An improved color image watermarking algorithm based on QR decomposition

Abstract

Most of the existing color image watermarking schemes use binary or grayscale image as watermark, and many of them belong to non-blind watermarking methods. In this paper, an improved color image watermarking algorithm based on QR decomposition is proposed to embed color watermark image into color host image. For embedding watermark, the 24-bits color host image is divided into non-overlapping 3 × 3 pixel blocks and each pixel block is decomposed by QR decomposition, and the 24-bits color watermark image is embedded into the color host image by modifying the relation between the second row first column coefficient and the third row first column coefficient of the orthogonal matrix Q. For extracting watermark, only the watermarked image is needed. Experimental results show that the proposed color image scheme not only has better watermark performance such as invisibility, robustness, security, capacity, and computation complexity, but also overcomes the false-positive detection problem.

1 Introduction

With the rapid development of computer networks and multimedia technology, humans can easily access or distribute any multimedia data from networks. Hence, multimedia security including illegal copying, tampering, modifying digital copyright etc., has been the focal point of considerable research activity in the last two decades. For solving the problem of copyright protection, digital watermarking technique has been considered as a powerful method [1, 8]. The main goal of image watermarking is to hide some important information into a host image. The important information includes pseudo random sequence, binary image, grayscale image and color image etc. In the field of image watermarking, researchers have been mainly focusing on grayscale watermarking technique so far, but color image watermark is an important challenge since the color image has more information than the grayscale image or binary image. In widespread multimedia applications, color images are the basic component of multimedia system [13]. Compared with the method of embedding binary or gray images into color image [7, 9, 16], the double color watermarking, i.e., embedding color watermark images into color host image [2, 4, 6, 10], not only has the features of more information and higher fidelity, but also can solve the copyright problems for some corporations which use the unique identifiable color logo as corporation mark.

Among the existing dual color image watermarking schemes, a blind color image watermarking was proposed based on singular value decomposition (SVD) in [6], in which one or more singular values must be modified to keep the order of singular values, which will degrade the quality of the watermarked image. Chou et al. [4] proposed a spatial-domain color image watermarking scheme by the quantization indices of the color host image in the uniform color space, unfortunately the robustness of the color watermark was weaker since this method was based on spatial-domain which has the common feature of lower robustness. Another watermarking method was proposed by Yin et al. [20], in which the singular values of original watermark were required to extract the embedded singular value, and the U and V orthogonal matrices of original watermark were also needed to recover the watermark. This method belongs to non-blind watermarking and causes the false-positive detection problem.

Since the QR decomposition is the major intermediate step in SVD, the feature highlights the use of QR decomposition in digital watermarking and a few of methods have been proposed in [11, 19]. Yashar et al. [19] proposed to modify coefficient for embedding watermark bit in the first row of R matrix after each 8 × 8 block based on QR decomposition. In [11], a 32 × 32 binary image was embedded into the 512 × 512 host image by modifying the elements in the Q matrix. The common feature in [11, 19] was to use binary image as the watermark.

There are two disadvantages in the aforementioned schemes, e.g., the scheme may be non-blind watermarking method, or the watermark image may be binary or grayscale image. In our previous work [14], the color digital watermark information was embedded into the 4 × 4 block of the color host image, that is, eight pixel values will be modified when embedding one watermark bit, which means the watermark invisibility will be affected. To overcome these drawbacks, a new double color image blind watermarking scheme based on QR decomposition is proposed in this paper. All elements in the first column of orthogonal matrix Q are very near. The smaller the difference between any two elements in the first column of orthogonal matrix Q, the smaller the modification for embedding watermark in the first column, which is helpful to improve the watermark invisibility. We use the similarity between q _2,1 and q _3,1 of Q matrix after performing QR decomposition on 3 × 3 pixels matrix to represent one kind of embedded watermark information. Hence, the number of the modified pixels will be reduced and the invisibility and robustness of watermark will be better than performing QR decomposition on 4 × 4 pixels matrix. Especially, the false-positive detection problem like in SVD-based algorithm will be overcome since all information of color image watermark can be embedded into the host image. The experimental results show that the proposed scheme can obtain better invisibility and stronger robustness against many common image processing attacks, and the comparison with the related algorithms also reveals the higher performance of the proposed algorithm.

