An entropy-based image watermarking scheme by improving robustness in shearlet and wavelet domain

The existing robust digital watermarking schemes mainly embed information in the fixed positions or with fixed embedding strength, while seldom considering adaptive adjustment based on the characteristics of the cover image, thus it reduces the imper-ceptibility and the robustness of watermarking. Aiming at these issues, we propose a scheme which can be able to dynamically adjust the watermark embedding position and strength. Furthermore, it guarantees the trade-off between robustness and imperceptibility. The appropriate embedding positions are dynamically selected for the watermark by comparing the image entropy, and the embedding strength of the image blocks are adaptively adjusted according to the entropy and the JND model in the HVS-based wavelet domain. SVD is performed on the image blocks to ensure the resistant ability of geometric attacks. The experimental results show that the scheme has good imper-ceptibility as well as strong robustness against various attacks. The robustness in common attacks is improved by at least 1% compared with similar watermarking schemes.


Introduction
Digital watermarking is an information hiding technology, it commonly used for copyright protection, ownership identification and authentication applications. From the anti-attack ability, the watermark technology can be classified into three categories: fragile, semi-fragile and robust. Among them, robust watermarking enables the watermarked image not only resistant to non-malicious attacks, but also malicious attacks within a certain range of distortions, even general image processing hardly affects the detection of the watermarks [1]. Imperceptibility and robustness are the two main characteristics of robust digital watermarking [2]. Imperceptibility refers to the visibility and quality of the watermarked image, while robustness is the ability of a watermark being effectively extracted and detected from a watermarked image under various attacks. Most watermarking schemes are difficult to achieve both features for they are mutually exclusive [3]. Therefore, robust watermarking researches generally focus on achieving a good balance between imperceptibility and robustness. The watermark can be embedded in spatial and transform domain of cover image. The former technology hides a large amount of data into the cover image, but its robustness is worse than the schemes based on transform domain [4]. The latter one embeds a watermark via converting cover image into a transform domain and then change its coefficients and shows good robustness under geometric attacks such as rotation, scaling, and panning [5]. Discrete Fourier transform (DFT), discrete wavelet transform (DWT) and discrete cosine transform (DCT) are commonly used in a watermark technology.
In recent years, some researchers consider to combine the plural transform methods for obtaining the advantage of each transform domain to improve the performance. Araghi et al. [6] utilized the advantage of the wavelet domain and SVD and proposed a hybrid robust digital watermarking scheme. Wavelet domain better reflects the HVS, and SVD has strong anti-geometric attack ability. Thanki et al. [7] proposed a digital watermarking scheme with redundant discrete wavelet transform (RDWT) and discrete curvelet transform (DCuT). In their scheme, after carrying out the DCuT of the original cover image, vertical wavelet coefficients are modified by two pseudo-random noise (PN) sequences according to scrambled watermark. Singh et al. [8] proposed a scheme based on DCT-SVD. The nonlinear chaotic map is used to randomly select the embedding position of the watermark in the DCT domain, it enhances the security of the watermark scheme and solves the false alarm detection problem by using a verification operation. Vali et al. [9] proposed an algorithm named self-adaptive differential evolution (SADE) based on RDWT and SVD to optimize the embedding strength. The scheme embeds a gray-scale watermark image into the singular values of RDWT subbands after multiplying by a scaling factor. Although the above watermarking schemes combine the advantages of multiple transformation, they embed a watermark with fixed embedding strength or in fixed position, and it is hard to achieve the trade-off between the robustness and imperceptibility.
