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A Novel Digital Image Covert Communication Scheme Based on Generalized FCM in DCT Domain

A Novel Digital Image Covert Communication Scheme Based on Generalized FCM in DCT Domain Fuzzy Inf. Eng. (2011) 2: 127-136 DOI 10.1007/s12543-011-0071-z ORIGINAL ARTICLE A Novel Digital Image Covert Communication Scheme Based on Generalized FCM in DCT Domain Li-yun Su · Feng-lan Li · Jiao-jun Li· Bo Chen Received: 11 January 2010/ Revised: 10 April 2011/ Accepted: 12 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract A novel covert communication method of digital image is presented, based on generalized fuzzy c-means clustering (GFCM), human visual system (HVS) and discrete cosine transform (DCT). Therefore, the original image blocks are clas- sified into two classes according to specified characteristic parameters. So one block is suited for embedding security information, but the other block is not. Hence the appropriate blocks can be selected in an image to embed the security information by selectively modifying the middle-frequency part of the original image in conjunction with HVS and DCT. Furthermore the maximal information strength is fixed based to the frequency masking. Also to improve performances of the proposed algorithm, the security information is modulated into the chaotic modulation array. The simulation results show that we can remarkably extract the hiding security information and can achieve good robustness with common signal distortion or geometric distortion and the quality of the embedded image is guaranteed. Keywords Covert communication· Digital image· Generalized fuzzy c-means clus- tering · Human visual system· Discrete cosine transform 1. Introduction The enormous popularity of the World Wide Web in the early 1990’s demonstrated the commercial potential of offering multimedia resources through the digital net-works. Li-yun Su () School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, P.R.China email: cloudhopping@163.com Feng-lan Li Library, Chongqing University of Technology, Chongqing 400054, P.R.China Jiao-jun Li School of Electronic Information and Automation, Chongqing University of Technology, Chongqing 400054, P.R.China Bo Chen School of Mathematics and Statistics, Southwest University, Chongqing 400715, P.R.China 128 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) Since commercial interests seek to use the digital networks to offer digital media for profit, they have a strong interest in transmitting their security information without any notice. Covert communication has been proposed as one way to accomplish this. The advent of the new IPv6 internet will bring a great chance for the development of multimedia network communication. So the information hiding techniques [1-4] of multimedia covert communication have become important gradually in communica- tion fields. Of course all of information hiding techniques should exhibit a number of desirable characteristics. At least, it should respect the following two issues for security information in digital image communications [5-7]: • Imperceptibility: The data embedding process should neither introduce any perceptible artifacts in the host image nor degrade the perceived quality of the host image; • Robustness: The digital security information is still present in the image after distorted attack and can be detected by security information detector, especially to the attack of compression or image processing. In this paper, we present a novel algorithm of using chaotic modulation array (CMA) according to chaotic time series [8,9], discrete cosine transform (DCT) and generalized fuzzy C-means clustering (GFCM). Because the chaotic sequence digi- tal signals are contained in coded means, we can not easily capture the signals from coded ones. Furthermore the GFCM algorithm is used to classify the original image blocks into two classes based on several characteristic parameters, with the features of human visual system (HVS). Therefore one is suited for embedding digital infor- mation, but the other is not. So we can select the appropriate blocks in an image to embed the digital information. The information is embedded in the original image by selectively modifying the middle-frequency part of the original image in conjunction with HVS and DCT. The maximal digital information strength is fixed according to the frequency masking. Hence the algorithm has good robustness. With common signal distortion or geometric distortion (such as addition of Gaussian noise, JPEG compression, cropping etc.), however the digital image covert communication sys- tems can achieve good robustness. 2. Chaotic System and Sequences Modulation There are many definitions about chaos, but they are accordant in nature. Here we give a kind of more intuitionistic definition. Supposed that V is a compact metric space, if continuous mapping f : V → V satisfies the following conditions: sensitiv- ity to initial value; topological transferring; the periodic point set of f is dense. One kind of dynamical system that is widely researched is Logistic mapping and the definition is given by x = μx (1− x ), (1) n+1 n n where x ∈ (0, 1) and when 3.5699456 <μ ≤ 4, we name the parameter region a chaotic region. In the region the system in chaotic state which two sequences by (1) with different initial value are non-periodic, non-convergent and non-correlative. Fuzzy Inf. Eng. (2011) 2: 127-136 129 The other kind of system is Chebyshev mapping that uses rank as parameter. The definition is given by x = cos(n(arccos(x )). (2) k+1 k By simple transforms, Logistic mapping also can be defined in (−1, 1). The formula is given by x = 1−λx,λ ∈ (0, 2). (3) k+1 In the surjection’s condition: λ=2, the two mappings are topological and conjuga- tive. They are regarded as the same dynamical system. At λ=2, it is the famous Ulam Von Neumann mapping. The alternate formula is Formula (3), where λ=2. Here, the probability distribution function (PDF) of sequences is given by 1/(π 1− x ), −1 < x < 1, ρ(x) = (4) 0, otherwise. So {x } is considered from the matrix of the PDF ρ(x). By the PDF, the chaotic sequences have δ− like model’s self-correlation function, zero’s mutual correlation function and zero-mean statistical characteristic [4]. Property I If two initial value x and y are selected independently, the correlation 0 0 function of two sequences is given by n−1 corr(x , y ) = lim (x − x)(y − y)/n = δ(x − y ). 