The rest of this paper is organized as follows. Section 2 describes the QR decomposition theory and its special features, and analyzes the proposed visual compensation method. Section 3 introduces the details of the watermark embedding procedure. Section 4 contains the description of our watermark extraction procedure. Section 5 gives the experimental results. Finally, Section 6 concludes this presentation.

2 Methodology

2.1 QR decomposition

The QR decomposition, which is a type of matrix decomposition, was formally introduced in [5], and its detailed definition was given as follows.

$$ \left[Q,R\right]=qr(A) $$

(1)

where R is an n × n upper triangular matrix (also called right triangular matrix) and Q is an n × n orthogonal matrix (its columns are orthogonal unit vector).

For example, suppose one matrix A of size 3 × 3 as follows

$$ A=\left[\begin{array}{ccc}\hfill 255\hfill & \hfill 255\hfill & \hfill 252\hfill \\ {}\hfill 254\hfill & \hfill 251\hfill & \hfill 244\hfill \\ {}\hfill 254\hfill & \hfill 247\hfill & \hfill 230\hfill \end{array}\right] $$

Using Eq. (1), its upper triangular matrix R and orthogonal matrix Q can be obtained as follows

$$ R=\left[\begin{array}{ccc}\hfill 440.5190\hfill & \hfill 434.7531\hfill & \hfill 419.1783\hfill \\ {}\hfill 0\hfill & \hfill 4.9753\hfill & \hfill 15.0644\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 1.6234\hfill \end{array}\right],Q=\left[\begin{array}{ccc}\hfill 0.5789\hfill & \hfill 0.6708\hfill & \hfill -0.4636\hfill \\ {}\hfill 0.5766\hfill & \hfill 0.0652\hfill & \hfill 0.8144\hfill \\ {}\hfill 0.5766\hfill & \hfill -0.7387\hfill & \hfill -0.3490\hfill \end{array}\right] $$

2.2 The special feature in unitary matrix Q

As can be seen from the above orthogonal matrix Q, all elements in the first column of orthogonal matrix Q are very near. The smaller the difference between any two elements in the first column of orthogonal matrix Q, the smaller the modification for embedding watermark in the first column, which is helpful to improve the watermark invisibility. In order to select better embedding elements in the first column of Q matrix on 3 × 3 image block, as shown in Fig. 1, we adopt the following steps to do experiment with many standard images from CVG-UGR image database [15].

Fig. 1

Compute the Correlation Coefficient (CC) between different elements in the first column of orthogonal matrix Q

• Step 1: The test image is divided into image blocks with size 3 × 3.

• Step 2: Each image block is decomposed by QR decomposition, and its Q matrix is obtained.

• Step 3: Matrix U _x is composed by the x-th row first column element q _x,1 of each Q matrix, where the size of matrix U _x is m × n, and x = 1,2,3 respectively.

• Step 4: Matrix V _y is composed by the y-th row first column element q _y,1 of each Q matrix, where the size of matrix V _y is m × n, y = 1,2,3 respectively, and y ≠ x.

• Step 5: Compute the Correlation Coefficient (CC) between the two matrices U _x and V _y by the Eq.(2) as follows.

  $$ CC\left({q}_{x,1},{q}_{y,1}\right)= \min \left(\frac{{\displaystyle \sum_{i=1}^n{\displaystyle \sum_{j=1}^m{U}_x\left(i,j\right){V}_y\left(i,j\right)}}}{\sqrt{{\displaystyle \sum_{i=1}^n{\displaystyle \sum_{j=1}^m{U_x}^2\left(i,j\right)}}}},\frac{{\displaystyle \sum_{i=1}^n{\displaystyle \sum_{j=1}^m{U}_x\left(i,j\right){\displaystyle \sum_{i=1}^n{\displaystyle \sum_{j=1}^m{V}_y\left(i,j\right)}}}}}{\sqrt{{\displaystyle \sum_{i=1}^n{\displaystyle \sum_{j=1}^m{V_y}^2\left(i,j\right)}}}}\right) $$

  (2)

  where m and n are the sizes of the U _x matrix or V _y matrix.

