) = 01 ) then if ( y (cid:54) = − ) then z = y + 2 else z = y − end if else if ( S ( j ) = 10 ) then if ( y (cid:54) = − ) then z = y + 1 else z = y − end if else if ( S ( j ) = 11 ) then z = y end if end if else if (cid:0) ( y ( j ) − (cid:1) then if ( S ( j ) = 00 ) then z = y else if ( S ( j ) = 01 ) then z = y − else if ( S ( j ) = 10 ) then z = y − else if ( S ( j ) = 11 ) then z = y + 1 end if else if ( S ( j ) = 00 ) then z = y + 2 else if ( S ( j ) = 01 ) then z = y + 1 else if ( S ( j ) = 10 ) then z = y else if ( S ( j ) = 11 ) then z = y − end if end if end if end if end for else Whole secret data cannot be embedded. Try short length secret data. end if return z to the embedded data. For this, we first encrypt this data by usingDES algorithm to obtain the encrypted secret data S (which is also a MeasurementStream [y i,u z i,v] MeasurementStream z i,v Extraction Rule
EncryptedExtractedSecret Data S'ExtractedSecret DataD'
Separation Decryption usingDES AlgorithmOriginal SecretData D
XORBit Error Rate(BER)
Algorithm 2
Fig. 2 : The Extraction Processsequence of s and s ) [1]. DES is a fairly standard algorithm usedfor data encryption [1]. Then, we represent S as a set of two-two bits,i.e. S = { S , S , . . . , S n } , 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 toembed two bits into the transform coefficients. The rule is designedin such a way so that the secret data could be extracted without anyloss, 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 corre-sponds to sparse vector coefficients of large values, and embeddingin it leads to degradation of image quality. Further, in y i,v , the secretdata is embedded selectively. We do not embed in y i,v with measure-ment value of − , and . This is because our embedding algorithmconcatenates the measurement values with integers from − to +3 ,and if these values are − , or , then we may end up getting many s after concatenation, which leads to difficulty in the extractionprocess. After embedding in other measurement values of y i,v , weobtain the modified y i,v , which is termed as z i,v . That is, z i,v = (cid:40) y i,v if y i,v = − , or y i,v + c otherwise , (13)where c ∈ {− , − , − , , , , } . We obtain our stego-data byconcatenating the measurements y i,u and z i,v as (cid:20) y i,u z i,v (cid:21) . The blockdiagram for this complete data embedding process is given in Fig. 1. Data Extraction
In this section, we explain the process of extracting embeddedsecret data from our stego-data. The steps of this extraction processare given below, which are exactly reverse to our data embeddingprocess.1. Separate the measurements z i,v from the stego-data, i.e. (cid:20) y i,u z i,v (cid:21) ,where i = 1 , , . . . , ( r × r /B is the block number, and u, v areindices available from the previous subsection.2. Extract only those measurements from z i,v whose values are notequal to − , or . The embedding rule ensures that the embeddeddata could be extracted without loss. In other words, Algorithm 1 ensures that no secret data is embedded in measurements with values − , and .3. Extract the encrypted message S (cid:48) from the measurementsobtained in the above step by applying Algorithm 2 .4. Decrypt this S (cid:48) by DES algorithm, and obtain the extracted secretdata D (cid:48) .Now, we check the correctness of this extracted secret data D (cid:48) bycomparing it with original secret data D . For this, we use the BitError Rate (BER), which is given as [17]Error Bits (EB) = D (cid:77) D (cid:48) , (14) BER = Number of ones in EBSize of D × , (15) pp. 1–xii v lgorithm 2 Extraction Rule
Input: • z : Sequence of modified linear measurements. These are z i,v that are not havingvalue equal to , or − . See extraction process in Section 3.2. Output: • S (cid:48) : Encrypted secret bit sequences. for j = 1 to length ( z ) do if ( y ( j ) = − or y ( j ) = 0 or y ( j ) = +1 ) then Continue else if ( z ( j ) %2 = 0 ) then if ( z ( j ) %4 = 0 ) then S (cid:48) ( j ) = 01 else S (cid:48) ( j ) = 11 end if else if (cid:0) ( z ( j ) − (cid:1) then S (cid:48) ( j ) = 00 else S (cid:48) ( j ) = 10 end if end if end if end for return S (cid:48) where (cid:76) denotes the bitwise XOR/ Exclusive OR operation. TheBER value for our steganography scheme is , i.e. we successfullyextract complete secret data without any error. This is the propertyof our embedding rule. The above extraction process is representedvia a block diagram in Fig. 2. Stego-Image Construction
