An Adaptive Embedding Strength Watermarking Algorithm Based on Shearlets’ Capture Directional Features

The discrete wavelet transform (DWT) is unable to represent the directional features of an image. Similarly, a fixed embedding strength is not able to establish an ideal balance between imperceptibility and robustness of a watermarked image. In this work, we propose an adaptive embedding strength watermarking algorithm based on shearlets’ capture directional features (S-AES). We improve the watermarking algorithm in the domain of DWT using non-subsampled shearlet transform (NSST). The improvement is made in terms of coping with anti-geometric attacks. The embedding strength is optimized by artificial bee colony (ABC) to achieve higher robustness under the premise of satisfying imperceptibility. The principle components (PC) of the watermark are embedded into the host image to overcome the false positive problem. The simulation results show that the proposed algorithm has better imperceptibility and strong robustness against multi-attacks, especially those of high intensity.


Introduction
With the development of the Internet, the amount of image data being transmitted has significantly increased. These images are easily copied, modified, subsequently infringing the copyright of the author. Digital image watermarking [1,2] is a technique to protect images' ownership. Watermarking with strong robustness is embedded into the image to protect the ownership of the image. The research on robust watermarking has achieved good research results. Most algorithms presented in the literature have the ability to resist common attacks effectively. In addition, these algorithms also have resistance against geometric attacks, such as rotation and translation attacks. However, the extracted watermarking becomes worse with the increase in the attacks' intensity in many existing algorithms. Therefore, making significant improvements in the resistance provided by robust watermarking against high intensity attacks is an important research problem.
The embedding strength of the digital watermarking algorithm is closely related to robustness and imperceptibility. The larger the value of embedding strength, the better the robustness and the worse the imperceptibility. On the other hand, a smaller value of embedding strength leads to better imperceptibility and worsens robustness [3,4]. Therefore, the appropriate embedding strength must be selected appropriately to maximize the robustness of the algorithm to enable resistance against high intensity attacks. In recent years, researchers optimized embedding strength on the basis of the genetic algorithm (GA) [5], particle swarm optimization (PSO) [6], and artificial bee colony (ABC) [7]. Cui [8] uses the differential evolution algorithm to optimize the embedding strength, which involves 1.
The watermarking algorithm in the domain of DWT is improved by using shearlet transform. The embedding position is selected on the basis of DWT and NSST to improve the robustness. 2.
The ABC algorithm and the improved optimized function is used to optimize the embedding strength to achieve higher robustness. 3.
The principle components of the watermark are embedded into the host image to solve the false positive problem of singular value decomposition.
The rest of this paper is organized as follows: In Section 2, we present the preliminaries of this work. In Section 3, we present the proposed watermark embedding and extracting algorithm. In Section 4, we present the experimental results and analysis. Finally, in Section 5, we conclude our work.

Discrete Wavelet Transform (DWT)
DWT is a multiresolution analysis tool [25,26]. DWT decomposes an image into three high frequency sub-bands which are low-high (HL), high-low (LH) and high-high (HH) and one low frequency sub-band low-low (LL). Among these bands, LL contains most of the information presented by the original image. The other three sub-bands, i.e., HL, LH, HH, represent the texture features and the edge information of the image. Figure 1 presents the two-level DWT decomposition of "Lena" [27]. watermarking algorithm. Therefore, S-AES achieves maximum robustness against various attacks under the premise of satisfying the imperceptibility. S-AES also has the ability to resist many kinds of attacks, especially strong intensity attacks. The main contributions of this work are as follows: 1. The watermarking algorithm in the domain of DWT is improved by using shearlet transform. The embedding position is selected on the basis of DWT and NSST to improve the robustness. 2. The ABC algorithm and the improved optimized function is used to optimize the embedding strength to achieve higher robustness. 3. The principle components of the watermark are embedded into the host image to solve the false positive problem of singular value decomposition.
The rest of this paper is organized as follows: In Section 2, we present the preliminaries of this work. In Section 3, we present the proposed watermark embedding and extracting algorithm. In Section 4, we present the experimental results and analysis. Finally, in Section 5, we conclude our work.

Discrete Wavelet Transform (DWT)
DWT is a multiresolution analysis tool [25,26]. DWT decomposes an image into three high frequency sub-bands which are low-high (HL), high-low (LH) and high-high (HH) and one low frequency sub-band low-low (LL). Among these bands, LL contains most of the information presented by the original image. The other three sub-bands, i.e., HL, LH, HH, represent the texture features and the edge information of the image. Figure 1 presents the two-level DWT decomposition of "Lena" [27]. The low frequency sub-band has good stability to resist different types of attacks because it contains most of the information of the image. S-AES selects low frequency sub-band of two-level DWT decomposition as the embedding position of the watermark.

