Image Watermark Based on Extended Shearlet and Insertion Using the Largest Information Entropy on Horizontal Cone

Extended discrete shearlet provides a directional multiresolution decomposition. It has been mathematically shown that extended discrete shearlet is a more efficient representation for the signals containing distributed discontinuities such as edges, compared to discrete wavelet.Multiresolution analyses such as curvelet and ridgelet share similar properties, yet their directional representations are significantly different from that of extended discrete shearlet. Taking advantage of the unique properties of directional representation of extended discrete shearlet, we develop an image watermark algorithm based on the largest information entropy. In proposed algorithm, firstly, 1-level extended discrete shearlet transform decomposes the test image into directional components on horizontal cone; each directional component reflects directional features and textured features differently. Next, the directional component whose information entropy is the highest is selected to carry watermark. Compared with related algorithms based on DWT and DCT, the proposed algorithm tends to obtain preferable invisibility when it is robust against common attacks.


Introduction
Image watermark is one of the most active and challenging subjects in the information hiding research because it is an efficient solution to protect the copyright of the digital media.Wavelets have been widely applied in image watermark owning to their perfect performance for piecewise smooth signals in one dimension [1,2].
However, wavelets are not actually optimal representation for signals in two dimensions or higher dimensions such as images [3][4][5][6][7].In fact, wavelets not only fail to preserve the detailed features but they also fail to capture the directional features [3,7].With the development of multiresolution analysis, various multiresolution methods have been proposed to overcome this limitation of wavelets, and it has been experimentally proved that these multiresolution methods perform well in image watermark [4,5,8].Among various multiresolution methods, discrete shearlet [6,9] is a relatively new multiresolution method, which has drawn considerable attention and has been applied in some fields of image processing such as image denoising, image approximations, and image inpainting for its superior performance in dealing with signals in high dimensions [6,[9][10][11].Extended discrete shearlet [9,11] is derived from discrete shearlet transform.It is based on a simple but rigorous mathematical framework which is composed of directional orthonormal bases; each orthonormal base exhibits high directional sensitivity and is highly anisotropic.The extended discrete shearlet transform attempts to obtain a sparse image representation which contains more directional information and preserves more detailed information than discrete wavelet.
Taking advantage of these unique properties of extended discrete shearlet, we develop an image watermark algorithm based on extended discrete shearlet transform and the largest information entropy [12,13].In proposed algorithm, by using extended discrete shearlet transform, an image is composed into directional components on the horizontal cone; each directional component is transformed into a set of shearlet coefficients.The watermark is embedded into the largest coefficients of the directional component whose information entropy is the highest.The proposed algorithm is compared with DCT-based algorithm and DWT-based algorithm.

Correlation with Shearlet
Transform.A 2D affine system with composite dilations [6,9,14] is described by where the parabolic scaling matrices   = (, 0; 0, √) ( > 0) are associated with scale transform and the shear matrices   = (1, ; 0, 1) are related to area-preserving geometrical transform.The generating function  is chosen like a tensor product by where  1 is continuous wavelet which is supported with and  2 is a bump function with certain weak additional properties and is supported with The associated continuous shearlet transform [6,[9][10][11]14] is defined by SH  (, , ) = ⟨,  ,, ⟩ . ( The shearlet transform is a function associated with , , and , whereas wavelet transform is a function which is merely associated with  and .The main idea of the shear parameter  is to capture directional features, such as orientations of curves [6,9,11].As a consequence, the shearlet is able to capture directional information.Figure 1 illustrates the ability of capturing directions of the shearlet transform.

Correlation with Extended Discrete Shearlet.
Extended discrete shearlet [6,11] is obtained by directly extending discrete shearlet.Yet its framework is slightly different from that of the discrete shearlet.Framework of the extended discrete shearlet is composed of directional orthonormal bases; each orthonormal basis provides anisotropic window functions elongated along the direction determined by shear matrix.In addition, the extended discrete shearlet framework uses a MRA approach to compute shearlet coefficients and uses an iterative method to approximate the inverse.Based on this construction, a given image is explicitly decomposed into variously directional components for different cones when extended discrete shearlet is associated with different shear matrices and direction numbers; each directional component reflects directional features and textured features differently (see Figure 2); besides, a stable reconstruction of a given image is obtained.
Actually, the extended discrete shearlet framework exhibits highly directional sensitivity; and, furthermore, the extended discrete shearlet framework is optimally sparse representation and is well localized in both the spatial and the frequency domains.Based on these properties, the extended discrete shearlet can be considered as an appropriate transform for image watermark.