When the stego-data is transferred over a communication media, theintruder can access this data from the public channel and can try toconstruct the stego-image. If the intruder obtains a high visual qual-ity image, then the goal of steganography is fulfilled. This is becausehe/ she will not be able to judge whether some data is hidden inthe image or not. Therefore, in this subsection, we give the stepsto construct the stego-image from the stego-data, which is equiva-lent to Section 2.3. We refer this process as construction rather thanreconstruction.1. Obtain the approximate sparse vector s (cid:48) from the stego-data andmeasurement matrices Φ u and Φ v as (recall (12)) s (cid:48) i,u = Φ − u y i,u , ands (cid:48) i,v = ADMM _ LASSO (cid:0) z i,v , Φ v (cid:1) . (16)Here, as discussed in Section 2.3, we use ADMM and LASSO toconstruct s (cid:48) i,v . The sparse vector s (cid:48) is obtained by concatenating s (cid:48) i,u and s (cid:48) i,v . Here, the size of s (cid:48) i,u , s (cid:48) i,v , and s (cid:48) is the same as that of s i,u , s i,v , and s , respectively.2. Convert each vector s (cid:48) i into a block of size B × B .3. Apply two-dimensional Inverse DCT (IDCT) to each of theseblocks to generate blocks x (cid:48) i of image. That is, recall (10), x (cid:48) i = IDCT (cid:0) s (cid:48) i (cid:1) . (17)4. Construct the stego-image of size r × r by arranging all theseblocks x (cid:48) i .The block representation of these steps is given in Fig. 3. We showin the Experimental Results section that image obtained from thisstego-data preserves the quality of the original image.As earlier, we term our proposed steganography scheme asCompressed-Sensing-Image-Steganography (CSIS) because we usecompressed sensing to enhance the embedding capacity of the imagesteganography scheme. MeasurementStream [y i,u z i,v] SeparatedMeasurementy i,u and z i,v Sparse Approximation
ApproximateSparse Blocks(size B X B) s i ' Blocks of Image Pixel(size B X B)x i Separation Inverse 2D-DCT
ReconstructedImage SI(Size r1 X r2)
Rearranging all Blocksand BlockFormation
Fig. 3 : Stego-Image Construction
Experiments are carried out in MATLAB on a machine with an IntelCore i3 processor @2.30 GHz and 4GB RAM. We use a set ofstandard grayscale images to test our CSIS. Sample test images areshown in Fig. 4 and Fig. 5. These images have the varying textureproperty and are taken from the miscellaneous category of USC-SIPIimage database [51] and two other public domain databases [52, 53].The miscellaneous category of USC-SIPI database consists of 24grayscale images. Some images, such as Lena, and Tiffany are nolonger available in this database. These images have played a sig-nificant role in image processing, and literature. Thus, we use otherpublic-domain test images databases [52, 53] for them. A total ofseven such images are chosen. Hence, we have a total of 31 grayscaleimages. Our CSIS is also applicable to color images, and we pick oneof them from USC-SIPI database.In this manuscript, we report average values of all the 31 imageswith detailed results for 10 images due to space limitations. This isfurther justified by the fact that the image processing literature hasused these 10 images or a subset of them.The size of each of test images is × , i.e. r × r . We takeblocks of size × , i.e. B × B . As earlier, the size of measurementmatrix Φ u is p × p . Recall from Section 3.1, p is the numberof coefficients with large values/ low frequency in the input sparsevector. For commonly used images, this value is between and [15, 54]. Since the measurement matrix cannot be different for everyinput matrix, we do experiments with three different values of p ( , and ) to find the optimal one here. Again from Section 3.1,the size of measurement matrix Φ v is ( | m | − p ) × p . We take | m | from the following range [15, 54]: { , , , , , , , } ,and as before, p = B × B − p (i.e. p = 64 − p ). For secretdata, we use randomly generated data, which is sequence of and bits. First , we check the embedding capacity of our proposed scheme.
