ℓ 1 SABMIS: ℓ 1 -minimization and sparse approximation based blind multi-image steganography scheme
JJuly 13, 2020 0:34 Manuscript (cid:96) SABMIS: (cid:96) -minimization and sparse approximation based blindmulti-image steganography scheme Rohit Agrawal
Data & Computational Sciences Laboratory, Indian Institute of Technology Indore,Indore, 453552, [email protected]
Steganography plays a vital role in achieving secret data security by embedding it intocover media. The cover media and the secret data can be text or multimedia, such asimages, videos, etc. In this paper, we propose a novel (cid:96) -minimization and sparse approx-imation based blind multi-image steganography scheme, termed (cid:96) SABMIS. By using (cid:96) SABMIS, multiple secret images can be hidden in a single cover image. In (cid:96) SABMIS,we sampled cover image into four sub-images, sparsify each sub-image block-wise, andthen obtain linear measurements. Next, we obtain DCT (Discrete Cosine Transform)coefficients of the secret images and then embed them into the cover image ' s linearmeasurements.We perform experiments on several standard gray-scale images, and evaluate embed-ding capacity, PSNR (peak signal-to-noise ratio) value, mean SSIM (structural similarity)index, NCC (normalized cross-correlation) coefficient, NAE (normalized absolute error),and entropy. The value of these assessment metrics indicates that (cid:96) SABMIS outper-forms similar existing steganography schemes. That is, we successfully hide more thantwo secret images in a single cover image without degrading the cover image signifi-cantly. Also, the extracted secret images preserve good visual quality, and (cid:96) SABMIS isresistant to steganographic attack.Keywords: Image Processing; Steganography; Sparse Approximation; Optimization; (cid:96) -Minimization.
1. Introduction
The security of digital data is essential for its transfer over the communication me-dia. To achieve this data security, in general, there are mainly two approaches used;cryptography and steganography. In cryptography, the encryption mechanismtransforms plain-text (i.e., secret data) into cipher-text using the encryption key.This cipher-text appeared as an unreadable form that attracts opponents to manipu-late its contents by using certain brute-force attacks. Nevertheless, steganographyavoids this situation.Steganography is derived from two Greek words; steganos means “covered orsecret,” and graphie means “writing”. The purpose of steganography is to hide thesecret data into some other unsuspected cover media so that the secret data be-comes visually imperceptible. In steganography, both the cover media and the secretdata can be text or multimedia. The media obtained after embedding secret datainto cover media is called stego-media. In this paper, we consider both the secret a r X i v : . [ c s . MM ] J u l uly 13, 2020 0:34 Manuscript Rohit Agrawal data and the cover media as images due to their heavy use in web-based applica-tions. The challenges here are: enhancing the embedding capacity, preserving thequality of the stego-image, and the scheme should be resistant to steganalysis a (i.e.,steganographic attacks). In the following paragraphs, first, we discuss different cat-egories of steganography schemes. Second, we discuss some of the existing schemesand their weaknesses. Finally, we discuss how the scheme proposed in this paperoutperforms the existing schemes.In general, image steganography can be categorized based on the domain inwhich embedding is performed, i.e., the spatial domain or the transformation do-main. In the spatial domain-based scheme, secret data is embedded directly into thecover image by some alteration in image pixels value.
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In the transformdomain-based scheme, initially, the cover image is transformed into frequency com-ponents by certain transformations, and then the secret data is embedded into thesecomponents. A few of those schemes are JSteg, F5, Outguess, etc.
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The steganography scheme can also be categorized based on their embedding mech-anisms, i.e., direct embedding or indirect embedding. In a direct embedding mech-anism, the secret data is embedded by flipping the LSB (Least Significant Bit) ofeither the pixel value (i.e., spatial domain-based schemes) or the transformed coef-ficients (i.e., transform domain-based schemes) of the cover image.
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In anindirect embedding mechanism, the pixel value or the transformed coefficient valuesare altered according to certain secret message bits.
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The spatial domain-based image steganography schemes outperform the trans-form domain one in-terms of embedding capacity. But they are not resistant tosteganographic attack. Transform-based schemes are resistant to these attacks andprovide good visual quality stego-image with limited embedding capacity. Further-more, if we try to increase the embedding capacity of these schemes, then thequality of stego-image degrades. Besides, the indirect embedding mechanisms makesteganography schemes more resistant to these attacks
12, 23, 26, 36, 39 as comparedto direct ones. Thus, some of the schemes mentioned above are not resistant tosteganographic attack.