• Step 6: Repeat the abovementioned steps to calculate CC (q _1,1, q _2,1), CC (q _1,1, q _3,1), CC (q _2,1, q _3,1).

  As can be seen from Table 1, the average CC (q _2,1, q _3,1) value of many standard test images is 0.9746, which shows q _2,1 and q _3,1 are the closest elements in the first column of 3 × 3 matrix Q, that is, the difference value between the two elements is very small, which means to slightly modify the two elements can represent one kind of watermark bit ‘1’ or’0’_. Since the image visual quality will be inevitably affected when modifying q _2,1 and q _3,1 to embed watermark, the compensation principle proposed in our previous work [14] is used to decrease the negative affection.

  Table 1 The CC values of different elements in first column of Q matrix after QR decomposition

2.3 The proposed watermarking scheme

In this paper, we propose a new double color image blind watermarking scheme based on QR decomposition, which includes the watermark embedding scheme and watermark extraction scheme.

2.3.1 Watermark embedding scheme

In the embedding process, the color watermark image is transformed to binary sequence at first. Then, the host color image is divided into non-overlapping block. Finally, the QR decomposition is performed on the image block and the watermark is embedded into Q matrix. Without loss of generality, let the original host image H be 24-bit color image with size of M × M and the watermark image be 24-bit color image W with size of N × N. The proposed watermark embedding scheme can be summarized as follows.

• Step 1: Transforming the color watermark image to binary formation

  Firstly, the original color watermark image W is also divided into three components R, G, B by dimension-reduction treatment. Then, the component watermarks are obtained corresponding to the R, G and B component, respectively. In order to enhance the security and the robustness of the embedded watermark, each component watermark is permuted by Arnold transform with private key KA _i (i = 1, 2, 3) [3, 18]. Arnold transform is also called Cat Face transfer, and it is given by

  $$ \left(\begin{array}{c}\hfill x\hbox{'}\hfill \\ {}\hfill y\hbox{'}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ {}\hfill 1\hfill & \hfill 2\hfill \end{array}\right)\left(\begin{array}{c}\hfill x\hfill \\ {}\hfill y\hfill \end{array}\right)\left( \mod N\right) $$

  (3)

  where (x,y) is the pixel of the component watermark image, (x’,y’) is the pixel of the component watermark image after permuting, N is order of watermark image matrix. Since the Arnold transform is periodic, the number of permuting can be considered as the key KA _i (i = 1, 2, 3) to enhance the security.

  After that, each pixel value is converted to 8-bit binary sequence. Finally, combining all 8-bit binary sequence to form the binary component watermark W _i (i = 1, 2, 3).

• Step 2: Processing the host image

  When more information is embedded into the color host image H, it is divided into three components H _i (i = 1, 2, 3), which represent the R, G and B component respectively, and each component image is divided into 3 × 3 non-overlapping blocks.

• Step 3: Selecting the embedding blocks

  In order to improve the watermark invisibility and robustness of the cropping attack, all embedded watermark will be uniformly distributed. According to the modified block numbers, the method of embedding interval selection based on private key K is adopted in this paper. The non-overlapping blocks 1, 1 + K, 1 + 2 × K,…, 1 + n × K in the odd lines and the non-overlapping blocks 2, 2 + K, 2 + 2 × K,…, 2 + n × K in the even lines are selected to embed watermark, where n is decided by the size of the host image (M × M), the size of the pixel block (B × B) and the private key K, that is, n = floor(M/(B * K)). The private key K can not only prevent the modified block from getting together, but also improve the invisibility of watermark, but ensure the security of watermark since the attacker cannot know the accurate position of embedding watermark and extracting watermark without the private key K.