How to dynamically select the suitable embedding position and strength is one of the challenges to optimize the watermarking scheme. To solve this issue, information entropy, which is commonly used to describe the randomness of image data, is introduced in this research. The higher entropy of an image, the higher the pixel complexity. In other words, selecting the region with higher information entropy as the embedding position can effectively improve the imperceptibility of the scheme. [10] applies the quantization method to control the embedding strength of watermark, selects some image blocks with high information entropy as the embedding positions, and utilizes just noticeable difference (JND) model to calculate the maximum embedding strength of each singular value matrix coefficient, thus it solves the trade-off between the imperceptibility and robustness. Experiments show that the scheme has good robustness owing to JND model. JND is the minimum perceptual threshold of the human eye when the image changes. Since the transform domain-based JND model has the advantages of low computational complexity and adaptability to HVS, it can achieve better robustness and imperceptibility than using a fixed embedding strength scheme, thus it is a suitable tool for controlling the embedding strength. Tikariha et al. [11] proposed a DWT-DCT-SVD robust digital watermarking scheme based on effective JND model. They combine many transformation domains and the JND model to improve the robustness of the overall watermarking scheme. Similarly, Hu et al. [12] proposed a robust blind digital watermarking scheme based on DCT-PSAMM. By combining the JND model and the LQI in the cosine domain, the watermarked image still cannot be observed obvious distortion even at a high embedding strength.
In the above digital watermarking schemes, the methods based on wavelet domain have achieved good performances. However, the two-dimensional discrete wavelet is composed of wavelet tensor accumulation, its basis function only has three directions of horizontal, vertical and diagonal, thus it cannot represent the edge information of image optimally. Therefore, some researchers put forward the multi-scale geometric analysis methods to develop an "optimal" representation method for high-dimensional data. The multi-scale geometric analysis methods applied in digital watermarking include ridgelet transform [13,14], contourlet transform [15][16][17], bandelet transform [18], shearlet transform [19][20][21][22][23] and so on. Among them, discrete shearlet transform (DST) [24] is a new multi-scale geometric transformation proposed in 2008. It overcomes the imperfections of limited direction selection of wavelet transform and anisotropy of base function, and has good directionality and multi-resolution representation for images. DST is widely used in image processing, and the research of digital image watermarking based on DST is gradually increasing in recent years [21][22][23]. However, most of the existing DST-based schemes have high computational complexity, which is not conducive to real-time embedding of the watermarks. Fast finite shearlet transform (FFST) [25] is a simple version of DST which translates the shearlets over the full grid at each scale and for each direction. And the use of fast Fourier transform speeds up the implementation of the algorithm, which makes it more suitable for the application of watermarking schemes. In consequence, we selected FFST to optimize the time complexity compared with traditional shearlet transforms.
In this paper, a robust hybrid watermarking scheme based on shearlet domain and wavelet domain is proposed. Table 1 shows the comparison between the proposed scheme and some state-of-the-art methods. The proposed scheme utilizes a multi-scale analysis method to analyze the cover image and get the optimal representation, improving the ability of watermark to resist general attacks and geometric attacks by using the existing DWT-SVD watermarking schemes. Information entropy is adopted in embedding position selection and improves the imperceptibility. Compared with the similar algorithms, it is not limited to select a part of the high entropy region as the embedding position. Instead, the whole region can be used to embed watermarks through quantitative comparison of the embeddable regions, so as to increase the capacity of watermarks. In terms of embedding strength, the HVS-based JND model is used to select the best embedding strength. The trade-off between robustness and imperceptibility is achieved by adjusting the information entropy. Experimental results show that the identifiable watermark can still be extracted after the watermarked image being not seriously distorted.
The rest of the paper is organized as follows. In Section 2, the preliminaries briefly describe the discrete shearlet transform, JND model and image encryption with ergodic matrix,. Section 3 describes the proposed watermarking scheme in detail. And the experimental results are presented in Section 4. Finally, Section 5 contains the conclusion of the paper.