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While some are resistant to these attacks.
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The purpose of the scheme proposed in this paper is to hide multiple secretimages in a single cover image. Hence, first, we discuss some recent image steganog-raphy schemes that hide multiple secret images in a single cover image. In, theauthors proposed a steganography scheme in which they hide two gray-scale secretimages in a single color image. This scheme is based on two-level DWT (DiscreteWavelet Transformation). In this, the quality of stego-image and extracted imagesare good, but the embedding capacity in bit per pixel is very less. In, the authorsproposed a steganography scheme in which they hide three binary secret images ina single gray-scale as well as in a color image. This scheme is based on LSB basedembedding approach, which is not resistant to steganographic attack. In this, thequality of stego-image is good, but the embedding capacity in bit per pixel is very a It is the study of detecting the secret data hidden using steganography. uly 13, 2020 0:34 Manuscript (cid:96) SABMIS less.Now, we discuss some recent image steganography schemes that hide only onesecret image in a single cover image. In, a gray-scale secret image is embedded in acolor image. This scheme is based on DWT and PSO (particle swarm optimization).In, a binary medical image is embedded in to a gray-scale cover image. This schemeis based on Redundancy Integer Wavelet Transform (RIWT), SVD (Singular ValueDecomposition), and Discrete Cosine Transformation (DCT). In, a binary medicalimage is embedded in a color cover image. This scheme is based on RIWT, DCT,and QR factorization. In all these schemes, the stego-image preserved good visualquality, but their embedding capacity is limited. And, if we try to increase thisembedding capacity, the quality of the stego-image degrades.To overcome this limitation, in this manuscript, we utilize other paradigms;optimization (i.e., (cid:96) -minimization) and sparse approximation. The steganographyscheme proposed in this manuscript is termed as (cid:96) SABMIS because we use the con-cept of (cid:96) -minimization and sparse-approximation for blind multi-image steganog-raphy scheme. (cid:96) SABMIS fulfills all the requirements of image steganography, i.e.,it has high embedding capacity, preserve good visual quality stego-image, and re-sistant to steganographic attacks. Here, first, the cover image is sampled into foursub-images. Next, each sub-image is sparsified using DCT, and then its linear mea-surements are obtained using a random measurement matrix. Next, we collect theDCT coefficients of all secret images. Now, we select fix number of these coeffi-cients for each secret image and then embed them into the permissible linear mea-surements using our proposed embedding rule (discussed in Section 3) and obtainmodified measurements. Finally, we generate the stego-image from these modifiedmeasurements by solving the (cid:96) -minimization problem (again, discussed in Section3). In (cid:96) SABMIS, as discussed earlier, we embed DCT coefficients of secret imagesinto linear measurements of the cover image, instead of embedding them directlyinto the transformed coefficients. These measurements act as encoded transformedcoefficients, and hence, also adds security to our proposed scheme. For performanceevaluation, we perform experiments on standard test images and evaluate embed-ding capacity, PSNR (Peak Signal-to-Noise Ratio) value, mean SSIM (StructuralSimilarity) index, NCC (Normalized Cross-Correlation) coefficient, NAE (Nor-malized Absolute Error, also called normalized average absolute difference), andentropy. We also show the visual comparison between the cover image & its corre-sponding stego-image, and between the secret image & its corresponding extractedsecret image. Moreover, we also show that our scheme outperforms existing imagesteganography schemes.The rest of the paper is organized as follows. Section 2 gives brief explanation ofeach of (cid:96) -minimization and sparse approximations. Section 3 explains our proposedsteganography scheme, including embedding and extraction of the secret image.Section 4 presents experimental results. Finally, Section 5 gives conclusions andfuture work.uly 13, 2020 0:34 Manuscript Rohit Agrawal
2. Background