• Step 4: Embedding watermark to the orthogonal matrix Q

  According to Eq. (1), each selected block is decomposed to obtain the orthogonal matrix Q by QR decomposition. The elements q _2,1 and q _3,1 in the first column of Q matrix will be modified as q ^'_2,1 and q ^'_3,1 by the following rules in Eqs. (4) and (5), respectively.

  $$ if\kern0.5em w=1\kern0.5em and\kern0.5em {q}_{2,1}-{q}_{3,1}<\left(2*T\right)/3, then\left\{\begin{array}{l}{q}_{2,1}^{\prime }={q}_{avg}+T/2\\ {}{q}_{3,1}^{\prime }={q}_{avg}+T/2\end{array}\right. $$

  (4)
  $$ if\kern0.5em w=0\kern0.5em and\kern0.5em {q}_{3,1}-{q}_{2,1}\le \left(2*T\right)/3, then\left\{\begin{array}{l}{q}_{2,1}^{\hbox{'}}={q}_{avg}-T/2\\ {}{q}_{3,1}^{\hbox{'}}={q}_{avg}-T/2\end{array}\right. $$

  (5)

  where w indicates the embedded binary watermark information bit and q _avg  = (q _2,1 + q _3,1)/2. It should be noted that the proposed method is similar to the method in our previous work [14], but the color digital watermark information was embedded into the 3 × 3 block of the host image, that is, only six pixel values instead of eight pixel values will be modified in the new proposed method.

• Step 5: Obtaining the watermarked image block

  The inverse QR operation is performed to obtain the watermarked image block according to Eq. (6).

  $$ A\hbox{'}=Q\hbox{'}\times R $$

  (6)

• Step 6: Repetition

  Repeating steps 4–5 until all watermark bits are embedded in the host image. Finally, the watermarked R, G, B components are reconstructed to obtain the watermarked image H '.

  In this watermark embedding procedure, all information of color image watermark has been embedded into the host image, which means the final extracted watermark will be decided by all watermark information bits instead of the partial watermark information, the false-positive detection problem like in SVD-based algorithm will not happen.

2.3.2 Watermark extraction scheme

In this paper, since the original host image and watermark image are not needed in the procedure, the watermark extraction procedure belongs to blind extraction method. The watermark extraction steps are described as follows.

• Step 1: Dividing the watermarked image

  The watermarked image H ' is divided into R, G, B component images, and then each component is further divided into watermarked blocks with size of 3 × 3 pixels, respectively.

• Step 2: Selecting the watermarked block

  The method of embedding interval selection based on private key K is used to select the watermarked blocks.

• Step 3: Extracting watermark

  Each watermarked block is decomposed by QR decomposition and its unitary matrix Q ' is obtained. The relation between q ^'_2,1 and q ^'_3,1 in the first column of Q ' is used to extract the watermark information w ', as shown in Eq. (7).

  $$ w\hbox{'}=\left\{\begin{array}{l}1,\kern0.5em if\kern0.5em {q}_{2,1}^{\prime}\ge {q}_{3,1}^{\prime}\\ {}0,\kern0.5em if\kern0.5em {q}_{2,1}^{\prime }<{q}_{3,1}^{\prime}\end{array}\right. $$

  (7)

• Step 4: Repetition

  The above steps 2–3 are repeated until all embedded image blocks are performed. These extracted bit values are partitioned into 8-bit groups and converted to decimal pixel values.

• Step 5: Reconstruction

  Each component watermark is transformed by the inverse Arnold transformation based on the private key KA _i (i = 1, 2, 3). Then, the final extracted watermark W ' is reconstructed from the extracted watermarks of the three components.