Non-Subsampled Shearlet Transform (NSST)
Shearlet transform is a multiscale analysis technique [28]. It has better multiresolution capabilities and stronger directional characteristics as compared to wavelet transform to represent the features of images or high-dimensional signals effectively. The non-subsampled shearlet transform is a discretization of shearlet transform. First, the image is multiscale decomposed by nonsubsampled pyramid filter (NSPF). Then, each scale sub-band image is multidirectional decomposed by shear filter (SF) [29]. Figure 2 presents the implementation of non-subsampled shearlet transform. The low frequency sub-band has good stability to resist different types of attacks because it contains most of the information of the image. S-AES selects low frequency sub-band of two-level DWT decomposition as the embedding position of the watermark.

Non-Subsampled Shearlet Transform (NSST)
Shearlet transform is a multiscale analysis technique [28]. It has better multiresolution capabilities and stronger directional characteristics as compared to wavelet transform to represent the features of images or high-dimensional signals effectively. The non-subsampled shearlet transform is a discretization of shearlet transform. First, the image is multiscale decomposed by non-subsampled pyramid filter (NSPF). Then, each scale sub-band image is multidirectional decomposed by shear filter (SF) [29]. Figure 2 presents the implementation of non-subsampled shearlet transform. Here, represents the original image. 1 and 1 are the high frequency and low frequency images after the first decomposition, respectively. Similarly, 2 and 2 represent the high frequency and the low frequency images after the second decomposition of the low frequency image, respectively. 2 is detail sub-band by directional decomposition of 2 . NSST decomposes an image into different directional sub-bands. Figure 3 presents the NSST decomposition of a zone plate. DWT does not possess the capability to represent the directional features of an image effectively. Contrarily, NSST is able to effectively capture the multidirectional features. Therefore, S-AES performs DWT and NSST on the image, and then selects the sub-band with the richest directional features as the embedding position to improve robustness.

Singular Value Decomposition (SVD)
Singular value decomposition is one of the useful numerical analysis tools in linear algebra, which can decompose an image into three matrices. For matrix A of size × , singular value decomposition is defined as Equation (1): Here, f represents the original image. f 1 d and f 1 a are the high frequency and low frequency images after the first decomposition, respectively. Similarly, f 2 d and f 2 a represent the high frequency and the low frequency images after the second decomposition of the low frequency image, respectively. f 2 s is detail sub-band by directional decomposition of f 2 d . NSST decomposes an image into different directional sub-bands. Figure 3 presents the NSST decomposition of a zone plate. Here, represents the original image. 1 and 1 are the high frequency and low frequency images after the first decomposition, respectively. Similarly, 2 and 2 represent the high frequency and the low frequency images after the second decomposition of the low frequency image, respectively. 2 is detail sub-band by directional decomposition of 2 . NSST decomposes an image into different directional sub-bands. Figure 3 presents the NSST decomposition of a zone plate. DWT does not possess the capability to represent the directional features of an image effectively. Contrarily, NSST is able to effectively capture the multidirectional features. Therefore, S-AES performs DWT and NSST on the image, and then selects the sub-band with the richest directional features as the embedding position to improve robustness.

Singular Value Decomposition (SVD)
Singular value decomposition is one of the useful numerical analysis tools in linear algebra, which can decompose an image into three matrices. For matrix A of size × , singular value decomposition is defined as Equation (1): DWT does not possess the capability to represent the directional features of an image effectively. Contrarily, NSST is able to effectively capture the multidirectional features. Therefore, S-AES performs DWT and NSST on the image, and then selects the sub-band with the richest directional features as the embedding position to improve robustness.

Singular Value Decomposition (SVD)
Singular value decomposition is one of the useful numerical analysis tools in linear algebra, which can decompose an image into three matrices. For matrix A of size M × M, singular value decomposition is defined as Equation (1): where U, V, and S are the real matrix of size N × N, represent the left singular matrix, the right singular matrix, and the singular matrix, respectively. S is a diagonal matrix, and the diagonal elements of S represent the image luminance. Matrix U and V represent the horizontal and vertical details of the image.
The singular values of the watermark are embedded into the host image to improve the resistance against various attacks because of its good stability. However, the matrices U and V contain the images' important information that the singular values are unable to represent. The watermarking algorithm using singular values will have the false position problem [30]. The principle components (PC) of the watermark are defined as Equation (2) [15]: The principal components represent the unique features of the image. Therefore, the principle components of the watermark are embedded into the host image to solve the false positive problem and improve the security in S-AES.