0 0 i i 0 0 n−→∞ i=0 Property II The mean of chaotic sequences generated with random initial value is given by n−1 x = lim ( x )/n = xρ(x)dx = 0. n→∞ −1 By Logistic sequences’ properties, the sequences’ statistical characteristics are the same with the white noise, it can be used in information encryption of digital commu- nication, multimedia and so on. The system has the following merits. Firstly, it only requires parameters of chaotic mapping and initial condition to create the sequences expediently and need not waste space to storage all sequences. Secondly, with initial condition we can obtain large numbers of chaotic sequences. Commonly, it is very difficult to deduce initial condition of the chaotic sequences from finite length, which is important for information security. 3. GFCM of Several Parameters of Image Blocks According to HVS, the background brightness and texture [10] of the image affect the visibility of the embedded information: the brighter of the background, the lower 130 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) visibility of the embedded information, which is so called brightness masking effect. The original image is divided into 8× 8 blocks, extracting four characteristic parame- ters of each block. Then according to the clustering theory the host image blocks are classified into two types: one is suitable to embedding information, the other is not. GFCM clustering algorithm [11,12] is a very popular fuzzy clustering theory cur- rently. Its basic idea is to establish a subject function for clustering, so that the dis- tances inside the type is the shortest, meanwhile the distance outside the type is the longest. The purpose of any fuzzy clustering algorithm is to classify the data into a number of known clusters. The clustering algorithms produce a degree of relationship between each data point to each cluster. There are many fuzzy c-means clustering, but the existing methods are seen to be very different in form. However, all of them can be unified into one model with the following GFCM objective function: n c c J (u, v) = [u ρ (d(x , v ))− ρ (d(v, v ))], (5) m i k i 0 i j ik k=1 i=1 j=1 where u = f for f ≥ 0 and ρ (x) is a continuous function of x ∈ [0,+∞) ik k k i i=1 satisfying its derivativeρ (x) > 0 for all x ∈ [0,+∞). Using Lagrange multipliers, the necessary conditions for minimum of J (u, v) can be obtained as follows: n c n 2γ m m v = u ρ (d(x , v ))x − ρ (d(v, v ))v / u ρ (d(x , v )) i k i k i j j k i ik i 0 ik i k=1 j=1 k=1 2γ − ρ (d(v, v )) (6) i j i=1 and −1/(m−1) −1/(m−1) u = f ρ (d(x , v )) / ρ (d(x , v )) . (7) ik k i k i j k j j=1 The iteration with update Equations (6) and (7) is called the GFCM algorithm. It can be argued that a more general model can be obtained if the Euclidean distance is replaced with the Mahalanobis distance or other general metric. However, it is too complicated to obtain further theoretical analysis results about GFCM when using non-Euclidean distance. To simplify the analysis, the Euclidean distance x − v is k i used for the dissimilarity measure d(x , v ) to all subsequent considerations. Thus, k i the GFCM algorithm can be described as follows: (0) Pick the initial prototype v , the number of clusters c and the termination limitε. Set l = 0. (l) Step 1: Find u by update Equation (7). ik (l+1) (l) Step 2: Find v by update Equation (6) with (u, v) . (l+1) (l) If max v − v <ε , then stop; Else l = l+ 1 and go to Step 1. i i The GFCM clustering algorithm is measured by the following four characteristics: 1) Brightness sensitivity: the gray mean of the image block, denoting the bright- ness of the image block: B = g ; ij N kt ij 64 Fuzzy Inf. Eng. (2011) 2: 127-136 131 2) Texture sensitivity: the variance of image gray, deciding the texture of image block: T = |g − B |; ij kt ij ij 3) Contrast sensitivity: the maximal distance among the image gray, representing the contrast of the image block: C = max (g − min (g ); ij N kt) N kt ij ij 4) Entropy sensitivity: the entropy of information theory, measuring the uncer- tainty of image block: E = − p log(p ), where g is the gray scale at position ij N kt kt kt ij (k, t), p = g / g . kt kt kt ij The bigger the four characteristic value is, the brighter the image background brightness is, and the more complex the image texture is, the more suited the cor- responding image blocks are for the digital information embedding. 