3 Simulation results

3.1 Metrics

The main performance of the watermarking methods under consideration is investigated by measuring their invisibility, robustness, execution time, security and embedding capacity. For proving the watermarking performance of the proposed method, all 24-bit 512 × 512 color images in the CVG-UGR image database are used as the host images [15], and two 24-bit 32 × 32 color images showed in Fig. 2a and (b) are used as original watermarks.

Fig. 2

Original watermark images: a PEUGEOT logo, b 8-color image; Original host images: c Lena, d Avion, e Peppers, f TTU

In order to evaluate the invisibility capability, not only is the peak signal-to-noise ratio (PSNR) used in this paper, but also the structural similarity (SSIM) index as a new method is also used to measure the similarity between the original color image I and the watermarked image I '.

PSNR as a good rule for measuring the watermark invisibility [12, 18], it is given by

$$ PSNR=10 \lg \frac{255^2}{MSE} $$

(8)

where MSE is the mean square error given by

$$ MSE=\frac{1}{3MN}{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x=1}^m{\displaystyle \sum_{y=1}^n{\left[I\left(x,y,j\right)-{I}^{\prime}\left(x,y,j\right)\right]}^2}}} $$

(9)

where, I(x, y, j), I ^'(x, y, j) present the value of pixel (x, y) in component j of the original image and the watermarked one, and m, n denote the width and height of the host images, respectively.

Moreover, the SSIM developed by Wang et al. [17] was considered to be correlated with the quality perception of the human visual system (HVS). The SSIM is designed by modeling image distortion that combines three factors: loss of correlation, luminance distortion and contrast distortion. The SSIM is defined as:

$$ SSIM\left(I,{I}^{\prime}\right)=l\left(I,{I}^{\prime}\right)c\left(I,{I}^{\prime}\right)s\left(I,{I}^{\prime}\right) $$

(10)

where

$$ \left\{\begin{array}{l}l\left(I,{I}^{\prime}\right)=\left(2{\mu}_I{\mu}_{I^{\prime }}+{C}_1\right)/\left({\mu}_I^2+{\mu}_{I^{\prime}}^2+{C}_1\right)\hfill \\ {}c\left(I,{I}^{\hbox{'}\prime}\right)=\left(2{\sigma}_I{\sigma}_{I^{\prime }}+{C}_2\right)/\left({\sigma}_I^2+{\sigma}_{I^{\prime}}^2+{C}_2\right)\hfill \\ {}s\left(I,{I}^{\prime}\right)=\left({\sigma}_{I{I}^{\prime }}+{C}_3\right)/\left({\sigma}_I{\sigma}_{I^{\prime }}+{C}_3\right)\hfill \end{array}\right. $$

(11)

The first term in Eq. (10) is the luminance comparison function which measures the closeness of the two images’ mean luminance (μ _I and ${\mu}_{I^{\hbox{'}}}$). This factor is maximal and equal to 1 only if ${\mu}_I={\mu}_{I^{\hbox{'}}}$. The second term is the contrast comparison function which measures the closeness of the contrast of the two images. Here the contrast is measured by the standard deviation σ _I and ${\sigma}_{I^{\hbox{'}}}$. This term is maximal and equal to 1 only if ${\sigma}_I={\sigma}_{I^{\hbox{'}}}$. The third term is the structure comparison function which measures the correlation coefficient between the two images I and I ^'. Note that ${\sigma}_{I{I}^{\hbox{'}}}$ is the covariance between I and I ^'. The positive values of the SSIM index are in [0, 1]. A value of 0 means no correlation between images, and 1 means that I = I ^'. The positive constants C ₁, C ₂ and C ₃ are used to avoid a null denominator.