Here, in the following subsections, we provide a brief explanation of (cid:96) -minimizationand sparse approximation. (cid:96) -Minimization The general form of the (cid:96) -minimization is given as
1, 5 min x (cid:107) x (cid:107) Subject to Ax = b, (2.1)where (cid:107)·(cid:107) is the (cid:96) -norm, x ∈ R N × , A ∈ R M × N , and b ∈ R M × . In manyapplications such as signal processing, pattern recognition, etc., a (cid:96) -minimizationor a minimum (cid:96) -norm solution is preferable. A few examples of these applicationsinclude data separation, face recognition, data clustering, image restoration, imageclassification, etc. Sparse Approximation
Many practical problems in digital image processing and other disciplines includefinding the best approximate solution to a system of linear equations. The theory ofsparse approximation corresponds with sparse solutions for linear equation systems.This theory has applications in many domains, such as signal/ image processing,machine learning, medical imaging, etc.Let, we have an unknown K -sparse signal x ∈ R N × , a measurement matrix Φ ∈ R M × N , and a measurement vector y ∈ R M × , such that y = Φ x . Reconstruction(or approximate reconstruction) of x from Φ and y , which is usually consideredas an inverse problem, can be obtained by solving the following (cid:96) -minimizationproblem: min x (cid:107) x (cid:107) Subject to Φ x = y, (2.2)where (cid:107)·(cid:107) is (cid:96) -norm, which measures the total number of non-zero elements in avector. This equation (i.e., (2.2)) is referred to as (cid:96) -norm minimization problem, which is a combinatorial and NP-hard problem. Since the solution of x is suffi-ciently sparse, we can substitute the (cid:96) -norm minimization by the (cid:96) -norm (i.e., theclosest convex norm) minimization problem. Hence, x is reconstructed from Φ and y by the solving (2.1), where A and b are equivalent to Φ and y , respectively.As discussed, the signal to be approximated (or reconstructed) should be sparse.However, there are cases when this signal is not sparse. So, we can sparsify it usingsome transformation. For example, the signal x , which is not sparse, can be sparsified using an orthogonal matrix (termed as sparsification matrix) Ψ ∈ R N × N as s = Ψ T x, (2.3)uly 13, 2020 0:34 Manuscript (cid:96) SABMIS where s ∈ R N × is the sparse representation of x . Sparse signal s can be recon-structed by solving (2.1), where the decision variable x is equivalent to s , A isequivalent to Θ = ΦΨ, and b is equivalent to y . After that, signal x is obtainedfrom s by inverse sparsification , i.e., x = Ψ s .The approach of reconstructing sparse signal by solving (2.1) is referred as aconvex optimization method. Some other approaches such as LASSO,
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OMP, CoSaMP, SpaRSA, etc. can be used to reconstruct the sparse signal from themeasurements.
3. Proposed Blind Multi-Image Steganography Scheme
Our proposed blind multi-image steganography scheme, which is based on (cid:96) -minimization and sparse approximation, consists of embedding secret images andtheir extraction from the generated stego-image. These parts are discussed in therespective subsections below. Secret Images Embedding
First, we perform sub-sampling on a cover image and obtain four sub-sampledimages (or sub-images). Let CI is the cover image of size N × N , then the foursub-images are obtain as CI ( n , n ) = CI (2 n − , n − ,CI ( n , n ) = CI (2 n , n − ,CI ( n , n ) = CI (2 n − , n ) ,CI ( n , n ) = CI (2 n , n ) , (3.1)where n , n = 1 , , . . . , N (in our case, N is completely divisible by 2); CI k , for k = { , , , } , are the four sub-images; and CI ( · , · ) is the pixel value at ( · , · ).Originally, these sub-images are not sparse; hence, next, we perform block-wisesparsification of each of these images. For this, we divide each sub-image into blocksof size b × b and obtain N × b blocks for each sub-image (in our case, b completelydivides N ). Now, we consider each block as a vector of size b ×