Second , we do the similarity analysis between the cover images andthe constructed stego-images by assessing .
Third , in the remainderof this section, we do security analysis, perform five comparisonswith existing steganography schemes, and also experiment with acolor image.
Embedding Capacity Analysis
Embedding capacity is defined as the maximum number of bitsembedded in the cover media, which is the image here. The embed-ding capacity of our proposed steganography scheme depends on thesampling rate (SR), which is given as SR = Total Linear MeasurementsTotal Pixels in Cover Image . (18)We have r × r total pixels in the cover image and | m | lin-ear measurements for each block with r × r B × B number of blocks.Therefore, our sampling rate is SR = (cid:18) | m | r × r (cid:19) × (cid:18) r × r B × B (cid:19) = | m | B × B . (19) pp. 1–xiivi rom this definition, it is evident that embedding capacity mainlydepends upon | m | , however, the compressed image quality dependsupon both p and | m | . Therefore, to maintain the quality of stego-image while enhancing embedding capacity, the combination ofthese parameters is critical.For different combinations of p and | m | , in Table 1, we givethe embedding capacity in bits of our proposed CSIS for the 10test images of Fig. 4 and Fig. 5 and the average capacity for all the31 images. We analyze the data of this table by comparing p and | m | − p instead of p and | m | because the former set directly mapsto the number of ordinary samples and compressed sensing samples,respectively. When p is constant, and | m | − p is increased, thenumber of compressed sensing samples increases, where the secretdata bits are embedded, leading to increased capacity. For example,consider columns and of Table 1, we can observe that the embed-ding capacity increases when p is constant, i.e. and | m | − p is increased from to . When | m | − p is constant and p is increased, the number of compressed sensing samples decreaseleading to decreased embedding capacity. For example, considercolumns and , we observe that embedding capacity decreaseswhen | m | − p is constant, i.e. and p is increased from to . Stego-image Quality Assessment
In general, when the embedding capacity increases, the visual qual-ity of stego-image degrades. Hence, with increased embeddingcapacity, preserving the visual quality of stego-image is also essen-tial. There is no universal metric to judge the quality of stego-image.However, we check the quality of stego-image by examining the sim-ilarity between cover images and their corresponding stego-images.This check is done in two ways. Initially we perform a visual orsubjective check. The subjective measure is a good way to assess thequality of stego-image, but it depends on many factors like viewingdistance, the display device, the lighting condition, viewer's visionability, and viewer's mood. Therefore, it is necessary to design math-ematical models to assess the quality of stego-images, which wediscuss next.
Subjective or Visual Measure:
Human observers arethe final arbiter of image quality. Therefore, the subjective measureis a perfect way of assessing the quality of the images. Here, weconstruct stego-images corresponding to different test images usedin our experiment for different combinations of p and | m | . Thisresult shows that the stego-images are almost similar to their cor-responding cover images. The same is true for their corresponding histograms also. As an example, we present the visual compari-son for ‘Pepper’ cover image for one set of parameters; p = 12 and | m | = 37 . Fig. 6 shows the (a) ‘Pepper’ cover image (b) ‘Pep-per’ cover image histogram (c) ‘Pepper’ stego-image (d) ‘Pepper’stego-image histogram. From these figures, we observe that thestego-image is almost similar to its corresponding cover image andtheir corresponding histograms are also very similar.We also construct the edge map diagrams for both the cover imageand its corresponding stego-image for this same example. Theseedge maps are shown in Fig. 7a and Fig. 7b, respectively. We cansee from these figures that both the edge maps are almost the same.Hence, the visual quality of the cover image and its correspondingstego-image is almost similar. Objective or Numerical Measures:
These measurescompare the cover images and their corresponding stego-imagesbased on some numerical criteria that do not require extensivesubjective studies. Hence, in recent times, these measures aremore commonly used for image quality assessment. These include;Peak Signal-to-Noise Ratio (PSNR), mean Structural Similarity(SSIM) index, Normalized Cross-Correlation (NNC) coefficient, andentropy. We discuss all of them below.