Besides, in order to measure the robustness of the watermark, we use the normalized correlation (NC) between the original watermark W and the extracted watermark W ^', which is denoted as follows.

$$ NC=\frac{{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x=1}^p{\displaystyle \sum_{y=1}^q\left(W\left(x,y,j\right)\times {W}^{\hbox{'}}\left(x,y,j\right)\right)}}}}{\sqrt{{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x=1}^p{\displaystyle \sum_{y=1}^qW{\left(x,y,j\right)}^2}}}}\sqrt{{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x=1}^p{\displaystyle \sum_{y=1}^q{W}^{\hbox{'}}{\left(x,y,j\right)}^2}}}}} $$

(12)

where W(x, y, j), W ^'(x, y, j) present the value of pixel (x, y) in component j of the original watermark and the extracted one, and P, Q denote the row and the column size of the original watermark image, respectively.

Generally, a larger PSNR or SSIM means that the watermarking method has higher invisibility. A higher NC reveals that the watermarking method has more robustness. Table 2 not only shows the watermark invisibility (reflected by average PSNR values) of the embedded host images, but also shows the robustness (measured by the average NC values) of extracted watermark after resisting many attacks with different threshold values T. The bigger the threshold T is, the worse the watermark invisibility is and the better the watermark robustness is, vice verse. Considering the tradeoff between the invisibility and robustness of watermarking, the threshold value T is set to 0.08.

Table 2 The PSNR and NC values with different threshold values T

3.2 Experiment result

3.2.1 The result of invisibility

For relatively fair comparison, the host images Fig. 2c and d used in [6, 11] and the host images Fig. 2e and f used in [4] are adopted in this experiment. When the different watermark images in Fig. 2a and b were respectively embedded into all 512 × 512 color images of the CVG-UGR image database, Fig. 3 not only gives the watermarked color images and their SSIM values, but also shows the extracted watermark and their NC and SSIM values.

Fig. 3

The watermarked color images and the extracted watermarks without any attacks

3.2.2 The result of robustness

In this sub-section, we give the extracted results from the attacked images Lena and F16 after the watermark of Fig. 2a was embedded into.

In the application of real life, storage and transmission of digital data are common operation. For this purpose, a lossy coding operation is often performed on the data to reduce the memory and increase efficiency. Since JPEG and JPEG 2000 standards are widely used formats in image compression, the proposed watermarking system is tested to investigate the robustness of the proposed watermarking system with JPEG and JPEG 2000 standards. JPEG is an image compression standard that supports lossy and lossless compression of grayscale as well as color images. In this experiment, the watermarked images are compressed with different compression factors from 10 to 100 increasing in steps of 10. Figure 4 gives the part of the results. Compared with the methods [6, 14], the proposed scheme achieves better robustness against the JPEG compression.

Fig. 4

The extracted watermarks via using different methods after the JPEG compression operation

Meanwhile, JPEG 2000 was developed by the Joint Photographic Expert Group in the aim of improving the properties of the JPEG standard. The watermarked images are also performed by JPEG 2000 compression with the compression ratio from 1 to 10 increasing in steps of 1. Figure 5 gives the NC values, SSIM values and visual-perception results with compression ratios of 5 and 10, respectively.

Fig. 5

The extracted watermarks via using different methods after the JPEG 2000 compression operation

In addition, a salt & peppers noising scheme is performed by generating noise to corrupt the watermarked images. The noising scheme generates 2 and 10 % noise, respectively, to degrade the watermarked images. Figure 6 shows the quantitative results in terms of the NC, SSIM and visual-perception after adding Salt & peppers noise, respectively. Moreover, the Gaussian noising is also added to corrupt the watermarked images, in which normally distributed noise with mean 0 and standard deviation σ was added to each of the watermarked images. The standard deviation σ of Gaussian parameter is set from 0.1 to 1.0, respectively. The extracted watermarks, NC values and SSIM values are shown in Fig. 7.

Fig. 6

The extracted watermarks via using different methods after the salt & peppers noising operation

Fig. 7

The extracted watermarks via using different methods after the Gaussian noising operation

In this experiment, the Butterworth low-pass filters are used to corrupt the watermarked images with cut-off frequency 100, and order from 1 to 10, respectively. Figure 8 shows the NC values, SSIM values and visual-perception results after low-pass filters attack.