1, and then sparsifythem (as in (2.3)) using discrete cosine transform matrix of size b × b as thesparsification matrix Ψ. That is, s i = Ψ T x i , (3.2)where i = 1 , , . . . , N × b , x i and s i are the i th original and sparse vector represen-tation of the respective blocks, and Ψ T is the transpose of Ψ. Now, we considerthese sparse vector in the zig-zag scanning order as given in one of our previouspaper. As a consequence of sparsification, each sparse vector has few coefficientsof significant values and the remaining coefficients of very small or zero values.Thus, we categories each vector into two groups s i,u ∈ R p and s i,v ∈ R p , where p and p are the number of coefficients having large values and small values (oruly 13, 2020 0:34 Manuscript Rohit Agrawal zero values), respectively, and p + p = b . Now, we project each sparse vector ontolinear measurements as y i = (cid:20) y i,u y i,v (cid:21) = (cid:20) s i,u Φ s i,v (cid:21) , (3.3)where Φ is the measurement matrices, which is a m × p (for our case, m > p )matrix of normally distributed random numbers; and y i ∈ R ( p + m ) × is the set oflinear measurements. Since the distribution of coefficients of the generated sparsevectors is almost the same for all blocks of an image, we use the same measurementmatrix for all blocks.Next, we perform processing in the secret images for embedding them into thecover image. In our proposed steganography scheme, we can embed a maximumof four secret images, one in each of the four sub-images of a single cover image.If we want to embed less than four secret images, we randomly select numbers ofsub-images equal to the number of secret images that we want to embed, from thetotal four sub-images. Let S k , for k = { , , , } , are the four secret images eachof the size M × M . First, we perform block-wise DCT to each of these imagesand obtain their corresponding DCT coefficients. Here, the size of each block is l × l , and hence, we have M l number of blocks for each secret image (in our case, l completely divides M ). Now, we consider these DCT coefficients as a vector in thezig-zag scanning order as given in. Let t ki ∈ R l × , for i = 1 , , . . . , M l , be thevector representation of the DCT coefficients of the S k secret image.Now, we perform embedding of the secret images in the cover image. We embed t i DCT coefficients from the secret image into y i linear measurements of one sub-image of the cover image. This embedding is given in Algorithm 1 that proposesthe embedding rule. In this, we show embedding of only one secret image into onesub-image.In this algorithm, p represents the number of coefficients having large values(as discussed above), p is the number of DCT coefficients from each t i that areembedded into the cover sub-image, and α , β , γ & c are the constants. Here, thesteps number 2, 3 and 5 shows the embedding of the first coefficient, the next c − M l will be less than or equal to N × b , p coefficientfrom t i can be embedded into y i ∈ R p + m × Algorithm 1 , and the stego-images & the extracted secret images preserve good visual quality. We discuss thevalues of all these parameters in Experimental Results section (i.e., in section 4).Finally, we construct the stego-image. As earlier, we can embed a maximumof four secret images into four sub-images of a single cover image. Hence, we firstconstruct four sub-stego-images and then perform inverse sampling to obtain astego-image from these four sub-stego-images. Let s (cid:48) i be the sparse vector of the i th uly 13, 2020 0:34 Manuscript (cid:96) SABMIS Algorithm 1
Embedding Rule
Input: • y : Sequence of linear measurements of the cover image. • t : Sequence of transform coefficients of the secret image. • The value of p , p , α , β , γ and c (discuss in Sections 3.1 and 4). Output: • y (cid:48) : The modified version of the linear measurements. Initialize y (cid:48) to y for i = 1 to N × b do y (cid:48) i ( p ) = y i ( p − c ) + α × t i (1) . for j = p − c + 1 to p − do y (cid:48) i ( j ) = y i ( j − c ) + β × t i ( j − p + c + 1) . end for for k = p + p + 1 to p + 2 × p − c do y (cid:48) i ( k ) = y i ( k − p + c ) + γ × t i ( k − p − p + c ) . end for end for return y (cid:48) block of a sub-stego-image (say k th ), then s (cid:48) i,u = y (cid:48) i (1 : p ) and s (cid:48) i,v = min s (cid:48) i,v (cid:13)(cid:13) s (cid:48) i,v (cid:13)(cid:13) Subject to Φ s (cid:48) i,v = y (cid:48) i ( p + 1 : p + m ) . (3.4)where y (cid:48) i ( a : b ) is the range from a th element to b th element of the vector y (cid:48) i . Forthe solution of the minimization problem of (3.4), we use LASSO (least absoluteshrinkage and selection operator) formulation of it and then solve it using ADMM(alternating direction method of multipliers) algorithm.