4. Selection of DCT Coefficients and Maximal Strength Human eyes don’t process information point by point, but code the image’s charac- teristics such as space, frequency and color by brain nerve. The characteristics of human visual apperceptions are different from information distribution in statistics meaning. Many characteristics which need a great deal of information in statistics meaning may be unimportant to human visual apperception. Human vision needn’t describe these characteristics in details. The frequency response functions of HVS: 0.554 23 H( f ) = 0.05exp( f ) for f < 7, exp(−9|log f − log9| ) for f ≥ 7, where f is the radial frequency against human vision. The unit is cycle per degree. f (cycle/degree) = f (cycle/pixel)× f (pixel/degree), d s 2 2 1/2 here f = (u + v ) ,(u, v) represents the frequency position in DCT domain, N represents the size of DCT blocks. f is the sampling function depending only on the distance of observation. Commonly we set f =48. From the above function, we should select those coefficients which have larger f value, in order to make good use of the characteristics of HVS frequency response. Here we choose the value of f between 12 and 22. Since the value of f varies the coefficients, so we can choose the value of f according to the image characteristics and reality requirements. The strongest strength is determined according to frequency occultation effects. To signal frequency occultation, Watson divides the image into 8 × 8 blocks, and defines the common frequency sensitivity for arbitrary DCT block. Every element in the form of sensitivity is the just noticeable difference (JND) p(u, v). Smaller value implies more susceptive to human vision. In addition, every characteristic block has a membership degree to being suitable for embedding information, the degree can be used for the strength of the whole block. For the arbitrary position (u, v)of each characteristic block N , the embedding strength includes two parts: one is the ij membership degree of characteristic block, the other is the JND p(u, v)at(u, v). The embedding method: F (u, v) = F (u, v)× (1+α· u · p(u, v)· W (u, v)), (8) N N ij cma where F (u, v) is the DCT coefficients at (u, v) in block N , α is the resilient param- N ij eter. W (u, v) is the embedded information array. cma 132 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) 5. Digital Information Embedding and Extracting The embedding algorithm is described in terms of the following steps: 1) In order to avoid the influences of noise, JPEG compression as well as cropping for the extracted information, chaotic modulation signal can be produced by using the chaotic sequence to modulate W (u, v) into the chaotic modulation signal. Then the chaotic modulation signal is arranged for chaotic modulation array W (u, v). cma 2) The original image blocks are classified into two categories based on brightness, texture, contrast and entropy sensitivity of image blocks with the features of HVS. One is suited for embedding digital information, but the other is not. According to the block classification, it embedded the information into the selected blocks. The 8× 8 DCT for M× N image is computed and the DCT coefficients are reordered into a Zig-Zag scan, such as in the JPEG compression algorithm. 3) The appropriate coefficients of the mid-frequency of suited blocks are selected according to the HVS frequency characteristic function in Section 4. 4) The chaotic modulation array is embedded into the appropriate mid-frequency coefficients according to the Formula (8). 5) The inverse DCT is applied to the embedded coefficient matrix to obtain the embedded image. Based on the embedding procedure, so the extracted algorithm is described in terms of the following steps: 1) Original image X and embedded image X are changed into image Y and Y utilizing the block DCT transform respectively. 2) The appropriate mid-frequency coefficients of suited blocks of image Y and Y are selected according to the HVS frequency characteristic function in Section 4. The chaotic modulation array W (u, v) is acquired according to the previous algorithm. cma 3) The embedded information is obtained by demodulating W (u, v). cma Similarity (SIM) between original information W and extracted information W is measured by means of formula as follows: ˆ ˆ SIM(W, W ) = W (i, j)· W (i, j)/ W (i, j). (9) If information is a sequence, detection bit rate (DBR) is measured by means of formula as follows: DBR=right number of detection / total number of information sequence. The distortion of embedded image X is measured by means of Peak Signal Noise Ratio (PSNR), which is defined as max(M, N)× max(X (i, j)) PS NR = 10log  . (10) (X(i, j)− X (i, j)) 6. Simulation Results and Analysis The binary image and the random sequence are generated, in order to test the new digital image covert communication algorithm. The standard image (‘Baboon’) is Fuzzy Inf. Eng. (2011) 2: 127-136 133 Table 1: Clustering center of the imageþBaboon’. Clustering center brightness texture contrast entropy V1 0.4558 0.6370 0.4615 0.8921 V2 0.4791 0.3455 0.2298 0.8973 then labeled, and several common signal processing techniques and geometric distor- tions are applied to the image to evaluate if the detector can reveal the presence of the image owner’s embedded information. To classify different sub-image blocks corresponding to disparate types of de-posit, each sub-image block is represented by its fuzzy features X (B, T, C, E) and the clas- sification methods will take place on this vector. The fuzzy clustering is a process of assigning a grade of membership to each object X (B, T, C, E) for any cluster. One of the most frequently used clustering algorithms which have been applied is the so-called GFCM. This algorithm assigns objects, described by several features, to different classes with different degrees of membership. An advantage of this method is that it provides an automatic method of forming the membership functions and does not require any initial knowledge about the structure in the feature vectors. The