Discrete shearlet transform:
In Multi-scale Geometric Analysis (MGA), the high-dimensional function is represented by the set of functions in the low-dimensional function space, and the optimal or "thinnest" function representation method is obtained. With more and more research and application, its ability to optimize the representation of images makes it important in the field of digital image processing. Discrete shearlet transform is a novel image analysis of MGA with the ability to perform multi-scale and multi-directional analysis of data [32].
The image decomposition based on shearlet transform is mainly divided into two steps: ①decompose the cover image by Laplacian pyramid to obtain the high-pass and low-pass sub-bands of the image. ②perform a suitable transform on the sub-bands to obtain the multi-scale and multi-directional representations, as shown in Fig. 1. Here, the details of the DST is described briefly as follows.
The criteria for discrete shearlet transformation are as shown in formula (3): Then apply the discrete Fourier transform, of which the target image ∈ 2 ( 2 ) of the N × N size can be defined as: and , ( ) ( ) is a window function located on a pair of trapezoids.

Fig. 1
The tiling of frequency plane and frequency support of a shearlet.

Just noticeable threshold model on wavelet domain
Barni et al. [33] proposed a noticeable threshold model in the wavelet domain, which has been considered as a suitable tool for controlling the embedding strength of watermark in each DWT coefficient of a cover image and a solution to balance the robustness and imperceptibility of the watermarking scheme. The model using three major characteristics of human visual system: Frequency masking , , Brightness masking ̃( , ) and Texture masking ( , ). The entire JND model is calculated according equation (5): Among them, , are respectively the decomposition scale and direction of the discrete wavelet transform. Three characteristics based on the human visual system are mathematically defined as equation (6)(7)(8)(9): ( , ) = 1 It can be seen from the above equations that the JND model of the wavelet domain is proposed by pixel-based masking. These characteristics are calculated in relatively small 2 × 2 neighborhood pixels, representing small local features of the image, and it ignores the effect of global features of the image on HVS. Block-based image processing is a common method that can be used to weigh the effects of global and local characteristics of target images on HVS. Therefore, a block-based digital image watermarking scheme is proposed in this research. The cover image and the watermark image are decomposed into non-overlapping image blocks before the embedding process. Then the entropy of each image block is calculated to represent the global features of the image block, and finally the imperceptibility of the watermarking scheme is improved by combining with JND.

Image Encryption with Ergodic Matrix
For ensuring the security of the digital watermarking scheme, an image encryption with ergodic matrix is applied before the watermark embedding process.
Ergodic matrix was introduced by Zhao et al. [34,35]. The basic idea can be briefly described as below: Let n×n q be a set of all n×n matrices over the finite field q , ( n×n q , +, ×) form a 1-ring, here + and × are addition and multiplication over q , respectively. We randomly generate two nonsingular matrices Q1, Q2 n×n q , then: ( n×n q , ) is a monoid, its identity element is × .
Ergodic matrix has the following definitions and properties [34,35]: is an ergodic matrix if and only if the order of Q is (q n -1) after the multiplication of Q over finite field q .
Theorem 2: Given Q n×n q is an ergodic matrix, then for ∀v n×1 From the theorems above, over finite field q , all n × n ergodic matrices have the same order and their generating sets have the same size, which are larger than that of any other n × n non-ergodic matrices. Take a random ergodic matrix Q 50×50 256 as an example, then the image of the matrix and the histogram is shown in Fig. 2.
(1) Image of a Random 50×50 Ergodic Matrix (2) Histogram of a Random Ergodic Matrix Zhou et al. [36] and Zhang et al. [37] apply ergodic matrix in the image encryption scheme and obtain a good result. It is demonstrated that an ergodic matrix with proper parameters [36,37] can be employed to completely shuffle the original image and has an immense key space of at least3.08×10 5898 . Therefore, the same method is used in watermark preprocessing in our research. For preprocessing is not our main concern, interested readers can refer to [36,37] .

The proposed scheme
In order to provide effective protection for digital images, a hybrid robust digital watermarking scheme based on shearlet and wavelet domain is proposed. The owner of the digital image is verified during the extraction process by using a secret key. The multiple transformation on the cover image is to improve the overall effect for the scheme. The embedding position and embedding strength are adaptive and optimized in this scheme, which guarantees the trade-off between the transparency and robustness.
In this section, the process of digital watermark embedding and extraction is illustrated in Sections 3.1 and 3.2, respectively. Section 3.3 uses an example to describe the proposed scheme in details.

Watermark embedding process
1. Read the cover image of size × and decompose it into sub-bands by -FFST, as shown in equation (10).
2. Calculate the entropy of each sub-band and select the sub-band with the largest entropy value for -DWT to get four sub-bands: , , and .
Where X, Y represent the row and column of the image in which the sub-block is located.
4. Read the watermark image , and the watermark image is encrypted by ergodic matrix (Section 2.3), and the encrypted watermark information ′ is also decomposed into small blocks , of size 2 +1 × 2 +1 by equation (13).
5. Calculate the entropy of the sub-block , and , separately. At the same time, in order to reduce the entropy error between sub-blocks, the entropy of each sub-block is divided by the minimum sub-block entropy. The specific embedding position of the watermark information is determined by the index after sorting the entropy values of , and , .
where is a scaling factor ranging from 0 to 1.