In the procedure of sharpening, the radii are 0.2 and 1.0, respectively. Figure 9 gives the result of sharpening attack. In the blurring attacks, two cases are simulated here to degrade the two watermarked images. The radius in the first case is 0.2, while the second case is 1.0. Figure 10 exhibits the results of visual comparison and quantitative values. It is evident that the proposed method is more robust than the methods [6, 14] in this case.

In addition, scaling operation is performed on the watermarked image from 25 to 800 %, respectively. Figure 11 shows the quantitative results and visual-perception results for the scaling with 400 % and 25 %, respectively. In the cropping attack, two cases are simulated here to crop the two watermarked images. The first case is cropped by 25 %, and the second case is cropped by 50 %. Figure 12 shows the results of anti-cropping attack.

In order to further prove the robustness, the proposed methods are also compared with the spatial domain method [4]. Two color images, as shown in Fig. 2e and f, were used as the host image and 8-color image of Fig. 2b was taken as the watermark image in [4]. For fair comparison, we also use these host images and watermark image to carry out the experiment under the same attack styles in [4]. Figures 13 and 14 show the comparison results using Peppers, TTU as the host image respectively.

3.2.3 The result of execution time

In these experiments, a laptop computer with a duo Intel CPU at 2.27GHZ, 2.00GB RAM, Win 7, MATLAB 7.10.0 (R2010a) is used as the computing platform. The embedding time and extraction time of the proposed methods is 0.546941, 0.410019 s, respectively. Table 3 shows the comparison of execution time between different methods.

Table 3 The comparison of execution time between different methods (Second)

4 Discussion

4.1 Invisibility discussion

By comparison, it can be seen from Fig. 3 that Song et al. [11] based on QR decomposition could not extract good watermark and the host image is obviously modified via embedding the watermark, which cannot meet the need of watermark invisibility. Thus, this method of Song et al. [11] is not suitable for embedding color watermark image into color host image. Golea et al. [6] and the proposed method have better watermark imperceptibility, but the former has lower NC and SSIM values than the latter. Thus, the proposed method not only has better watermark invisibility, but also can effectively extract the embedded watermark. Based upon this experiment, we draw the conclusion that the proposed method can effectively embed and extract color watermark image.

4.2 Robustness discussion

In this paper, various attacks such as JPEG compression, JPEG 2000 compression, cropping, adding noise, scaling, filtering, rotation, blurring et al. are performed on the watermarked image to estimate the robustness of the proposed method, and compare the proposed method with the related method [6] based on SVD and method [14] based on QR decomposition.

As can be seen from Figs. 4 and 5, the proposed scheme, compared with the methods [6, 14], achieves better robustness against the compress attacks such as JPEG compression and JPEG 2000 compression.

It is evident from Fig. 6 that the proposed method is more robust against salt & peppers noise attacks than the methods [6, 14]. However, the bigger the Gaussian parameter is, the weaker the robustness is in Fig. 7. Hence, it must be further improved in the future work. Relatively, the robustness of the proposed method is superior to that of other methods under adding Gaussian noising in Fig. 7.

Figure 8 shows the proposed method has better robustness against the low-pass attack. Moreover, the results of visual comparison and quantitative values given in Figs. 9 and 10 prove that the proposed method outperforms the methods [6, 14] in aspects of sharpening and blurring, respectively.

Fig. 8

The extracted watermarks via using different methods after the low-pass filter operation

Fig. 9

The extracted watermarks via using different methods after the sharpening operation

Fig. 10

The extracted watermarks via using different methods after the blurring operation

In addition, the proposed method has better robustness as respect of scaling operation and cropping in Figs. 11 and 12, respectively. Because the watermark used in [6] is not permuted, the position and size of cropping can fully affect the watermark in the cropped region. In Fig. 12, there is a black region in the extracted watermark, which means the watermark information in this region is fully deleted by cropping attack. Moreover, the method of embedding interval selection replaces the Hash pseudo-random selection in method [14]. Hence, the quality of the proposed method outperforms that of the methods [6, 14].

