Texture Directionality-Based Digital Watermarking in Nonsubsample Shearlet Domain

. Digital watermarking is a technique used to protect an author’s copyright and has become widespread due to the rapid development of multimedia technologies. In this paper, a novel watermarking algorithm using the nonsubsample shearlet transform is proposed, which combines the directional edge features of an image. A shearlet provides an optimal multiresolution and multidirectional representation of an image based on distributed discontinuities such as edges, which ensures that the embedded watermark does not blur the image. In the proposed algorithm, the nonsubsample shearlet transform is used to decompose the cover image into directionalsubbands,wheredifferentdirectionalsubbandsrepresentdifferentdirectionalandtexturedfeatures.Thesubbandwhose texturedirectionalityisstrongestisselectedtocarrythewatermarkandisthussuitableforthehumanvisualsystem.Next,singular valuedecompositionisperformedontheselectedsubbandimage.Finally,thewatermarkisembeddedinthesingularvaluematrix, whichisbeneficialforthewatermarkingrobustnessandinvisibility.Incomparisonwithrelatedwatermarkingalgorithmsbasedon discretewavelettransformsandnonsubsamplecontourlettransformdomains,experimentalresultsdemonstratethattheproposed schemeishighlyrobustagainstscaling,cropping,andcompression.


Introduction
Given the rapid development of digital data and associated technologies, copyright protection of multimedia has become a challenging problem.In order to protect an author's ownership, the method of digital watermarking (a digital data hiding technique) provides an effective solution.A good digital watermarking algorithm must ensure that (1) the embedded watermark is visually imperceptible, (2) the reconstructed watermarked image has good visual quality, and (3) the watermark is resilient to attacks.In other words, a watermarking technique should resist watermark degradation due to common signal processing and geometric attacks.
A watermarking algorithm is regarded optimal if the scheme can balance the quality index of watermark robustness, imperceptibility, and capacity as proposed by Fridrich [1].Watermarking can be performed in either the spatial domain (e.g., least significant bit) [2] or the transform domain (e.g., discrete Fourier transform (DFT) or discrete cosine transform (DCT)) [3,4].Moreover, the discrete wavelet transform (DWT) [5,6] has the ability to capture 1D signal characteristics at different scales or resolutions, which makes the method widely applicable for image processing applications such as denoising and compression.However, DWT is unable to effectively represent multidimensional signals, and its limited directional filtering restricts watermarking capacity.
In order to overcome the limitations of the wavelet transform, several multiscale geometric analysis (MGA) tools such as ridgelet, directionlet, and contourlet transforms have been developed and successfully applied in image processing applications [7][8][9][10][11][12].Shearlet, a multiresolution method, provides a tremendously effective representation of images containing edges [13,14] and is widely used in watermarking algorithms [15][16][17][18].Spread spectrum image watermarking in the discrete shearlet domain [15] employs the multiresolution 2 Mathematical Problems in Engineering representation characteristics of the DST to facilitate texture modeling for a given image.Zhao et al. [16] proposed a watermarking scheme based on the extended shearlet, where the watermark was embedded into the subband with the largest information entropy.Based on previous studies [17], a watermarking algorithm was proposed using the shearlet transform and bidiagonal singular value to achieve balance between robustness and imperceptibility.
In this paper, a nonsubsample shearlet transform (NSST), along with a combination of Laplace and shear filters, is used to decompose an image into different scales and directional subbands.The watermark can then be embedded into a special directional subband that achieves the best result.The proposed watermarking scheme has high robustness and imperceptibility depending on texture directionality in the NSST domain.The texture of an image with strong directionality reflects the contours and directional edges of the image, and the directional filter bank (DFB) of the NSST can effectively capture high-frequency content.The selection of a subband with strong texture directionality plays a significant role in the embedding and extraction steps of the watermarking algorithm.
The rest of this paper is organized as follows.Basic shearlet theory and related data are presented in Section 2. The proposed novel watermarking embedding and detection scheme is presented in Section 3. Experimental results are provided and discussed in Section 4, while important findings and conclusions are summarized in Section 5.

Correlation with Basic Theory
2.1.Shearlet Transform.Shearlets, or composite wavelets, provide an effective way to achieve a sparse directional image representation based on affine transformations and a combination of geometry and multiple scales.For dimension  = 2, the affine system with composite dilations is described by where  ∈  2 ( 2 ), A and B are 2 × 2 invertible matrices, and | det | = 1.When A is the set   = (, 0; 0, √) ( > 0), it is related to the scale transform, and the shear matrices   = (1, ; 0, 1) are related to the area-preserving geometrical transform.The transform is described as [14] We select  such that where ψ1 is a continuous wavelet for which ψ1 ∈  ∞ () Sampling SH() on the discrete set of , , and  is done as follows [13,16]: The associated discrete shearlet transform is described by As observed in Figure 2, image directionality is strongly apparent after the shearlet transform.Different directional subbands represent the directional information related to different image details.Finally, the subband with the maximum texture directionality is selected for watermark embedding.