classification results of GFCM algorithm with the image ‘Baboon’ are shown in Ta- ble 1 (normalized data). The sub-image blocks with larger feathers values are suited for embedding digital information, the other is not, according to the HVS masking characteristics. We choose the lager values of all kinds of feathers as the appropriate sub-image blocks, marked off according to the threshold T . The membership de- th grees are used as the embedded strength to realize the adaptive digital image covert communication systems. Experimental results obtained on the standard image ‘Baboon’ with 256 × 256 pixels. The original and binary image with 28 × 56 pixels are shown in Figure 1 (left) and in Figure 2 (left), respectively. The embedded image is shown in Figure 1 (right). From the figure, obviously, human eyes can not apperceive the existence of embedded information. The experimental results are also listed in the following figures and tables. 1) Gaussian noise: The Baboon image was corrupted by the addition of Gaussian noise, thus obtaining the embedded image. A zero-mean Gaussian noise with vari- ance σ =0.5 was used. Though the image degradation is so heavy that it can not be accepted in practical applications, the information is still easily recovered as shown in Figure 2 (right). Furthermore the other similar results are shown in Table 2. 2) JPEG compression: The JPEG compression algorithm is one of the most im- portant and strong attacks. However the embedding schemes should be resistant to JPEG coding with smoothing and decreasing quality applied to the signed image. Ob- viously, when the JPEG compressed image quality decreases, so does the extracted information; however, the embedded information is well extracted even though the 134 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) Fig.1 Original and embedded images: 1 (left), test image; 2 (right), embedded image, PSNR=45.33, SIM=1, DBR=1 Fig.2 Original information image and retrieved secret image: 1 (left), original image; 2 (right), extracted information image, SIM=0.9219 quality factor is 40, and the image is visibly distorted (Table 3). 3) Geometric distortion: Virtually, all practical applications call for the informa- tion to be immune to geometric manipulations such as cropping. Experiments have been carried out to prove that the information contained in a sub-image is still suf- ficient to extract the information. In particular, suppose that the sub-image can be replaced at the same position which occupied in the original picture, the proposed system can extract the information. Even though the cropped part is 75% of the orig- inal image, DBR is still 0.7360. Table 2: Gaussian noise attacks. Variance 0.01 0.04 0.08 0.1 0.2 0.3 0.4 0.5 PSNR 21.21 18.31 16.05 14.77 13.91 13.39 10.79 9.09 DBR 0.976 0.954 0.934 0.949 0.947 0.947 0.943 0.908 Fuzzy Inf. Eng. (2011) 2: 127-136 135 Table 3: JPEG compression attacks. JPEG factor 40 50 60 70 80 90 DBR 0.8120 0.9944 0.9955 0.9962 0.9990 1 7. Conclusion In this paper, a novel covert communication algorithm is proposed for digital images. The presented adaptive method has good performance due to security information operating in the frequency domain. In order to avoid easy extraction of hiding infor- mation, a chaotic modulation array of a binary image or a pseudo-random sequence of real numbers having normal distribution with zero mean and unity variance is embed- ded in the appropriate blocks of the selected DCT coefficients. Modulating the binary image by chaotic sequence can get modulation array. The embedded coefficients are produced by arranging the DCT coefficients in a zig-zag scan and by extracting the co- efficients according to GFCM and HVS. After embedding hiding information, which is adapted to image being signed by exploiting the membership degrees of blocks, to further ensure the information invisibility and the robustness. Experimental results demonstrate that the meaningful information is robust to several signal processing techniques and geometric distortions, including JPEG compression, Gaussian noise as well as cropping, and the quality of the embedded image is guaranteed. Acknowledgments This work was partly supported by Chongqing Science & Technology Commission (CSTC) foundation of China (No. CSTC2010BB2310), Chongqing Municipal Edu- cation Commission(CMEC) foundations of China (No. KJ080614, No. KJ100810, No. KJ100818), Chongqing University of Technology(CQUT) foundation of China (No. 2007ZD16). Authors would like to thank referees for their helpful comments. References 1. Cox I J, Kilian J, Leighton T, Shamoon T (1997) Secure spread spectrum watermarking for multime- dia. IEEE Transactions On Image Processing 6(12): 1673-1687 2. Yang E H, Sun W (2008) On information embedding when watermarks and covertexts are correlated. IEEE transactions on Information Theory 54(7): 3340-3345 3. Hung K M (2008) A novel robust watermarking technique using int DCT based AC prediction. WSEAS Transactions on Computers 7(1): 16-24 4. Tang S F, Ma H, Song E B, Su L Y (2006) An adaptive blind scheme for readable watermark based on ALE. International Conference on Signal Processing Proceedings 1: 2982-2986 5. Damien D, Benoit M (2006) Watermarking relying on cover signal content to hide synchronization marks. IEEE Transactions on Information Forensics and Security 1(1): 87-101 6. Su L Y, Ma H, Tang S F (2006) Adaptive image digital watermarking with DCT and FCM. Wuhan University Journal of Natural Sciences 11(6): 1657-1660 7. Peng J, Su L Y, Li Z, Tang S F (2007) A novel algorithm of covert image communication with wavelet transform and information fusion. International