Fig. 11

The extracted watermarks via using different methods after the scaling operation

Fig. 12

The extracted watermarks via using different methods after the cropping operation

It can be seen from Figs. 13 and 14 that the proposed algorithm is more robust. In most cases, the performance of the proposed method is superior to the method in [4] except adding Gaussian noise with larger parameter. In [4], the changed color values for various attacks have directly affected the mapping relation between the original color value and the color table, which results in the degraded quality of extracted watermark.

Fig. 13

The watermarks extracted from the attacked Peppers images via using different methods

Fig. 14

The watermarks extracted from the attacked TTU images via using different methods

4.3 Execution time discussion

As can be seen from Table 3, the average execution time of the proposed method is less than that of [6]. This means that SVD is more complex than QR decomposition because the QR decomposition is an intermediate step in SVD and the time complexity for SVD is typically worse than for QR decomposition. Meanwhile, the average execution time of the proposed algorithm is also less than that of [4] because the host image needs to be transformed to the CIE-Lab color space for the color quantization and its inverse-transformation is also required in [4]. The Hash pseudo-random selection used in [14] needs more time than the interval selection used in this paper. Therefore, one should notice that the proposed algorithm requires much less number of computations.

4.4 Security discussion

In this paper, in order to enhance the watermarking security, each component watermark is permuted by Arnold transform with private keys KA _i (i = 1, 2, 3) and the watermark embedding block is decided by the private key K. Firstly, the position of embedding or extraction based on the private key K is unknown for anybody without the right key K, which makes it difficult to extract and tamper the embedded watermark; Secondly, when the component watermark is extracted from the watermarked image, nobody can extract and rearrange the watermark without the private keys K and KA _i (i = 1, 2, 3). Kirchhoff’s criterion is often used to assess the usability of cryptograph methods and systems: all algorithms must be public, and only the key is secret. The security of our algorithm only relies on the private keys KA _i (i = 1, 2, 3) and K, which meets Kirchhoff’s criterion. Figure 15 gives the watermark image extracted from the watermarked image Lena. As can be seen from the figure, the right watermark can be extracted only if the private keys K and KA _i (i = 1, 2, 3) are right. Moreover, the attacker without the right keys can only try all possible forms of the keys until he believes that the original image has been obtained through exhaustive manner, which is an NP-hard problem.

Fig. 15

The extracted watermark with different keys

4.5 Embedding capacity discussion

In this paper, the 24-bit 32 × 32 color image is embedded into the 24-bit 512 × 512 color host image. The proposed method and the methods [6, 14] have the same watermark capacity 0.03125 bpp (bit per pixel). The watermark capacity of Chou et al. [4] can achieve 0.48bpp (bit per pixel) which is more than that of others. In this proposed method, the embedding block with size 3 × 3 is smaller than that in [14] with size 4 × 4, in theory, the proposed embedding capacity will be 1.333 times of the [14]. Since the watermarking capacity, invisibility and robustness are in conflict with each other, the watermarking capacity of the proposed method is acceptable in the premise of ensuring the watermarking invisibility and robustness.

5 Conclusions

Based on QR decomposition, an improved color image watermarking scheme is proposed in this paper. For embedding watermark, the 24-bits color host image is divided into non-overlapping 3 × 3 pixel blocks and each pixel block is decomposed by QR decomposition, and the 24-bits color watermark image is embedded into the color host image by modifying the relation between the second row first column coefficient and the third row first column coefficient of the orthogonal matrix Q. When extracting watermark, only the watermarked image is needed. Experimental results have shown that this proposed algorithm not only attains higher invisibility of watermarking, but also has stronger robustness in various distortion operations. In future, we will extend the proposed method to color video watermarking.