Singular Value Decomposition (SVD)
. Singular value decomposition (SVD) is a numerical statistical analysis tool that is used to effectively deal with matrices.Assume that an   image is a nonnegative matrix.The singular value decomposition of matrix AA is defined as where  ∈   *  and  ∈   *  are orthogonal matrices,  ∈   *  is a diagonal matrix, and   is the transpose of V.The diagonal elements of S satisfy where  is the rank of AA and is equal to the number of nonnegative singular values.An important property of SVD is that the singular values (SVs) of an image possess significant stability because these values do not vary rapidly with small image perturbations.Therefore, a watermark embedded in a singular matrix exhibits high robustness [19][20][21][22][23].

Proposed Watermarking Scheme
3.1.Texture Directionality.Texture is a property that exists in many areas.In the scope of image processing, texture denotes a trend or pattern that reflects a change in the intensity and direction of image pixels.Directionality is one of the most significant characteristic features.This paper proposes a method based on statistical measurements for calculating image directionality from the directional histogram [24,25].
For a given image pixel, if the outputs of the horizontal and vertical edge operators are Δ and Δ, respectively, then the corresponding gradient vector Δ is Thus, the modulus |Δ| and angle  of the vector are obtained.The direction edge histogram   is then calculated by quantizing  and counting the ratio of effective pixels.A direction edge histogram for the Lena test image is shown in Figure 3(a).
Assuming that the variation trend denotes  peaks in the histogram, Figure 3(b) shows one of these peaks.Let   be a collection of bins from the previous valley  1 to the next valley  2 for each peak , and let   be the angle position of the peak.  () is the bin height at angle position .Texture directionality is calculated as [24] The image texture direction is computed as a numerical value based on the above description.Figure 4 shows reconstructed images for eight shearlet direction coefficients, and the corresponding texture directionality values are shown in Table 1.The fourth subband with the largest value represents strong directionality and is selected to carry the watermark.The texture of an image with strong directionality reflects the contours and directional edges, and the directional filter bank of NSST can effectively capture high-frequency content.Further, due to its scale-invariant nature, stability does not change with scaling, rotation, or noise attacks.Thus, this property is an advantage for watermark hiding and robustness.

Proposed Watermarking Algorithm.
Consider that the cover (host) image is a grayscale image (, ) with size ×, while the watermark is a binary image W with size × and elements (, ).The selection of a suitable location, effective coefficients to embed, and the embedding mechanism is of crucial importance in any watermarking scheme.In the proposed scheme, we select the eighth directional subband of level 3 with maximum directionality.This suggests that the subband has significant values to indicate the presence of directional edges, which contribute to watermark imperceptibility.Details on the embedding and extraction process are provided below.

Watermark Embedding
Step 1.In order to increase watermarking security, the encrypted watermark  * is obtained through the Arnold scrambling transform: where key represents scrambling times and is held to the decrypted watermark as the secret key.
Step  Step 3. The subband with maximum texture directionality is divided into nonoverlapping blocks of size 64 × 64, equal to the size of the watermark.
Step 4. SVD is performed on the block that has the largest energy  8 , and the singular value matrix  of size 64 × 64 is obtained to achieve watermark embedding based on the rule where  represents the singular value matrix containing watermarking information and  is the embedding factor used to control imperceptibility.With an increasing value of , the amount of the embedded watermark is increased, which means that the values of SSIM (see ( 14)) become larger and that the distortion is more pronounced.Table 2 shows the relationship between values of  and the watermark imperceptibility in terms of SSIM.
Step 5.The watermarked image   is obtained by the recombination of modified singular values and inverse NSST.