A Novel Watermark Algorithm Based on Extended Discrete Shearlet
The most important issue in extended discrete shearletbased watermark algorithm is how to choose the watermark embedding location or the effective coefficients to be embedded, since the choice of watermark embedding location has an effect on both invisibility and robustness of watermark algorithm to some extent [1,2,4,5,8].In this section, a novel algorithm for image watermark is proposed, which is based on combining the analysis of extended discrete shearlet coefficients [6,11] and the basic theory of the largest information entropy to search the watermark embedding location for different images.Each directional component reflects directional features and preserved textured features differently; therefore, the occurring probabilities of different pixel values in each directional component are different from those of all others and the probability distribution of each directional component's information is different from that of all others in confusion degree.The higher information entropy [12,13] of the directional component is, the more uncertain occurring probabilities of its different pixel values are, and probability distribution of its information is more disordered and more chaotic; that means it can hide more watermark bits.Based on this analysis, the directional component whose information entropy is the largest is selected to carry watermark.The information entropy of the th directional component on horizontal cone is computed as follows: ( (3) Calculate the information entropy of the th directional component by where  ( = 1, . . ., ).
Generally, watermark insertion is realized by modifying some special values of pixels or transforming domain coefficients [1,2,4,5].Here watermark insertion is realized by modifying the largest  ×  coefficients of selected directional component in extended discrete shearlet domain.

Watermark Insertion Using the Largest Information
Entropy.For an image  used as watermark, in order to uniformly and randomly spread bits over the host image, it needs to be transformed into a binary image by Here threshold Tr is selected by OSTA method.After processing the watermark image , a binary image  is obtained and then reshaped to binary sequence  with length  × .According to the above analysis in Section 3.1, the directional component, whose information entropy is the highest, can hide more watermark bits and is selected to carry watermark.To obtain directional components on horizontal cone, a l-level extended discrete shearlet decomposition is applied to test image  with size  × , which is associated with shear matrix  0 and direction numbers .Then, to find out the directional component whose information entropy is the highest, information entropy of each directional component on horizontal cone is computed by (9).
Here we suppose the th (1 ≤  ≤ ) directional component's information entropy is the highest.Then the binary sequence  is embedded into the largest  ×  coefficients of the th directional component by where 1 ≤  ≤  × ,  is used to control embedding strength, and  is the threshold.Finally, the watermarked image   is obtained after 1-level extended inverse discrete shearlet transform by using these modified shearlet coefficients.

Watermark Extraction.
The watermark can be extracted correctly from the watermarked image without original test image.The extracting process is as follows, which is mirroring embedding process.
To extract embedded watermark bits from the watermarked image, 1-level extended discrete shearlet decomposition is firstly applied to the watermarked image, where the shear matrix is selected as  0 and the number of direction is ; then shearlet coefficients of every directional component on horizontal cone are obtained.
Then, the watermarked coefficients are found out according to watermark size and the location key of embedding watermark.The extracted watermark bits are recovered into a sequence ⃗  by where 1 ≤ ,  ≤ , 1 ≤  ≤  × , and  is the threshold.Finally, the extracted watermark   is recovered by reshaping ⃗ .

Invisibility of Watermark.
Invisibility is an evaluative measure of perceptual quality of the watermarked image.In a satisfactory image watermark algorithm, watermark should not cause much degradation of perceptual quality of the watermarked image.In the proposed algorithm, a 64 × 64 watermark image is embedded into different test images ("Lena, " "Boat, " and "Barbara" with 512 × 512) to test invisibility, where the number of directions is 9 and the embedding strength is 0.1.As shown in Table 1, there are not much visual differences between original test images and their corresponding watermarked images.In addition, the PSNR value of each watermarked image is above 53 dB, and NC value of corresponding extracted watermark image is above 0.98.The extracted watermarks are all easily distinguishable.Whether from a subjective view or from the objective view, the proposed algorithm has preferable invisibility.
Furthermore, by analyzing the absolute difference between the test image and the watermarked image, it is not difficult to find that the watermark tends to spread on these regions with strong edges and extreme brightness.In particular, it is noted that the watermarked image is magnified a hundred times.