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The reason for this isthat it has a wide application in the image processing domain. The sparse vector s (cid:48) i is the concatenation of s (cid:48) i,u and s (cid:48) i,v . After that, we perform inverse sparsificationand obtain non-sparse vectors as x (cid:48) i = Ψ s (cid:48) i . Now, we covert each vector x (cid:48) i into blockof size b × b , and then construct the sub-stego-image of size N × N by arrangingall these blocks. In the end, we perform inverse sampling (see (3.1)) and obtain asingle stego-image from the four sub-stego-images.uly 13, 2020 0:34 Manuscript Rohit Agrawal
Secret Images Extraction
In this subsection, we discuss the process of extraction of the secret images from thestego-image. Initially, we perform sampling (as done in subsection 3.1 using (3.1))onto the stego-image to obtain four sub-stego-images. Let T k , for k = { , , , } ,are the four sub-stego-images. Since the extraction of all the secret images is similar,here, we discuss the extraction of only one secret image from one sub-stego-image.In this process, first, we perform block-wise sparsification of the sub-stego-image.For this, we divide this image into blocks of size b × b and then consider each blockas a vector of b ×
1. Here, we have a total of N × b blocks for a sub-stego-image.Next, we sparsified each vector (say x (cid:48)(cid:48) i ), as done in subsection 3.1, using the samesparsification matrix and (3.2), and then obtain sparse vector (say s (cid:48)(cid:48) i ).Next, as earlier, we consider these sparse vector in the zig-zag scanning order,and then categories each vector into two groups s (cid:48)(cid:48) i,u ∈ R p and s (cid:48)(cid:48) i,v ∈ R p , whereas earlier, p and p are the number of coefficients having large values and smallvalues (or zero values), respectively. After that, we project each sparse vector ontolinear measurements (say y (cid:48)(cid:48) i ∈ R ( p + m ) × ), as done in subsection 3.1, using thesame measurement matrix Φ ∈ R m × p and (3.3). This y (cid:48)(cid:48) have DCT coefficients ofthe secret image that is extracted by the extraction rule given in Algorithm 2 .This extraction rule is reverse of the embedding rule, given in
Algorithm 1 .In
Algorithm 2 , the size of t (cid:48) i is same as the size of t i , i.e., l ×
1. After extractingDCT coefficients t (cid:48) from the stego-image, we convert each vector t (cid:48) i into the blocksof size l × l , and then perform block-wise inverse discrete cosine transformation(IDCT) to obtain secret image pixels. Finally, we obtain extracted secret image ofsize M × M by arranging all these blocks.As mentioned earlier, this steganography scheme is a blind multi-image steganog-raphy scheme because it does not require any cover image data at the receiver sidefor the extraction of secret images.
4. Experimental Results
Experiments are carried out in
M AT LAB on a machine with Intel Core i3 [email protected] GHz and 4GB RAM. We use a set of standard gray-scale images to test our (cid:96) SABMIS. Some sample test images used in our experiments are shown in Fig.1. These images are taken from the USC-SIPI image database, and have varyingtexture property. In this paper, we take all these ten images as the cover images,and four images (Fig. 1a, Fig. 1b, Fig. 1e and Fig. 1j) as the secret images for ourexperiments. However, we can use any of the ten images apart from these four asthe secret image.Though the images shown in Fig. 1 seems to be of same size, however for ourexperiments, the size of each cover image is kept as 1024 × N × N ),and the size of each secret image is kept as 512 ×
512 (i.e., M × M ). We takeblocks of size 8 × b × b and l × l ). Recall from subsection 3.1, the size of measurement matrix Φ is m × p ,uly 13, 2020 0:34 Manuscript (cid:96) SABMIS Algorithm 2
Extraction Rule
Input: • y (cid:48)(cid:48) : Sequence of linear measurements of the stego-image. • The value of p , p , α , β , γ and c (discuss in Sections 3.1 and 4). Output: • t (cid:48) : Sequence of transform coefficients of the extracted secret image. Initialize t (cid:48) to zeros for i = 1 to N × b do t (cid:48) i (1) = y (cid:48)(cid:48) i ( p ) − y (cid:48)(cid:48) i ( p − c ) α . for j = p − c + 1 to p − do t (cid:48) ( j − p + c + 1) = y (cid:48)(cid:48) i ( j ) − y (cid:48)(cid:48) i ( j − c ) β . end for for k = p + p + 1 to p + 2 × p − c do t (cid:48) ( k − p − p + c ) = y (cid:48)(cid:48) i ( k ) − y (cid:48)(cid:48) i ( k − p + c ) γ . end for end for return t (cid:48) (a) Lena (b) Peppers (c) Boat (d) Goldhill (e) Zelda(f) Tiffany (g) Liv. room (h) Tank (i) Airplane (j) Cameraman Fig. 1: Test images used in our experimentsuly 13, 2020 0:34 Manuscript Rohit