Conference on Image and Graphics 1: 307-312 136 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) 8. Su L Y (2010) Prediction of multivariate chaotic time series with local polynomial fitting. Computers & Mathematics with Applications 59(2): 737-744 9. Su L Y, Liu R H, Li F L, and Li J J (2010) Prediction comparative study of complex multivariate systems with AGA-BP. International Conference on Computer Application and System Modeling 10: 50-54 10. Jayant N H , Nstok J (1993) Signal compression based on model of human perception. Proceeding of the IEEE 81(10): 1385-1422 11. Yu J, Yang M S (2005) Optimality test for generalized FCM and its application to parameter selection. IEEE Transactions on Fuzzy Systems 13(1): 164-176 12. Yu J, Yang M S (2007) A generalized fuzzy clustering regularization model with optimality tests and model complexity analysis. IEEE Trans. on Fuzzy Systems 15(5): 904-915 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

A Novel Digital Image Covert Communication Scheme Based on Generalized FCM in DCT Domain

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Fuzzy Inf. Eng. (2011) 2: 127-136 DOI 10.1007/s12543-011-0071-z ORIGINAL ARTICLE A Novel Digital Image Covert Communication Scheme Based on Generalized FCM in DCT Domain Li-yun Su · Feng-lan Li · Jiao-jun Li· Bo Chen Received: 11 January 2010/ Revised: 10 April 2011/ Accepted: 12 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract A novel covert communication method of digital image is presented, based on generalized fuzzy c-means clustering (GFCM), human visual system (HVS) and discrete cosine transform (DCT). Therefore, the original image blocks are clas- sified into two classes according to specified characteristic parameters. So one block is suited for embedding security information, but the other block is not. Hence the appropriate blocks can be selected in an image to embed the security information by selectively modifying the middle-frequency part of the original image in conjunction with HVS and DCT. Furthermore the maximal information strength is fixed based to the frequency masking. Also to improve performances of the proposed algorithm, the security information is modulated into the chaotic modulation array. The simulation results show that we can remarkably extract the hiding security information and can achieve good robustness with common signal distortion or geometric distortion and the quality of the embedded image is guaranteed. Keywords Covert communication· Digital image· Generalized fuzzy c-means clus- tering · Human visual system· Discrete cosine transform 1. Introduction The enormous popularity of the World Wide Web in the early 1990’s demonstrated the commercial potential of offering multimedia resources through the digital net-works. Li-yun Su () School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, P.R.China email: cloudhopping@163.com Feng-lan Li Library, Chongqing University of Technology, Chongqing 400054, P.R.China Jiao-jun Li School of Electronic Information and Automation, Chongqing University of Technology, Chongqing 400054, P.R.China Bo Chen School of Mathematics and Statistics, Southwest University, Chongqing 400715, P.R.China 128 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) Since commercial interests seek to use the digital networks to offer digital media for profit, they have a strong interest in transmitting their security information without any notice. Covert communication has been proposed as one way to accomplish this. The advent of the new IPv6 internet will bring a great chance for the development of multimedia network communication. So the information hiding techniques [1-4] of multimedia covert communication have become important gradually in communica- tion fields. Of course all of information hiding techniques should exhibit a number of desirable characteristics. At least, it should respect the following two issues for security information in digital image communications [5-7]: • Imperceptibility: The data embedding process should neither introduce any perceptible artifacts in the host image nor degrade the perceived quality of the host image; • Robustness: The digital security information is still present in the image after distorted attack and can be detected by security information detector, especially to the attack of compression or image processing. In this paper, we present a novel algorithm of using chaotic modulation array (CMA) according to chaotic time series [8,9], discrete cosine transform (DCT) and generalized fuzzy C-means clustering (GFCM). Because the chaotic sequence digi- tal signals are contained in coded means, we can not easily capture the signals from coded ones. Furthermore the GFCM algorithm is used to classify the original image blocks into two classes based on several characteristic parameters, with the features of human visual system (HVS). Therefore one is suited for embedding digital infor- mation, but the other is not. So we can select the appropriate blocks in an image to embed the digital information. The information is embedded in the original image by selectively modifying the middle-frequency part of the original image in conjunction with HVS and DCT. The maximal digital information strength is fixed according to the frequency masking. Hence the algorithm has good robustness. With common signal distortion or geometric distortion (such as addition of Gaussian noise, JPEG compression, cropping etc.), however the digital image covert communication sys- tems can achieve good robustness. 