Watermark Extraction.
Watermark extraction is the inverse process of watermark embedding.Note that the original image is used here to extract the watermark.The steps of watermark extraction are provided below.
Step 1.A 2-level NSST is applied to the watermarked image.
Step 2. The subband with maximum directionality is divided into nonoverlapping blocks.
Step 3. SVD is applied to the block with the largest energy   , and the singular value matrix   is obtained.
Step 4. The singular value matrix  is used to acquire the encrypted watermark Ŵ * as follows: Step 5.The extracted watermark   is obtained by the inverse Arnold transform in combination with key:

Experimental Analysis
A large number of varied test images that reflect different scenes were used to test the effectiveness of the scheme.Lena, Snow, Snow2, Plane, Baboon, Woman, Lake, Crowd, Peppers, Scenery, Man, and Bridge (each with size 512 × 512) were selected as host images and a binary logo with size 64 × 64 was selected as the watermark (Figure 5).The "pkva" filters of NSST are used with the number of directional subbands set as 8 and 16.Also, the values of  and the secret key are set to 0.7 and 10, respectively, after numerous experimental trials.Structural similarity index measurement (SSIM) is used to measure visual quality and watermark robustness is assessed by normalized correlation (NC) and bit error rate (BER).
where   and   are the averages of  and , where (, ) represents the elements of the original watermark and   (, ) represents the elements of the extracted watermark.
where  is the number of erroneously detected bits for a × watermark image.

Watermarking Invisibility.
The performance of the proposed algorithm was tested under no attacks.Watermark imperceptibility in terms of SSIM is shown in Table 3. SSIM values between the host and watermarked images for the proposed algorithm are greater than 0.99 and reached 1 in some images, which shows good watermark invisibility.Further, the watermark can be extracted with little distortion.NC values of the extracted watermarks are greater than 0.99 and BER values are near 0, which illustrates effective watermark extraction.3 shows the values of SSIM, NC, and BER for scaling attacks with scaling parameters 1/4, 1/2, 2, and 4. Figure 6 shows the extracted watermark from the attacked images (Snow, Plane, and Crowd) for a scaling parameter of 1/2.

Watermarking
(a) For these test images, the extracted watermarks are clearly distinguishable.NC is greater than 0.98 and average BER is 0.8.In comparison with the other images, the result for Crowd exhibits limited robustness, but its NC value reaches 0.96, and the watermark can be extracted clearly.Therefore, the proposed algorithm can resist scaling attacks.

Robustness to JPEG Compression Attacks. JPEG compression is a commonly used attack in image processing.
Table 5 shows the values of NC and BER for JPEG compression with different quality factors.Results suggest that, irrespective of compression factor, NC values are greater than 0.94.Therefore, the proposed algorithm is robust against   JPEG compression.Three perfectly extracted watermarks from a JPEG attack (quality factor  = 40) are shown in Figure 7.

Robustness to Noise Attacks.
In order to evaluate the robustness of the proposed scheme against noise, every watermarked image is attacked by "Gaussian" and "Salt and Pepper" noise with variances of 0.001, 0.005, 0.01, and 0.05.Results are shown in Tables 6 and 7.For a "Gaussian" noise attack, noise points are found to appear in the extracted watermarks when the noise variance increases.However, the extracted watermark is still distinguishable in all test images.Similarly, for a "Salt and Pepper" noise attack, the algorithm shows good robustness.BER is 0 for Woman and Lake with a noise variance of 0.001, which illustrates tremendous robustness to noise attacks.Figures 8 and 9 show watermarks extracted from images attacked using "Gaussian" and "Salt and Pepper" noise with a variance of 0.001.

Conclusions
A novel watermarking algorithm based on texture directionality was proposed by combining the advantages of the nonsubsample shearlet and singular value decomposition.The strongest directionality subband, which provides the most significant texture information of the image, was selected for embedding the watermark since it preserves perceptual image quality.The proposed scheme is found to exhibit noteworthy robustness for most image processing attacks such as noise, rotation, cropping, and filtering.A comparison with two existing hybrid watermarking techniques showed better performance of the proposed method for many of the tested attacks.

Figure 1 :Figure 2 :
Figure 1: Diagram of shearlet transform: (a) illustration of succession of Laplacian and directional filtering and (b) example of frequencies by the shearlet transform.

Figure 3 :
Figure 3: Directionality of texture: (a) direction edge histogram of Lena and (b) a peak in the edge histogram.

Figure 12 :
Figure 12: Comparison of NC for four algorithms on Lena image.

Figure 13 :
Figure 13: Comparison of NC for four algorithms on Baboon image.

Table 2 :
The relationship between  and watermarking imperceptibility.
2  and  2  are the variances of  and , and   is the covariance of  and .

Table 3 :
Experimental results of invisibility.

Table 4 :
Experimental results for the scaling attack.

Table 5 :
Experimental results for the JPEG compression attack.

Table 6 :
Experimental results for the "Gaussian" white noise attack.

Table 7 :
Experimental results for the "Salt and Pepper" noise attack.

Table 10 :
Comparison of NC values on Lena image.

Table 11 :
Comparison of NC values on Baboon image.