PSNR : We compute the
PSNR value to evaluate the imperceptibilityof stego-images. That is,
P SNR = 10 log R MSE dB, (20)where
MSE represents the mean square error between the coverimage I and the stego-image SI , R is the maximum intensity ofpixel, which is for grayscale images, and dB refer to decibel.The MSE is calculated as
MSE = (cid:80) r i =1 (cid:80) r j =1 ( I ( i, j ) − SI ( i, j )) r × r , (21)where r and r represent the row and column numbers of thedigital image, respectively, and I ( i, j ) and SI ( i, j ) represent thepixel value of the cover image and the constructed stego-image,respectively.A higher PSNR value indicates the higher imperceptibility of thestego-image. In general, a value higher than 30 dB is considered tobe good since human eyes can hardly distinguish the distortion inthe stego-image [16, 55]. The PSNR values of the stego-images cor-responding to 10 test images of Fig. 4 and 5, and average for all 31images for different combination of p and | m | are given in Table 2.(a) Lena (b) Peppers (c) Boat (d) Goldhill (e) Zelda Fig. 4 : Test images used in our experiments(a) Tiffany (b) Living room (c) Tank (d) Airplane (e) Camera man
Fig. 5 : Continued from Fig. 4; test images used in our experiments pp. 1–xii vii able 1
Embedding capacity (in bits) obtain by proposed CSIS for different parameters and for different test imagesTest image Parameters p = 10 | m | = 32 p = 10 | m | = 35 p = 12 | m | = 37 p = 12 | m | = 40 p = 12 | m | = 42 p = 12 | m | = 47 p = 14 | m | = 36 p = 14 | m | = 39 Lena 171087 194519 194265 217491 232924 272130 170361 193679Peppers 173091 196725 196357 219641 235265 274890 172304 196193Boat 171563 194819 194559 217665 233430 272162 170738 194167Goldhill 174359 198019 197477 221155 236888 276297 173674 197031Zelda 170447 193811 193635 216639 232441 270830 170080 192951Tiffany 170457 193717 193291 216419 231924 270386 169747 192739Living room 174534 198336 198216 222186 238076 277904 174402 198336Tank 174961 198933 198395 222165 238276 277972 174564 198223Airplane 167255 189865 189195 212003 227341 265207 165822 188313Camera man 161201 183181 180375 202601 215917 251596 157618 177801Avg. of 10 images 170895 194192 193576 216796 232248 270937 169931 192943Avg. of 31 images 152786 176645 174678 198080 214135 251989 150023 173564 (a) ‘Pepper’ cover image (b) Cover image histogram(c) ‘Pepper’ stego-image (d) Stego-image histogram
Fig. 6 : ‘Pepper’ cover image, its stego-image, and their correspond-ing histogram using parameter p =12 and | m | =37.(a) Cover image edge map (b) Stego-image edge map Fig. 7 : Edge maps of ‘Pepper’ cover image and its stego-imageusing parameter p =12 and | m | =37.From this table, we can easily observe that this value is higher than30 dB for all combinations of parameters and for all images. Means SSIM Index : It is an image quality assessment metric usedto measure the structural similarity between two images [56]. Thismeasure is based on the assumption that the human visual system(HVS) is more adapted to the image's structural information. Themean SSIM (MSSIM) index is given as
SSIM ( x, y ) = (2 µ x µ y + C )(2 σ xy + C )( µ x + µ y + C )( σ x + σ y + C ) , (22) MSSIM ( I, SI ) = 1 M M (cid:88) j =1 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 coverimage I and stego-image SI , i.e. for the overall image quality. Here, µ x is the weighted mean of x , µ y is the weighted mean of y , σ x isthe weighted standard deviation of x , σ y is the weighted standarddeviation of y , σ xy is the weighted covariance between x and y , C & C are arbitrary constants, i j & si j are the content of the coverimage and stego-image, respectively, at the j th local window, and M is the number of local windows. We took the values of all theseparameters according to [56]. The value of the mean SSIM indexlies between and , where the value indicates that there is nosimilarity between the two images, and the value indicates that theimages are exactly similar.The mean SSIM index values between the stego-images and theircorresponding cover images for different combination of p and | m | are given in Table 3. As earlier, 10 images from 4 and 5 are exten-sively analyze and average of 31 images is reported. From this table,we observe that all these values are close to , which representsthat the stego-images are very much similar in structure to theircorresponding cover images. NCC Coefficient : Normalized correlation (NC) metric measuresthe degree of similarity between two images, and when the twoimages are independent, this correlation is called normalized cross-correlation (NCC) [54]. The NCC coefficient is given as