Robustness of the Proposed Algorithm.
Robustness is a default measure which is used to evaluate the watermarked image's resistance against attacks.In a satisfactory algorithm for image watermark, the watermark would not be easily

Robustness to JPEG Compression
Attack.JPEG lossy compression is the most common attack during the transformation of the watermarked image.In order to evaluate robustness of proposed algorithm, every watermarked image is compressed using JPEG with different quality factors.From extracted watermarks in Table 3, it is not hard for anyone to observe that extracted watermark is still distinguishable until the quality factor is set to 30.In addition, for all test images, PSNR values are all above 30 dB; NC values are all above 0.8.Therefore, the proposed algorithm is very robust against JPEG compression.

Robustness to Filtering Attack.
As shown in Table 4, whether Gaussian low-pass filtering, average filtering, or median filtering, the extracted watermarks are still distinguishable for the "Lena" image and the "Barbara" image.In addition, NC values are all above 0.855 for the "Lena" image, 0.77 for the "Boat" image, and 0.86 for the "Barbara" image.However, the extracted watermarks are imperfect for the "Boat" image.

Robustness under Various Noises Attack.
The watermarked image is easily disturbed by various noises.In the test, we add "Salt and Pepper" (Table 5) noise and "Gaussian" white noise (Table 6) to the watermarked images.For the watermarked image with "Salt and Pepper" noise, with the increase of noise attack strength, more and more noise points appear in the extracted watermark, but the extracted watermark is still distinguishable and complete until the attack strength is set to 0.05.When the attack strength is set to 0.05, the extracted watermark is not perfect but distinguishable; the NC value is 0.8262 for the Lena image, 0.7866 for the Boat image, and 0.7861 for the Barbara image, respectively.
Similarly, for the watermarked image under the attack of "Gaussian" white noise, when the noise mean is set to zero, noise points increasingly appear in extracted watermark with the increase of noise variance.The extracted watermarks are  The extracted watermark still distinguishable and perfect for all test images until the noise variance is set to 0.0005.When the noise variance is set to 0.0005 and 0.001, the extracted watermark is incomplete but distinguishable for all test images; the NC values are all above 0.8.Therefore, the proposed algorithm can resist noise attack to some extent.

The Comparison Experiments of Related Algorithm.
To verify the effectiveness of the proposed algorithm, the proposed algorithm is compared with related algorithms based on DWT and DCT.DCT-based watermark algorithm is performed by embedding watermark in the largest  ×  coefficients of the first-level DCT subband.Watermarking based on DWT is performed by embedding watermark in the largest  ×  coefficients of the first-level subband, where a one-level DWT decomposition is performed using "Haar" wavelet.
To be fair, both the watermark image and the test images are the same to all algorithms.Table 7 shows the results of comparison on "Lena" image.As shown in Table 7, the proposed algorithm outperforms other algorithms in invisibility.Meanwhile, in robustness against these attacks, such as scaling attack and cropping attack, the proposed algorithm gets advantage over other algorithms.
In Table 2, although the NC values of proposed algorithm are not higher than other algorithms except no attack and scaling attack, it is noted that PSNR values of proposed algorithm are higher than other algorithms except cropping middle (50%).In other words, watermark which is embedded in host image cause much degradation of perceptual quality of the watermarked image, yet the quality of the watermarked image for proposed algorithm is better.

Conclusion
This paper tries to implement an image watermark by taking advantage of the unique advantages of extended discrete shearlet and selecting watermark embedding location based on the highest information entropy.The unique advantages of extended discrete shearlet made the watermark system more invisible.Experimental results demonstrate that visual performance of the watermarked images is preferable to that of related algorithms based on DWT and DCT; meanwhile, the proposed watermark algorithm can resist common attack.Therefore, the extended discrete shearlet can be considered as an appropriate transform for digital watermark.

3. 1 .
Analysis of Extended Discrete Shearlet Coefficients.After 1-level extended discrete shearlet decomposition associated with shear matrix  0 and direction numbers , the test image of size  ×  is decomposed into  directional components on horizontal cone (see Figure 2(b)); each of directional components is transformed into a set of shearlet coefficients.

Figure 2 :
Figure 2: The extended discrete shearlet transform results corresponding to different directions and shear matrices.

Table 1 :
The experimental results of invisibility.
removed from the watermarked image after common and deliberate attacks.In this paper, we simulate the real situation of disguised attacks and check out robustness for the proposed algorithm, where the number of orientations is 9 and the embedding strength is 0.1.

Table 2 :
Experimental results under the scaling attack.

Table 3 :
Experimental results under the attack of JPEG lossy Compression.

Table 4 :
Experimental results under filtering attack.

Table 5 :
Experimental results under the attack of "Salt and Pepper" noise.

Table 6 :
Experimental results under the attack of "Gaussian" white noise (mean = 0).