Decompose
, into three matrices by SVD: Left singular matrix , , singular value matrix , and right singular matrix , . Then change the value of , by the following equation (19) to embed the watermark information into the cover image: 8. The changed singular value matrix ̅ , is again decomposed by SVD into ̿ , , ̿ , and ̿ , by equation (20), and then used the following equation (21) to get the image block after embedding the watermark: 9. In the end, -IDWT and -IFFST are performed on the combined coefficient matrix ̅ , ( , ) = , ( , ) + , • , ( , ) in order to obtain the image after the watermark is embedded. is denoted as equation (22). Fig. 3 and 4 show the embedding and extraction processes of watermark information, respectively. The extraction process is the reverse process of the embedding process, the steps are as follows:

Watermark extraction process
1. The watermarked image ′ is processed with -FFST and -DWT, and four subbands are obtained as in steps 1 and 2 of the embedding process.
2. Decompose the four sub-bands into 2 +1 × 2 +1 small blocks ′ , , and find the corresponding embedding position of the watermark information by using the index obtained in step 5 of the embedding process.
3. Perform SVD on the watermarked image ′ , which may have been attacked.
Then put its singular value matrix ′ , with the previous ̿ , and ̿ , to perform I-SVD (inverse SVD). The equation (23-24) is as follows: 4. Extract encrypted watermark block information according to the following equation (25): 5. The watermark information is decrypted using the secret key to obtain the final watermark information.

Fig. 4
The watermark extraction procedure.

Discussion and analysis
To elaborate the proposed scheme, two examples are given and they are discussed in more details. In Example 1, for a more convenient representation, a 4 × 4 matrix is selected as a cover image to perform FFST-DWT. , and are three matrices of after SVD, where the singular value matrix is used for watermark embedding. Ergodic matrix encrypts the original watermark of size 4 × 4, and the obtained matrix ′ is taken as the information to be embedded.
is an embedded intensity matrix of size 4 × 4 calculated by the formula (17). Using the equation (19), the ′ is inserted into the singular value matrix , and the singular value matrix ̅ containing the watermark information is obtained. Then SVD is used to obtain the left singular matrix ̿ , the right singular matrix ̿ and the singular value matrix ̿ . The coefficient matrix ′ after embedding the watermark by I-SVD is used on ̿ with the previous and . In this example, the initial values of the coefficient matrices for positions (1,1), (1,2), and (1, 3) are 4.1277, 4.0456, and 3.5664, respectively. And they change to 4.2753, 4.0236 and 3.4439 after embedding the watermark. It can be seen that the impact of the watermark information on the original coefficient matrix is minimal, and all the coefficients have changed to a certain extent, which increases the imperceptibility of the watermark and reduces the blockiness of the image after embedding the watermark.
Example 2 shows how to extract watermark information from the coefficient matrix after the watermark is embedded. For the extracting process, the secret key, the embedding position and the embedding strength of the watermark information are indispensable. Therefore, the illegal extractor often lacks the proper data information and cannot extract the information correctly ′ ̿ is a singular value matrix obtained by performing SVD on ′ and multiplied by ̿ and ̿ obtained ′′ ̿̿̿̿ when the watermark is embedded. It can be seen, without the watermarked image being attacked, all the values in ′′ ̿̿̿̿ are equal to the value of ̅ and no change has taken place. ̿ in Example 1 is subtracted by ′′ ̿̿̿̿ and divided by the embedded intensity matrix to obtain the encrypted watermark ′ .
is the watermark of the inverse of encrypted watermark, and the matrix coefficient is completely equal to the embedded watermark . In attack simulation, may be an arbitrary floating-point number. In this research, we take a floating-point number of ≥0.5 as an integer 1, and a floating-point number of <0.5 as an integer 0.

Experimental results
The scheme was tested on Matlab R2016a and a computer with i7-4710MQ and 8GB of memory. The experimental results are presented and analyzed in this section. Eight nature images are selected for discussion in this section, which are Airplane, Baboon, Boat, House, Lena, Man, Peppers and Sailboat. Fig. 5 shows the cover images and a binary image used as a watermark image. The size of the cover image and the watermark image are 512 × 512 and 128 × 128, respectively. The watermark embedding parameters of our algorithm are =1, =3 and =0.12.

The imperceptibility test of proposed scheme
One of the most commonly used objective evaluation criterions to evaluate the image quality is peak signal to noise ratio (PSNR). The other one is SSIM, this serves as a subjective evaluation standard for images based on HVS. In this study, the imperceptibility of the watermarking scheme is discussed in combination with the above two evaluation criteria.
PSNR considers pixel-level errors and can be defined as equation (26)(27): Where and ′ represent the original image and the image after embedding the watermark, the size is × .
SSIM reflects people's subjective feelings relative to PSNR, and similarity evaluation is performed on three aspects of two images: brightness, contrast and structure. Its equation is as equation (28):
In most cases, if the PSNR value is not less than 40 dB and the SSIM value is not lower than 0.9, it indicates that there is no obvious difference between the two images. Fig. 6 shows that the PSNR value is larger than 45 dB and the SSIM value is not lower than 0.99. Therefore, Human beings cannot perceive changes in digital images through the naked eye. It indicates the proposed scheme satisfies the requirement of the imperceptibility. In addition, we compared the histograms of the four original cover images with the watermarked images (Fig. 7), the results prove that the grayscale distribution of the two images is similar. In summary, we can clearly conclude that the watermarking scheme has good imperceptibility.