NCC = (cid:80) r i =1 (cid:80) r j =1 I ( i, j ) SI ( i, j ) (cid:80) r i =1 (cid:80) r j =1 I ( i, j ) , (24)where r and r represent the row and column numbers of the digitalimage, respectively. I ( i, j ) and SI ( i, j ) represent the pixel value ofthe cover image and the constructed stego-image, respectively. Thevalue equal to 1 indicates that both the images are exactly similar.For our experiments, the values of NCC are given in Table 4. The setof images used are same as for PSNR and SSIM. We observe that allthese values are close to 1, which means that the stego-images arealmost identical to their corresponding cover images. Entropy : In general, entropy is defined as the measure of averageuncertainty of a random variable, which here is the average numberof bits required to describe the random variable. In the context ofan image, it is a statistical measure of randomness that can be usedto characterize the texture of the image [57]. For a grayscale image,entropy is given as
Entropy = − (cid:88) i =0 ( p i log p i ) , (25)where p i is the probability of value i pixel of the image. Table 5gives the entropy values for the cover images and their correspondingstego-images for different combinations of p and | m | . The set ofimages used are same as for PSNR, SSIM, and NCC. From this table,we observe that for all these combinations of p and | m | , the entropyof the cover images and their corresponding stego-images are almostsimilar. pp. 1–xiiviii able 2 Value of PSNR (in dB) obtain by proposed CSIS for different parameters and for different test imagesTest image Parameters p = 10 | m | = 32 p = 10 | m | = 35 p = 12 | m | = 37 p = 12 | m | = 40 p = 12 | m | = 42 p = 12 | m | = 47 p = 14 | m | = 36 p = 14 | m | = 39 Lena 34.34 35.11 35.62 36.15 36.71 37.31 36.33 36.91Peppers 34.05 34.35 35.23 35.76 36.21 36.98 35.44 35.81Boat 32.67 33.07 33.84 34.25 34.72 36.84 34.37 34.81Goldhill 32.69 33.61 34.06 34.54 35.12 35.32 34.33 34.93Zelda 39.31 39.61 40.10 41.32 40.02 42.67 40.73 42.46Tiffany 33.64 33.96 34.69 35.88 36.49 37.23 35.73 36.37Living room 30.94 31.29 32.07 32.98 33.31 33.78 33.48 33.64Tank 34.27 34.32 35.13 35.62 35.98 36.98 35.36 35.87Airplane 32.89 34.15 34.88 35.78 36.39 37.91 34.43 35.42Camera man 35.71 36.89 37.52 38.86 39.38 40.86 40.04 40.65Avg. of 10 images 34.051 34.636 35.314 36.114 36.433 37.588 36.024 36.687Avg. of 31 images 34.245 34.883 35.593 36.379 36.668 37.921 36.282 36.901
Table 3
Value of Mean SSIM index obtain by proposed CSIS for different parameters and for different test imagesTest image Parameter p = 10 | m | = 32 p = 10 | m | = 35 p = 12 | m | = 37 p = 12 | m | = 40 p = 12 | m | = 42 p = 12 | m | = 47 p = 14 | m | = 36 p = 14 | m | = 39 Lena 0.9308 0.9394 0.9475 0.9518 0.9562 0.9672 0.9512 0.9558Peppers 0.9203 0.9225 0.9291 0.9463 0.9424 0.9547 0.9333 0.9383Boat 0.9211 0.9356 0.9444 0.9517 0.9575 0.9663 0.9484 0.9545Goldhill 0.9011 0.9122 0.9236 0.9343 0.9421 0.9532 0.9227 0.9359Zelda 0.9512 0.9563 0.9613 0.9657 0.9694 0.9768 0.9628 0.9678Tiffany 0.9239 0.9315 0.9357 0.9434 0.9501 0.9596 0.9369 0.9437Living room 0.9012 0.9092 0.9211 0.9332 0.9384 0.9460 0.9341 0.9382Tank 0.8835 0.8857 0.9006 0.9094 0.9197 0.9388 0.9086 0.9131Airplane 0.9463 0.9525 0.9605 0.9646 0.9673 0.9755 0.9607 0.9692Camera man 0.9677 0.9752 0.9839 0.9864 0.9871 0.9907 0.9838 0.9868Avg. of 10 images 0.9198 0.9272 0.9360 0.9445 0.9492 0.9598 0.9398 0.9463Avg. of 31 Images 0.9206 0.9276 0.9365 0.9449 0.9498 0.9601 0.9412 0.9471
Table 4
