J. Cent. South Univ. Technol. (2011) 18: 116−124 DOI: 10.1007/s11771−011−0668−8
Chaotic system and QR factorization based robust digital image watermarking algorithm SONG Wei(宋伟)1, 2, HOU Jian-jun(侯建军) 1, LI Zhao-hong(李赵红) 1, HUANG Liang(黄亮) 1 1. School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing 100044, China; 2. School of Information Engineering, Minzu University of China, Beijing 100081, China © Central South University Press and Springer-Verlag Berlin Heidelberg 2011 Abstract: In order to protect copyright of digital images, a new robust digital image watermarking algorithm based on chaotic system and QR factorization was proposed. The host images were firstly divided into blocks with same size, then QR factorization was performed on each block. Pseudorandom circular chain (PCC) generated by logistic mapping (LM) was applied to select the embedding blocks for enhancing the security of the scheme. The first column coefficients in Q matrix of chosen blocks were modified to embed watermarks without causing noticeable artifacts. Watermark extraction procedure was performed without the original cover image. The experimental results demonstrate that the watermarked images have good visual quality and this scheme is better than the existing techniques, especially when the image is attacked by cropping, noise pollution and so on. Analysis and discussion on robustness and security issues were also presented. Key words: digital watermarking; QR factorization; pseudorandom circular chain; logistic mapping
1 Introduction The rapid growth of Internet have changed our society and daily lives. However, the protection of intellectual property right (IPR) of digital multimedia becomes a crucial problem. As an effective technique, digital watermarking has been proposed. And it can be classified into fragile digital watermark technique, semi-fragile digital watermark technique, and robust digital watermark technique. In the fragile watermark algorithms [1−4], sensitive characteristics of host images are often used to embed watermarks, such as prominent intensity pixel value of image blocks and least significant bit (LSB). They are so sensitive to the sorts of operations where these algorithms are used to verify the integrity of digital images. The semi-fragile watermark algorithms [5−9] are robust to some distortions, but fragile to other operations. For example, ZHANG and YANG [8] used JPEG invariant characteristic to embed watermarks by qualifying DCT coefficients. This scheme can resist JPEG compression, but it is fragile to other attacks. Semi-fragile watermark schemes are used to authenticate the protected images even these images suffer from some attacks. The main purpose of the robust the digital watermarking techniques [10−18] is the ownership
authentication or the copyright protection. These algorithms can resist attacks and extract watermarks successfully from the distorted images. There are two main approaches: schemes in spatial domain [10−11], and schemes in frequency domain [12−18]. In spatial domain, watermarks are embedded by modifying the pixel value directly. In the frequency domain, the protected images must first be transformed into frequency domain by using a frequency-oriented mechanism, such as discrete Fourier transforms (DFT) [12], discrete cosine transforms (DCT) [13], and discrete wavelet transforms (DWT) [14−18], or other techniques. Some of these algorithms use human visual system (HVS) to improve the visual quality of the watermarked images, and use quantization and statistical method to enhance robustness. In this work, the invariant characteristics of QR factorization were investigated and a novel robust digital image watermarking scheme based on chaotic system and QR factorization was proposed. QR factorization was used to embed and extract watermarks for its good performances, and logistic mapping (LM) was used to generate pseudorandom circular chain (PCC) in order to enhance the security of this scheme.