Agrawal where p + p = b (here, b = 64). Usually, for most images, more than half of thecoefficient in a DCT sparsified vector has value either very small or zero. Hence, inour experiments, we take p = p = 32, and m = 10 × p (i.e, we over-sampled linearmeasurements). In general, the DCT coefficients can be divided into three sets, lowfrequencies, middle frequencies, and high frequencies. Low frequencies are associatedwith the illumination, middle frequencies are associated with the structure, and highfrequencies are associated with the noise or small variation details. We discardedthese high-frequency coefficients, which are usually half of the total coefficients.Hence, we take p = 32. Due to the property of DCT, the first coefficient havethe highest value, then a few coefficients have values lower than the first one, andremaining coefficients have values lowest. These three category of coefficients areembedded in the step numbers 2, 3 and 5 of Algorithm 1 , respectively. Hence,to obtain good quality extracted secret image, we take α = 0 . β = 0 . γ = 1,and c = { , } (the results are reported for only one value of c that gave betterperformance).A successful steganography scheme should have high embedding capacity, andshould not distort the cover media significantly (i.e., the distortion should be visuallyimperceptible). Thus, we evaluate the performance of our proposed (cid:96) SABMIS byanalyzing these two metrics. As this paper proposes a multi-image steganographyscheme, we evaluate the performance of (cid:96) SABMIS for the cases where all the fourimages and less than four images are embedded.The embedding capacity (or embedding rate) is the number (or length) of secretinformation bits that can be embedded in each pixel of the cover image. It is mea-sured in bits per pixel (bpp). Thus, we have embedding capacities of 2 bpp, 4 bpp, 6bpp, and 8 bpp for embedding one, two, three, and four secret images, respectively.Next, we evaluate the quality of the stego-image. Imperceptibility is the measureof the invisibility of the secret image that is hidden in the generated stego-image.There is no universal criterion to determine imperceptibility. However, we evaluateit by visual and numerical (PSNR, MSSIM, NCC, NAE, and Entropy) metrics.We construct stego-images corresponding to different test images used in our ex-periments and then check the distortion visually. We also check their correspondingedge map diagrams. Here, we present the visual comparison only for ‘Zelda’ coverimage. This comparison is given in Fig. 2. In this figure, 2a shows ‘Zelda’ coverimage, 2b shows stego-image, 2c shows the edge map diagram of cover image, and2d shows the edge map diagram of stego-image. From these sub-figures, we observethat the stego-image is almost similar to its corresponding cover image. Also, theircorresponding edge maps are almost the same.The numerical metrics evaluate imperceptibility by comparing the cover imagesand their corresponding stego-images based on some numerical criteria. These in-clude; PSNR value, mean SSIM index, NNC coefficient, NAE, and entropy. PSNR value is evaluated in decibel (dB), and its higher value indicates the higher imper-ceptibility of the stego-image. In general, value above 30 dB is considered to be gooduly 13, 2020 0:34 Manuscript (cid:96) SABMIS (a) cover image (b) stego-image (c) CI edge map (d) SI edge map Fig. 2: Visual quality analysis between ‘Zelda’ cover image (CI) and its correspond-ing stego-image (SI)for the quality of the stego-image.
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For more details, please see. The PSNRvalues of the stego-images corresponding to different test images are given in Fig. 3and Fig. 4. In Fig. 3, we show the PSNR values for the four cases, i.e., embeddingone, two, three, and four images. As we have four secret images, we have the choiceof embedding any one of them and present its corresponding PSNR value. How-ever, we embed all four images individually, obtain their PSNR values, and thenpresent the average of these four PSNR values. Similarly, the average PSNR valuesare presented for the cases when we embed two and three images. For the case ofembedding four images, there is only one choice, and hence its corresponding PSNRvalue is presented.In Fig. 4, we show the PSNR values of all the stego-images when all the foursecret images are embedded separately. In this figure, we observe the highest PSNRvalue (i.e., 47.32 dB) when ‘Zelda’ secret image is hidden in ‘Zelda’ cover image,while the lowest PSNR value (i.e., 40.21 dB) when ‘Peppers’ secret image is hiddenin ‘Boat’ cover image. Also, we observe that for all the cases, we obtain PSNRvalues higher than 30 dB, and hence, considered good.