Value of normalized cross-correlation obtain by proposed CSIS for different parameters and for different test imagesTest image Parameter p = 10 | m | = 32 p = 10 | m | = 35 p = 12 | m | = 37 p = 12 | m | = 40 p = 12 | m | = 42 p = 12 | m | = 47 p = 14 | m | = 36 p = 14 | m | = 39 Lena 0.9985 0.9988 0.9989 0.9991 0.9991 0.9992 0.9991 0.9993Peppers 0.9982 0.9983 0.9985 0.9987 0.9988 0.9989 0.9985 0.9988Boat 0.9979 0.9983 0.9985 0.9987 0.9988 0.9989 0.9987 0.9989Goldhill 0.9976 0.9981 0.9983 0.9986 0.9987 0.9988 0.9982 0.9986Zelda 0.9991 0.9993 0.9994 0.9995 0.9995 0.9997 0.9994 0.9995Tiffany 0.9992 0.9993 0.9994 0.9995 0.9995 0.9996 0.9994 0.9995Living room 0.9962 0.9961 0.9962 0.9971 0.9972 0.9964 0.9970 0.9982Tank 0.9987 0.9988 0.9992 0.9991 0.9992 0.9993 0.9993 0.9991Airplane 0.9989 0.9991 0.9993 0.9994 0.9994 0.9995 0.9993 0.9995Cameraman 0.9989 0.9991 0.9994 0.9994 0.9994 0.9995 0.9995 0.9996Avg. of 10 images 0.9983 0.9985 0.9989 0.9989 0.9989 0.9989 0.9988 0.9991Avg. of 31 images 0.9983 0.9985 0.9990 0.9989 0.9989 0.9989 0.9990 0.9991
Table 5
Entropy comparison of cover images and their corresponding stego-images obtain by proposed CSIS using different parametersTestimage Coverimage Stego-image using different parameters p = 10 | m | = 32 p = 10 | m | = 35 p = 12 | m | = 37 p = 12 | m | = 40 p = 12 | m | = 42 p = 12 | m | = 47 p = 14 | m | = 36 p = 14 | m | = 39 Lena 7.4456 7.4552 7.4581 7..4569 7.456 7.4545 7.4534 7.4551 7.4536Peppers 7.5715 7.5924 7.5924 7.5908 7.5911 7.5901 7.5889 7.5897 7.5898Boat 7.1238 7.1323 7.1339 7.1322 7.1334 7.1337 7.1331 7.1277 7.1304Goldhill 7.4778 7.4653 7.4686 7.4704 7.4723 7.4719 7.4731 7.469 7.4717Zelda 7.2668 7.2625 7.2635 7.2638 7.2643 7.2649 7.2652 7.2633 7.2642Tiffany 6.6015 6.6076 6.6063 6.6046 6.606 6.6074 6.607 6.6096 6.6076Living room 7.2950 7.4200 7.4200 7.4253 7.4260 7.4261 7.4262 7.4267 7.4278Tank 5.4957 6.3614 6.3728 6.3771 6.3829 6.3846 6.3871 6.3709 6.3815Airplane 6.7025 6.773 6.7637 6.7535 6.7501 6.7468 6.7396 6.7614 6.7454Camera man 7.0482 7.0743 7.0763 7.0738 7.0703 7.0683 7.0664 7.0726 7.0661Avg. of 10 images 7.0028 7.1144 7.0691 7.07678 7.1152 7.1148 7.1140 7.1145 7.1138Avg. of 31 images 6.9985 6.6451 7.7132 6.7124 6.6476 6.6462 6.6447 6.6469 6.6448
Security Analysis
Since the proposed CSIS is a transform domain based technique andit employs indirect embedding strategy, i.e. it does not follow theLSB flipping method,and hence, it is immune to statistical attacks [24, 58]. Also, CSIS does not lead to the shrinkage effect. Thatmeans, after embedding, the nonzero coefficients do not modify tozero value, and hence attacks against F5 [25, 58] are not considered. pp. 1–xii ix ena Peppers Boat Goldhill Zelda Living room Tiffany Tank Airplane Camera man
Images B E R ( % ) BER when Correct Secret Key is usedBER when Wrong Secret Key is used
Fig. 8 : BER with the correct and with a wrong secret-key (i.e.measurement matrix)(a) Original measurements(b) Modified measurements
Fig. 9 : Distribution of measurements for ‘Peppers’ imageMoreover, in CSIS, the measurement matrix Φ is considered asthe secret-key, which is shared between the sender and the legit-imate receiver. If the eavesdropper intercepts the stego-image bya randomly generated measurement matrix, he cannot not enterthe embedding domain without the original secret-key. Hence, weachieve increased security in our proposed system. To justify this,we extract the secret data in two ways, i.e. by using the correct mea-surement matrix and by using a measurement matrix that is veryclose to the original one, and obtain the BER (discussed in Section3.2) between the original secret data the extracted one.In Fig. 8, we present this BER for earlier discussed 10 coverimages, and for the parameter p =12 and | m | =37. In this figure, wesee that for the correct secret-key, the BER is 0, and for a tiny dif-ference in the measurement matrix, i.e. wrong secret-key, the BERis very high, which is 35% to 40%. That is, even a small change inthe secret-key will lead to an extreme shift in accuracy between theoriginal secret data and the extracted one.In addition to the above security analysis, we also measure thesecurity by analyzing the distribution of the measurements andtheir corresponding modified measurements, i.e. after embedding thesecret data. For ‘Pepper’ image with parameter p =12 and | m | =37,this distribution of the original measurements and the modified mea-surements