2. Chaotic System and Sequences Modulation There are many definitions about chaos, but they are accordant in nature. Here we give a kind of more intuitionistic definition. Supposed that V is a compact metric space, if continuous mapping f : V → V satisfies the following conditions: sensitiv- ity to initial value; topological transferring; the periodic point set of f is dense. One kind of dynamical system that is widely researched is Logistic mapping and the definition is given by x = μx (1− x ), (1) n+1 n n where x ∈ (0, 1) and when 3.5699456 <μ ≤ 4, we name the parameter region a chaotic region. In the region the system in chaotic state which two sequences by (1) with different initial value are non-periodic, non-convergent and non-correlative. Fuzzy Inf. Eng. (2011) 2: 127-136 129 The other kind of system is Chebyshev mapping that uses rank as parameter. The definition is given by x = cos(n(arccos(x )). (2) k+1 k By simple transforms, Logistic mapping also can be defined in (−1, 1). The formula is given by x = 1−λx,λ ∈ (0, 2). (3) k+1 In the surjection’s condition: λ=2, the two mappings are topological and conjuga- tive. They are regarded as the same dynamical system. At λ=2, it is the famous Ulam Von Neumann mapping. The alternate formula is Formula (3), where λ=2. Here, the probability distribution function (PDF) of sequences is given by 1/(π 1− x ), −1 < x < 1, ρ(x) = (4) 0, otherwise. So {x } is considered from the matrix of the PDF ρ(x). By the PDF, the chaotic sequences have δ− like model’s self-correlation function, zero’s mutual correlation function and zero-mean statistical characteristic [4]. Property I If two initial value x and y are selected independently, the correlation 0 0 function of two sequences is given by n−1 corr(x , y ) = lim (x − x)(y − y)/n = δ(x − y ). 0 0 i i 0 0 n−→∞ i=0 Property II The mean of chaotic sequences generated with random initial value is given by n−1 x = lim ( x )/n = xρ(x)dx = 0. n→∞ −1 By Logistic sequences’ properties, the sequences’ statistical characteristics are the same with the white noise, it can be used in information encryption of digital commu- nication, multimedia and so on. The system has the following merits. Firstly, it only requires parameters of chaotic mapping and initial condition to create the sequences expediently and need not waste space to storage all sequences. Secondly, with initial condition we can obtain large numbers of chaotic sequences. Commonly, it is very difficult to deduce initial condition of the chaotic sequences from finite length, which is important for information security. 3. GFCM of Several Parameters of Image Blocks According to HVS, the background brightness and texture [10] of the image affect the visibility of the embedded information: the brighter of the background, the lower 130 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) visibility of the embedded information, which is so called brightness masking effect. The original image is divided into 8× 8 blocks, extracting four characteristic parame- ters of each block. Then according to the clustering theory the host image blocks are classified into two types: one is suitable to embedding information, the other is not. GFCM clustering algorithm [11,12] is a very popular fuzzy clustering theory cur- rently. Its basic idea is to establish a subject function for clustering, so that the dis- tances inside the type is the shortest, meanwhile the distance outside the type is the longest. The purpose of any fuzzy clustering algorithm is to classify the data into a number of known clusters. The clustering algorithms produce a degree of relationship between each data point to each cluster. There are many fuzzy c-means clustering, but the existing methods are seen to be very different in form. However, all of them can be unified into one model with the following GFCM objective function: n c c J (u, v) = [u ρ (d(x , v ))− ρ (d(v, v ))], (5) m i k i 0 i j ik k=1 i=1 j=1 where u = f for f ≥ 0 and ρ (x) is a continuous function of x ∈ [0,+∞) ik k k i i=1 satisfying its derivativeρ (x) > 0 for all x ∈ [0,+∞). Using Lagrange multipliers, the necessary conditions for minimum of J (u, v) can be obtained as follows: n c n 2γ m m v = u ρ (d(x , v ))x − ρ (d(v, v ))v / u ρ (d(x , v )) i k i k i j j k i ik i 0 ik i k=1 j=1 k=1 2γ − ρ (d(v, v )) (6) i j i=1 and −1/(m−1) −1/(m−1) u = f ρ (d(x , v )) / ρ (d(x , v )) . (7) ik k i k i j k j j=1 The iteration with update Equations (6) and (7) is called the GFCM algorithm. It can be argued that a more general model can be obtained if the Euclidean distance is replaced with the Mahalanobis distance or other general metric. However, it is too complicated to obtain further theoretical analysis results about GFCM when using non-Euclidean distance. To simplify the analysis, the Euclidean distance x − v is k i used for the dissimilarity measure d(x , v ) to all subsequent considerations. Thus, k i the GFCM algorithm can be described as follows: (0) Pick the initial prototype v , the number of clusters c and the termination limitε. Set l = 0. (l) Step 1: Find u by update Equation (7). ik (l+1) (l) Step 2: Find v by update Equation (6) with (u, v) . (l+1) (l) If max v − v <ε , then stop; Else l = l+ 1 and go to Step 1. i i The GFCM clustering algorithm is measured by the following four characteristics: 1) Brightness sensitivity: the gray mean of the image block, denoting the bright- ness of the image block: B = g ; ij N kt ij 64 Fuzzy Inf. Eng. (2011) 2: 127-136 131 2) Texture sensitivity: the variance of image gray, deciding the texture of image block: T = |g − B |; ij kt ij ij 3) Contrast sensitivity: the maximal distance among the image gray, representing the contrast of the image block: C = max (g − min (g ); ij N kt) N kt ij ij 4) Entropy sensitivity: the entropy of information theory, measuring the uncer- tainty of image block: E = − p log(p ), where g is the gray scale at position ij N kt kt kt ij (k, t), p = g / g . kt kt kt ij The bigger the four characteristic value is, the brighter the image background brightness is, and the more complex the image texture is, the more suited the cor- responding image blocks are for the digital information embedding. 