Lena Pepper Boat Goldhill Zelda Tiffany Livingroom Tank Airplane Cameraman
Images PS NR V a l u e Fig. 3: PSNR value of the stego-images when different number of images are hiddenuly 13, 2020 0:34 Manuscript Rohit Agrawal
Lena Pepper Boat Goldhill Zelda Tiffany Livingroom Tank Airplane Cameraman
Images PS NR V a l u e Lena secret image is hiddenPepper secret image is hiddenCameraman secret image is hiddenZelda secret image is hidden
Fig. 4: PSNR value of the stego-images when only 1 secret image is hiddenWe measure the structural similarity between the cover images and their cor-responding stego-images by the metric of mean SSIM. The value of mean SSIMindex lies between 0 and 1, where the value 0 implies that there is no similaritybetween the cover image and stego-image, and the value 1 implies that the coverimage is exactly similar to its corresponding stego-image. We measure the degreeof similarity between the cover images and their corresponding stego-images byanother metric called normalized cross-correlation (NCC) coefficients. Similar tomean SSIM, the value of NCC equal to 1 implies that the cover image is exactly sim-ilar to its corresponding stego-image. We also measure the quality of stego-image byevaluating NAE between the cover images and their corresponding stego-images.A value close to 0 indicates that the stego-image is almost similar to their corre-sponding cover image. Moreover, we also measure the entropy of the cover imagesand their corresponding stego-images. Entropy is a statistical randomness measure,which can be used to characterize the texture of an image. In Table 1, we give thevalues of all these metrics for our (cid:96) SABMIS when hiding all the four secret images.We do not present the values for the cases of embedding less than four secret imagesas their results will be better than those given in Table 1.From this table, we observe that all the values of mean SSIM index are equal to1. Hence, the cover images and their corresponding stego-images are similar in struc-ture. The values of NCC coefficients are also close to 1. The values of NAE are closeto 0. The values of entropy of stego-images are almost similar to their correspond-ing cover images. Thus, we can say that the stego-images and their correspondingcover images are almost identical. In addition to high embedding capacity with goodquality stego-image, (cid:96) SABMIS is also resistant against steganographic attacks. Thereason for this is that (cid:96) SABMIS is a transformed domain-based technique. And, inthis scheme, we embed secret images by an indirect embedding strategy (i.e., LSBflipping based embedding is not adopted).uly 13, 2020 0:34 Manuscript (cid:96) SABMIS Table 1: Mean SSIM (MSSIM) index, NCC coefficient, NAE, and entropy of thestego-image.
CoverImage MSSIM NCC NAE EntropyOriginalimage Stego-image
Lena 1 0.9992 0.009 7.443 7.463Pepper 1 0.9997 0.012 7.573 7.598Boat 1 0.9998 0.012 7.121 7.146Goldhill 1 0.9998 0.013 7.471 7.486Zelda 1 0.9964 0.011 7.263 7.272Tiffany 1 0.9999 0.006 6.602 6.628Livingroom 1 0.9996 0.013 7.431 7.438Tank 1 0.9998 0.014 6.372 6.405Airplane 1 0.9972 0.015 6.714 6.786Cameraman 1 1.0000 0.009 7.055 7.123
Average 1 0.9991 0.011 7.104 7.1343
Performance Comparison
In this subsection, we compare the performance of our (cid:96) SABMIS with the existingsteganography schemes. These results are given in Table 2.Table 2: Performance comparison of our (cid:96) SABMIS with various other steganogra-phy schemes.
SteganographyScheme EmbeddingCapacity(in bpp) PSNR(in dB) Resistant toSteganographicAttacks? (cid:96) SABMIS Hide 1 Image 2 44.98 YesHide 2 Images 4 42.54Hide 3 Images 6 39.83Hide 4 Images 8 38.98
In this table, the first column represents various steganography schemes, andthe remaining columns represent the metrics used for the comparison. As earlier, asuccessful steganography scheme should have high embedding capacity with a con-siderably good quality stego-image. Also, it should be resistant to steganographic at-tacks. Hence, we use embedding capacity, PSNR value, and checking which schemesare resistant to steganographic attacks or not as the performance metrics for thecomparison. In this table, the embedding capacity is represented in bit per pixel,uly 13, 2020 0:34 Manuscript Rohit Agrawal where each pixel is considered as a gray-scale. For these existing schemes, the datais not available for all the test images used in this experiment. Hence, in this table,we report the average PSNR values of all the images given in the respective papers.From this table, we observe that except and, our (cid:96) SABMIS outperforms allother existing steganography schemes. The scheme proposed in is based on LSBbased embedding, which is not resistant to steganographic attacks. The schemeproposed in embeds single secret image in a color image, which is different fromour goal. Also, it has limited embedding capacity. Hence, we can say that out of allthese schemes, (cid:96) SABMIS embeds multiple secret images with high embedding ca-pacity and good quality stego-image. Also, (cid:96) SABMIS is resistant to steganographicattacks.As discussed earlier, we also calculate the mean SSIM index, NCC coefficients,NAE, and entropy values for the performance evaluation. However, these results arenot compared with other techniques because the data for the same are not availablefor these other techniques.