The robustness test of proposed scheme
When the watermarked image is transmitted over the Internet, the watermark may be changed under general attacks and geometric attacks. Noise, filtering, JPEG compression, and histogram equalization are general attacks, while geometric attacks include scaling, rotation, clipping, and panning. The robustness of the scheme under various attacks is measured by the normalized correlation coefficient (NC) and the bit error rate (BER). The high NC and low BER represent the high similarity between the original watermark information and the extracted watermark information. Where NC can be calculated by equation (29): Where and ′ denote the original watermark information and the extracted watermark information, respectively. Both of their size is × . When the NC value is closer to 1, the two images are more similar. Fig. 8 shows the NC values of eight images under general attacks and geometric attacks.
The equation for BER is as equation (30): Where ⨂ represents an exclusive OR operation, and ′ represent the original watermark and the extracted watermark of size × , respectively. When the BER value is closer to 0, the similarity between the two watermark pictures is higher. Table  2 shows the watermarked image of Lena after 25 attacks and the extracted watermark, as well as its BER and NC values.  The process of data transmission is inevitably affected by noise. The noise may be caused by the source sensor equipment or added artificially. As a result, the ability to resist noise attacks is a basic requirement for a robust digital watermarking scheme.
Common noises include Gaussian noise, salt and pepper noise, and speckle noise. Salt and pepper noise, also known as impulse noise, affects the extraction of watermark information by randomly changing the partial pixel value of the image to black and white. Gaussian noise refers to a kind of noise whose probability density function obeys Gaussian distribution, and speckle noise is common in radar and sonar image processing. Fig. 9(a) shows the robustness of the scheme under noise attack. It can be seen from the Fig. 9(a) that the BER value of the watermark information extracted under the noise attack is not more than 0.04. That is, at most 4 bits out of 100 watermark bits are incorrect, so the human eye can easily distinguish the extracted watermark information.
In image processing, filtering is a common method for preprocessing images, it can eliminate image noise and achieve image smoothing and blurring. General filter pro-  Fig. 9(b).
Geometric attacks make it impossible to do relevant watermark detection or restore the embedded watermark by destroying the synchronization of the carrier data and the watermark. Common geometric attacks include zoom, rotate, cut, and pan. It's easy to know that the proposed scheme has good robustness under the resizing, rotation and translation attacks in Fig. 9(c). Moreover, the NC under cropping is still as high as 0.7, and the information contained in the watermark image can still be correctly recognized by the naked eye.
JPEG compression is the most vulnerable artificial attack when images are propagated in the network by removing redundant information from the image to affect the extraction of watermark. From the experimental results in Fig. 9(d), the NC values of the proposed scheme under histogram equalization, brightness reduction and JPEG compression are all above 0.98, which means that the scheme has good robustness against these attacks.

Experimental comparison
In order to prove the superiority of the proposed scheme in terms of robustness, several latest robust watermarking schemes [26,9,16,31] are selected for comparison.
The comparison results of NC values under various attacks is shown in Fig. 10 and Table 3, and the cover image is Lena. As can be seen from Table 3, the robustness in JPEG compression, Gaussian noise, salt and pepper noise, speckle noise, median filtering 3×3, Gaussian filtering, resizing (0.5,2) is significantly better than other schemes.
Moreover, the NC value is not lower than 0.95 under most attacks, which illustrates that the overall robustness of the scheme is very good, and it can meet the robustness requirements under most conditions.

Conclusion
In this paper, a robust entropy-based image watermarking scheme in shearlet and wavelet domain is proposed. The multi-scale analysis method FFST is used to decompose the cover image, and the sub-band with the maximum entropy value is selected for DWT transformation, thus to improve the imperceptibility. By changing the coefficient of singular value matrix, the watermark is embedded to improve its ability to resist geometric attacks. Adaptive embedding position and embedding strength is optimized to make the watermarking scheme achieve "optimal". Information entropy is adopted as the criterion for embedding position selection, and the dynamic embedding intensity is obtained for each image block by combining the JND model of wavelet domain, which balances the imperceptibility and the robustness.

Availability of data and material
The datasets used or analysed during the current study are available from the corresponding author on reasonable request.