4. Selection of DCT Coefficients and Maximal Strength Human eyes don’t process information point by point, but code the image’s charac- teristics such as space, frequency and color by brain nerve. The characteristics of human visual apperceptions are different from information distribution in statistics meaning. Many characteristics which need a great deal of information in statistics meaning may be unimportant to human visual apperception. Human vision needn’t describe these characteristics in details. The frequency response functions of HVS: 0.554 23 H( f ) = 0.05exp( f ) for f < 7, exp(−9|log f − log9| ) for f ≥ 7, where f is the radial frequency against human vision. The unit is cycle per degree. f (cycle/degree) = f (cycle/pixel)× f (pixel/degree), d s 2 2 1/2 here f = (u + v ) ,(u, v) represents the frequency position in DCT domain, N represents the size of DCT blocks. f is the sampling function depending only on the distance of observation. Commonly we set f =48. From the above function, we should select those coefficients which have larger f value, in order to make good use of the characteristics of HVS frequency response. Here we choose the value of f between 12 and 22. Since the value of f varies the coefficients, so we can choose the value of f according to the image characteristics and reality requirements. The strongest strength is determined according to frequency occultation effects. To signal frequency occultation, Watson divides the image into 8 × 8 blocks, and defines the common frequency sensitivity for arbitrary DCT block. Every element in the form of sensitivity is the just noticeable difference (JND) p(u, v). Smaller value implies more susceptive to human vision. In addition, every characteristic block has a membership degree to being suitable for embedding information, the degree can be used for the strength of the whole block. For the arbitrary position (u, v)of each characteristic block N , the embedding strength includes two parts: one is the ij membership degree of characteristic block, the other is the JND p(u, v)at(u, v). The embedding method: F (u, v) = F (u, v)× (1+α· u · p(u, v)· W (u, v)), (8) N N ij cma where F (u, v) is the DCT coefficients at (u, v) in block N , α is the resilient param- N ij eter. W (u, v) is the embedded information array. cma 132 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) 5. Digital Information Embedding and Extracting The embedding algorithm is described in terms of the following steps: 1) In order to avoid the influences of noise, JPEG compression as well as cropping for the extracted information, chaotic modulation signal can be produced by using the chaotic sequence to modulate W (u, v) into the chaotic modulation signal. Then the chaotic modulation signal is arranged for chaotic modulation array W (u, v). cma 2) The original image blocks are classified into two categories based on brightness, texture, contrast and entropy sensitivity of image blocks with the features of HVS. One is suited for embedding digital information, but the other is not. According to the block classification, it embedded the information into the selected blocks. The 8× 8 DCT for M× N image is computed and the DCT coefficients are reordered into a Zig-Zag scan, such as in the JPEG compression algorithm. 3) The appropriate coefficients of the mid-frequency of suited blocks are selected according to the HVS frequency characteristic function in Section 4. 4) The chaotic modulation array is embedded into the appropriate mid-frequency coefficients according to the Formula (8). 5) The inverse DCT is applied to the embedded coefficient matrix to obtain the embedded image. Based on the embedding procedure, so the extracted algorithm is described in terms of the following steps: 1) Original image X and embedded image X are changed into image Y and Y utilizing the block DCT transform respectively. 2) The appropriate mid-frequency coefficients of suited blocks of image Y and Y are selected according to the HVS frequency characteristic function in Section 4. The chaotic modulation array W (u, v) is acquired according to the previous algorithm. cma 3) The embedded information is obtained by demodulating W (u, v). cma Similarity (SIM) between original information W and extracted information W is measured by means of formula as follows: ˆ ˆ SIM(W, W ) = W (i, j)· W (i, j)/ W (i, j). (9) If information is a sequence, detection bit rate (DBR) is measured by means of formula as follows: DBR=right number of detection / total number of information sequence. The distortion of embedded image X is measured by means of Peak Signal Noise Ratio (PSNR), which is defined as max(M, N)× max(X (i, j)) PS NR = 10log  . (10) (X(i, j)− X (i, j)) 6. Simulation Results and Analysis The binary image and the random sequence are generated, in order to test the new digital image covert communication algorithm. The standard image (‘Baboon’) is Fuzzy Inf. Eng. (2011) 2: 127-136 133 Table 1: Clustering center of the imageþBaboon’. Clustering center brightness texture contrast entropy V1 0.4558 0.6370 0.4615 0.8921 V2 0.4791 0.3455 0.2298 0.8973 then labeled, and several common signal processing techniques and geometric distor- tions are