is shown in Fig. 9a and Fig. 9b, respectively. The greenand blue colors are automatically added by Matlab and do not haveany significance here. From these figures, we see that the distributionfor both cases is almost the same. We also check these distributionsfor all the images and obtain the same results. We do not includethese in this manuscript due to space limitations.The preservation of distribution of measurements in the earliertwo histograms can also be justified by the probability of additionand subtraction operation decided by our algorithm. In Fig. 10, weplot this probability. From this figure, we see that the lines of prob-abilities of addition and subtraction operation oscillate around 0.5. Here, the minimum and maximum deviation to 0.5 are 0.02 and0.07, respectively, i.e. for proposed CSIS, the probabilities of boththe addition and the subtraction are nearly the same. The distribu-tion of measurements and the probability of addition & subtractionoperation as discussed have justified that for our proposed CSIS,the likelihood of detecting data embedding by an eavesdropper issignificantly low. Lena Peppers Boat Goldhill Zelda Living room Tiffany Tank Airplane Camera man
Images P r ob a b ili t y Probability of additionProbability of subtraction
Fig. 10 : Probability of addition and subtraction operation
Performance Comparison
In this subsection, we compare the performance of the proposedCSIS with the existing steganography schemes. This result is givenin Table 6. In this table, the first column represents the comparisonmetrics, and the remaining columns give the metric data for differentsteganography schemes.In the first row of Table 6, we compare the average embeddingcapacity over all the 31 images. We report these embedding capacityfor the parameter p = 12 & | m | = 37 . In this table, we do not com-pare these results for all the images because the existing schemes'data are not available for all the images. From the first row of thistable, we observe that on an average our steganography scheme hasapproximately . , . , . , . , . , . , . and . times embedding capacity as compared to references [6], [12], [13],[14], [16], [17], [18], and [19], respectively. Here, we can see thatour proposed scheme has a higher embedding capacity compared toall schemes except the one, which is [6]. The reason for this is thatthis scheme is based on embedding secret data in the spatial domain.As discussed in the Introduction, spatial domain based embeddingtechniques have a higher embedding capacity, but they are prone tosecurity issues. Also, these techniques are not based on compression,which is the main motivation of this manuscript. Further, as evidentfrom Table 1, for a set of parameters p = 12 and | m | = 47 , CSIShas 270937, and 251989 bits embedding capacity for the average of10 and 31 images, respectively. Hence, for this set of parameters,CSIS has approximately the same embedding capacity as that of [6].In the second row of this table, for our scheme we report the rangeof PSNR values when considering all sets of parameters and againall 31 images. From the second row of this table, we observe thatsimilar to existing steganography schemes, our CSIS also has PSNRvalues greater than 30 dB, which is considered good [16, 55].The purpose of the proposed CSIS is to embed secret data in thecompressed domain. Hence, in the third row of Table 6, we checkwhich schemes are based on compression and which are not. Fromthis row, we observe that except [6, 18], our CSIS and all otherschemes are based on compression. Finally, from the fourth row tothe sixth row of Table 6, we compare the security of these schemesby 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 [17] schemes passall the three security tests. Hence, we can conclude that out of allthese schemes, only CSIS fulfills all the goals of steganography withhigher embedding capacity. 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 pp. 1–xiix able 6
Performance comparison between proposed CSIS and various other steganography schemes
Metrics Steganography SchemesCSIS
Ref. [6] Ref. [12] Ref. [13] Ref. [14] Ref. [16] Ref. [17] Ref. [18] Ref. [19]Capacity (in bits)
Yes