Quality Assesment of Secret Recovered Image
As earlier, PSNR value can be used to assess the quality of the secret recoveredimage, and the value greater than 30 dB is considered good. However, this is nottrue for every case. For example, the steganography scheme proposed in has PSNRvalues greater than 30 dB for the extracted secret images. However, these imageshave black colored and white colored alternate horizontal and vertical lines. Hence,we only report other metrics to evaluate the quality of the extracted/ recoveredsecret image.As human is the final spectator of the extracted secret image, human observersare considered the final arbiter to assess the quality of these images. In Fig. 5aand Fig. 5b, we show the ‘Pepper’ secret image and the extracted secret imagefrom ‘Zelda’ stego-image. From these figures, we observe that there is very littledistortion in the extracted image. Besides this, we also show their correspondingedge maps diagram in Fig. 5c and 5d, respectively. Again, we observe very littlevariation in their corresponding edge maps diagrams.Moreover, we also evaluate the mean SSIM index, NCC coefficient, NAE, andentropy to measure the quality of the extracted secret images. The values of thesemetrics are given in Table 3. From this table, we observe that for all the images,the values of mean SSIM index is 1, NCC coefficient is close to 1, NAE is close to0, and the entropy of the original secret images and their corresponding extractedsecret images are almost the same. Thus, we see that the extracted secret imagepreserve good quality.
5. Conclusions and Future Work
We present a blind multi-image steganography scheme based on (cid:96) -minimizationand sparse approximation. Here, we can embed a maximum of four secret im-uly 13, 2020 0:34 Manuscript (cid:96) SABMIS (a) ‘Peppers’ se-cret image (b) Extractedimage (c) Secret imageedge map (d) Extractedimage edge map Fig. 5: Visual quality analysis between ’Peppers’ secret image and ‘Peppers’ ex-tracted image from ‘Zelda’ stego-image.Table 3: Mean SSIM (MSSIM) index, NCC coefficient, NAE, and entropy of theextracted/ recovered secret image.
SecretImage MSSIM NCC NAE EntropyOriginalImage RecoveredImage
Lena 1 0.9973 0.026 7.446 7.627Pepper 1 0.9946 0.329 7.571 7.624Cameraman 1 0.9953 0.027 7.048 7.258Zelda 1 0.9972 0.028 7.267 7.284
Average 1 0.9961 0.103 7.333 7.449 ages in a single cover image. Initially, we perform sampling in the cover imageand obtain four sub-images. Next, we sparsify each sub-image and then obtainits linear measurements using a random measurement matrix. Finally, using ourembedding rule, we embed DCT coefficients of the secret images into the linearmeasurements. The stego-image is obtained from the modified measurements bysolving a (cid:96) -minimization problem.We perform experiments on several standard gray-scale images that vary in tex-ture. For performance evaluation, we calculate embedding capacity, PSNR value,mean SSIM index, NCC coefficient, NAE, and entropy. Experiments show that our (cid:96) SABMIS hides four secret images in a cover image without distorting it signifi-cantly and has PSNR values greater than 35 dB, which is usually considered good.Also, the mean SSIM index is equal to 1, the NCC coefficient is close to 1, and thevalue of NAE is close to 0, which shows that the stego-images are almost identicalto its corresponding cover images. We obtain approximately the same entropy valuefor both the cover images and their corresponding stego-images. We also obtain al-most same values for all the metrics (i.e., mean SSIM, NCC coefficients, NAE, andentropy) as above for the extracted secret images. This indicates that the extractedsecret images preserve good visual quality. Finally, (cid:96) SABMIS is also resistant touly 13, 2020 0:34 Manuscript Rohit Agrawal steganographic attacks.In the future, we plan to embed the secret images in other media such as audio,video, etc. We also plan to apply optimization techniques to calculate the values ofparameters α, β, γ , etc. used in our embedding and extraction algorithms.
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