applied to the image to evaluate if the detector can reveal the presence of the image owner’s embedded information. To classify different sub-image blocks corresponding to disparate types of de-posit, each sub-image block is represented by its fuzzy features X (B, T, C, E) and the clas- sification methods will take place on this vector. The fuzzy clustering is a process of assigning a grade of membership to each object X (B, T, C, E) for any cluster. One of the most frequently used clustering algorithms which have been applied is the so-called GFCM. This algorithm assigns objects, described by several features, to different classes with different degrees of membership. An advantage of this method is that it provides an automatic method of forming the membership functions and does not require any initial knowledge about the structure in the feature vectors. The classification results of GFCM algorithm with the image ‘Baboon’ are shown in Ta- ble 1 (normalized data). The sub-image blocks with larger feathers values are suited for embedding digital information, the other is not, according to the HVS masking characteristics. We choose the lager values of all kinds of feathers as the appropriate sub-image blocks, marked off according to the threshold T . The membership de- th grees are used as the embedded strength to realize the adaptive digital image covert communication systems. Experimental results obtained on the standard image ‘Baboon’ with 256 × 256 pixels. The original and binary image with 28 × 56 pixels are shown in Figure 1 (left) and in Figure 2 (left), respectively. The embedded image is shown in Figure 1 (right). From the figure, obviously, human eyes can not apperceive the existence of embedded information. The experimental results are also listed in the following figures and tables. 1) Gaussian noise: The Baboon image was corrupted by the addition of Gaussian noise, thus obtaining the embedded image. A zero-mean Gaussian noise with vari- ance σ =0.5 was used. Though the image degradation is so heavy that it can not be accepted in practical applications, the information is still easily recovered as shown in Figure 2 (right). Furthermore the other similar results are shown in Table 2. 2) JPEG compression: The JPEG compression algorithm is one of the most im- portant and strong attacks. However the embedding schemes should be resistant to JPEG coding with smoothing and decreasing quality applied to the signed image. Ob- viously, when the JPEG compressed image quality decreases, so does the extracted information; however, the embedded information is well extracted even though the 134 Li-yun Su· Feng-lan Li · Jiao-jun Li · Bo Chen (2011) Fig.1 Original and embedded images: 1 (left), test image; 2 (right), embedded image, PSNR=45.33, SIM=1, DBR=1 Fig.2 Original information image and retrieved secret image: 1 (left), original image; 2 (right), extracted information image, SIM=0.9219 quality factor is 40, and the image is visibly distorted (Table 3). 3) Geometric distortion: Virtually, all practical applications call for the informa- tion to be immune to geometric manipulations such as cropping. Experiments have been carried out to prove that the information contained in a sub-image is still suf- ficient to extract the information. In particular, suppose that the sub-image can be replaced at the same position which occupied in the original picture, the proposed system can extract the information. Even though the cropped part is 75% of the orig- inal image, DBR is still 0.7360. Table 2: Gaussian noise attacks. Variance 0.01 0.04 0.08 0.1 0.2 0.3 0.4 0.5 PSNR 21.21 18.31 16.05 14.77 13.91 13.39 10.79 9.09 DBR 0.976 0.954 0.934 0.949 0.947 0.947 0.943 0.908 Fuzzy Inf. Eng. (2011) 2: 127-136 135 Table 3: JPEG compression attacks. JPEG factor 40 50 60 70 80 90 DBR 0.8120 0.9944 0.9955 0.9962 0.9990 1 7. Conclusion In this paper, a novel covert communication algorithm is proposed for digital images. The presented adaptive method has good performance due to security information operating in the frequency domain. In order to avoid easy extraction of hiding infor- mation, a chaotic modulation array of a binary image or a pseudo-random sequence of real numbers having normal distribution with zero mean and unity variance is embed- ded in the appropriate blocks of the selected DCT coefficients. Modulating the binary image by chaotic sequence can get modulation array. The embedded coefficients are produced by arranging the DCT coefficients in a zig-zag scan and by extracting the co- efficients according to GFCM and HVS. After embedding hiding information, which is adapted to image being signed by exploiting the membership degrees of blocks, to further ensure the information invisibility and the robustness. Experimental results demonstrate that the meaningful information is robust to several signal processing techniques and geometric distortions, including JPEG compression, Gaussian noise as well as cropping, and the quality of the embedded image is guaranteed. Acknowledgments This work was partly supported by Chongqing Science & Technology Commission (CSTC) foundation of China (No. CSTC2010BB2310), Chongqing Municipal Edu- cation Commission(CMEC) foundations of China (No. KJ080614, No. KJ100810, No. KJ100818), Chongqing University of Technology(CQUT) foundation of China (No. 2007ZD16). 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Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Jun 1, 2011

Keywords: Covert communication; Digital image; Generalized fuzzy c-means clustering; Human visual system; Discrete cosine transform

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