No Yes Yes Yes Yes Yes No YesResistant toChi-square
Yes
No No Yes Yes Yes Yes Yes YesResistant toShrinkageEffect
Yes
NA Yes No Yes Yes Yes Yes YesSecret Key
Yes
No No No No No Yes No No color image. For this we only use ‘Pepper’ color image of resolution × , and perform experiments for p = 12 and | m | = 37 aswell as p = 14 and | m | = 36 .Fig. 11 shows the subjective/ visual measure for ‘Pepper’ colorimage for p = 12 , | m | = 37 . From this figure, we observe thatthe cover image and its corresponding stego-image are almost sim-ilar. Table 7 gives the results for other measures like embeddingcapacity, PSNR values, mean SSIM index, NCC coefficients forthe different color components, and entropy for both cover imageand stego-image. We can observe from this table that the embed-ding capacity of our color image is approximately three times theembedding capacity of ‘Pepper’ grayscale image for the same set ofparameters. Please see columns and of Table 1. This is becauseof 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 , 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. Wesee that entropy for both these images is almost the same. We present an enhanced-embedding capacity image steganographyscheme based on compressed sensing technique. Here, we combinethree components to achieve increased embedding capacity withoutdegrading the quality of stego-images, as well as making it resistantto steganalysis attacks.
First , we use compressed sensing to sparsifycover image block-wise and obtain its linear measurements usinga matrix. We uniquely select a large number of permissible mea-surements. Hence, we achieve a high embedding capacity. Sincethe measurement matrix is a secret-key that is shared between thesender and the legitimate receiver, this adds extra security to ourscheme. Also, we encrypt the secret data using the DES algorithmand then embed two bits of secret data into each permissible mea-surement instead of embedding one bit per measurement.
Second ,we propose a technique of data extraction that is lossless and recov-ers our secret data entirely.
Third , we use ADMM solution of theLASSO formulation of the obtained optimization problem in thestego-image construction. The reason for selecting them is that theyhave broad applicability in the field of image processing, requireless assumptions on the property of the objective function, have fastconvergence, and are easy to implement.We initially perform experiments on several standard grayscaleimages that vary in texture, and with different sets of parameters andrandomly generated binary data as our secret data. For performanceevaluation, we calculate embedding capacity, PSNR value, meanSSIM index, NCC coefficient, and entropy. Experiments show thatour proposed CSIS achieves higher embedding capacity than exist-ing steganography schemes that follow compression. We achieve1.53 times more embedding capacity as compared to the most recentscheme of the similar category. PSNR values coming out of ourscheme are more than 30 dB, which is considered good. Both meanSSIM index and NCC coefficients values are close to one, whichshows that the cover images and their corresponding stego-imagesare almost similar. This similarity is further supported by the factthat we obtain approximately the same entropy value for both thecover images and their corresponding stego-images. Further, we also show the applicability of CSIS on a color image. Again, the resultsobtained are almost the same as that of grayscale images. However,we get approximately three times higher embedding capacity for thecolor image because of the presence of the three component in colorimages.In future, we plan to embed the secret data in text, audio, andvideo. Other future works include extending this work for a real-time application such as hiding fingerprint data, iris data, medicalinformation of patients, and personal signature. As mentioned inthe Introduction, another line of work is embedding images insideimages. Since a lot of work has been done in embedding a singleimage, we will focus on hiding multiple secret images and multilevelimage steganography scheme.
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