2 Basic theories 2.1 QR factorization In QR factorization [19], the orthogonal triangular
Foundation item: Project(2007AA01Z241-2) supported by the National High-tech Research and Development Program of China; Project(2006XM002) supported by Beijing Jiaotong University Science Foundation, China; Project(0910KYZY55) supported by the Fundamental Research Funds for the Central University in China Received date: 2009−05−19; Accepted date: 2010−05−28 Corresponding author: SONG Wei, PhD; Tel: +86−10−51688396; E-mail:
[email protected]
J. Cent. South Univ. Technol. (2011) 18: 116−124
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decomposition of a matrix is performed, which is defined as A=[a1, a2, ···, an]=qr(A)=QR
(1)
where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning QTQ=I) and R is an upper triangular matrix (also called right triangular matrix). In QR factorization, the elements of first row in the upper triangular matrix is bigger than other rows, so power of the matrix is distributed averagely on top right corner. The first column of Q represents the relationship between pixels. So, Q can resist attacks for its orthogonal characteristic. And two conclusions can be gotten as follows: 1) The first row of R matrix has determined the complexity of the original matrix. 2) The coefficients in first column of Q may be modified for robust QR-based watermarking algorithm. Here, an example is given to illustrate the relationship between the Q component coefficients. The original image and the distorted image are shown in Figs.1(a) and (b), respectively. Table 1 shows the pixel values of the original block and the corresponding luminance-enhanced block, and their Q components, respectively. From Table 1, we can see that the
magnitude relationships (i.e., |−0.507 2|>|−0.490 5| and |−0.505 3|>|−0.493 0|) between the coordinates (1, 2) and (1, 3) can be preserved well even though the luminance processing is performed. The ratio of invariant symbol in each component to total number blocks is also tested when the host image is attacked by JPEG compression (Factor: 70), noise (0.03), luminance- enhancement (50%), contrast-enhancement (50%), blurring, and sharpening. In Fig.2, Ci (i=1, 2, 3, 4) represents the i-th column component. It can be observed that only the symbol in first column of Q is invariant. So, watermarks can be embedded by modifying the first column of Q. 2.2 Logistic mapping system and pseudorandom circular chain (PCC) Logistic mapping (LM) system is a chaotic system, and it is often used in watermarking technique for its simpleness and good performance [10]. The definition is as follows: xn+1=λxn(1−xn), λ∈[0, 4], xn∈(0, 1)
(2)
where λ is the control parameter of logistic mapping system. Pseudorandom circular chain (PCC) can be generated by logistic mapping system [4]. PCC has the following
Fig.1 Original (a) and luminance-enhanced (b) images Table 1 Relationship of QR factorization transformed Q component coefficients between original block and its corresponding luminance-enhanced block Original block Luminance-enhanced block 122 118 118 118 164 160 160 160 122 122 115 117 164 164 156 158 118 113 114 119 160 153 155 161 119 113 117 119 161 153 158 161 −0.507 2 −0.507 2 −0.490 5 −0.494 7
Q component of original block 0.047 6 0.171 4 −0.815 1 0.064 8 0.290 1 −0.808 4 0.499 1 0.559 3
−0.843 2 0.272 1 0.147 1 0.439 5
Q component of luminance-enhanced block −0.505 3 −0.122 2 0.719 3 −0.460 7 −0.505 3 −0.755 8 −0.295 6 0.293 1 −0.493 0 0.371 4 −0.605 2 −0.502 6 −0.496 1 0.525 2 0.169 9 0.670 1
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J. Cent. South Univ. Technol. (2011) 18: 116−124
Fig.2 Ratio of invariant symbol in each column of Q component
characteristics, which meet the need of the security of the algorithm: 1) The mapping relationship in PCC is one to one, and it is non-overlapped; 2) PCC is decided by key, and different keys can generate different PCC; 3) The neighbor elements in PCC are far away from each other; 4) Because PCC is generated by chaotic system, the key space is large enough.
3 Proposed algorithm 3.1 Watermarks embedding procedure In watermark-embedding procedure, watermark image Wp×q was embedded by modifying the first column component of Q matrix in QR factorization of selected blocks by PCC. The detail of watermark embedding procedure is described as follows. Step 1: Host image Io with size of N×M was partitioned into non-overlap blocks with size of (N/n)× (M/m): ⎡ I (1, 1) L I (1, m) ⎤ ⎥ ⎢ Io = ⎢ M O M ⎥ ⎥ ⎢ ⎥ ⎢I L I ( n, m ) ⎦ ⎣ ( n, 1)
∆1=|Q(i, j)(2, 1)|−|Q(i, j)(3, 1)|, ∆2=−∆1; if ∆1≥T,
Q(′i , j ) (2,1) = Q(i , j ) (2,1), Q(′i , j ) (3,1) = Q(i , j ) (3,1); if ∆2≥T,
Q(′i , j ) (2,1) = Q(i , j ) (3,1), Q(′i , j ) (3,1) = Q(i, j ) (2,1); (3)
Step 2: QR factorization was performed on each block: [Q(i, j), R(i, j)]=qr(I(i, j)), (i=1, 2, ···, n; j=1, 2, …, m) (4) where qr represents QR factorization function, I(i,j) represents the block (i, j), Q(i, j), R(i, j) stand for Q and R matrix in QR factorization of block I(i, j), respectively. Step 3: PCC was built by key k: PCC=f (LM (k, n×m))
Step 5: Two-dimensional watermark image Wp×q was transformed into one-dimensional sequences: wl (l=1, 2, ···, p×q). Step 6: Watermarks were embedded as follows: The modification of Q component coefficients might alter the original pixel values and degraded the quality of the watermarked image. The larger the modification of Q component was, the worse the distortion of image quality was, but the stronger the robustness of the scheme was. On the other hand, the smaller modification implied better image quality and weaker robustness. Therefore, a tradeoff between robustness and quality should be selected. Assume that watermark wi corresponded to the selected block (i, j). When wi was 1, it was embedded as follows:
(5)
Step 4: Embedding blocks were selected by PCC.
if 0≤∆1