Artificial Bee Colony (ABC)
Artificial bee colony [15] is a global optimization algorithm proposed by Karaboga in 2005. This algorithm estimates the optimal value by maximizing or minimizing the objective function. ABC is widely used for its various advantages, such as fast convergence speed, less control parameters, and easy implementation. S-AES designs a reasonable objective function and uses ABC to find out the embedding strength.

The Proposed Scheme
In order to achieve higher robustness under the premise of meeting imperceptibility, an adaptive embedding strength watermarking algorithm based on shearlets' capture directional features (S-AES) is proposed. DWT and NSST are adopted to select the embedding position. ABC is used to optimize the embedding strength. Embedding the principle components of the watermark into the host image solves the false positive problem caused by SVD.

The Watermark Embedding Scheme
The watermark embedding process is presented in Figure 4. This process comprises seven steps, which are presented below. where , , and are the real matrix of size × , represent the left singular matrix, the right singular matrix, and the singular matrix, respectively. is a diagonal matrix, and the diagonal elements of represent the image luminance. Matrix and represent the horizontal and vertical details of the image.
The singular values of the watermark are embedded into the host image to improve the resistance against various attacks because of its good stability. However, the matrices and contain the images' important information that the singular values are unable to represent. The watermarking algorithm using singular values will have the false position problem [30]. The principle components (PC) of the watermark are defined as Equation (2) [15]: The principal components represent the unique features of the image. Therefore, the principle components of the watermark are embedded into the host image to solve the false positive problem and improve the security in S-AES.

Artificial Bee Colony (ABC)
Artificial bee colony [15] is a global optimization algorithm proposed by Karaboga in 2005. This algorithm estimates the optimal value by maximizing or minimizing the objective function. ABC is widely used for its various advantages, such as fast convergence speed, less control parameters, and easy implementation. S-AES designs a reasonable objective function and uses ABC to find out the embedding strength.

The Proposed Scheme
In order to achieve higher robustness under the premise of meeting imperceptibility, an adaptive embedding strength watermarking algorithm based on shearlets' capture directional features (S-AES) is proposed. DWT and NSST are adopted to select the embedding position. ABC is used to optimize the embedding strength. Embedding the principle components of the watermark into the host image solves the false positive problem caused by SVD.

The Watermark Embedding Scheme
The watermark embedding process is presented in Figure 4. This process comprises seven steps, which are presented below.  Step  Step 1 The color image of size N × N is transformed from RGB color space to YCbCr color space, which extracts I Y , I Cb , I Cr of the image, as demonstrated in Equation (3) [31]: According to human visual system, the human eyes are not sensitive to luminance components [32]. In order to guarantee better imperceptibility, S-AES embeds the watermark to luminance component I Y .
Step 2 Apply two-level DWT on I Y to obtain low frequency sub-band I LL as the embedding position. The low frequency sub-band contains most of the information to resist against different types of attacks to guarantee robustness.
Step 3 Apply NSST on low frequency sub-band I LL to obtain four different directional sub-bands. Calculate the entropy of these sub-bands, which is referred to as Equation (4): where n represents the number of gray levels and p(a i ) represents the probability of gray level a i of the image. The larger the values of entropy, the richer the directional features of the sub-band. The sub-band Subband max with the maximum entropy is selected to embed the watermark to ensure better imperceptibility against attacks.
Step 4 Apply k times Arnold transform on directional sub-band Subband max to obtain Subband, which is referred to as Equation (5): where, (x, y) represents the original coordinates of the pixel and (x , y ) represents the new coordinates of the transformed pixel. The size of directional sub-band is M × M. Arnold transformation breaks the spatial continuity between image pixels, which is conducive to the security of the algorithm.
Step 5 Apply SVD on scrambled directional sub-band Subband referred to as Equation (6): Apply SVD on watermark image Water, and extract principle components of the watermark PC water , as demonstrated in Equations (7) and (8): The principle components, which represent the unique features of the image, are embedded into the host image to avoid the false positive problem in the S-AES algorithm.
Step 6 Embed the PC of the watermark into the singular matrix of the host image, as demonstrated in Equation (9): where α is embedding strength which is optimized by ABC. Please note that it is necessary to apply SVD on PC mark to maintain good stability of the singular matrix because PC mark is no longer a diagonal matrix. In [3], the authors perform inverse SVD using the singular matrix embedded with PC (e.g., sPC mark ), which leads to the worse ability of anti-geometric attacks, such as rotation.
Step 7 Apply inverse SVD to obtain the directional sub-band Subband mark containing the watermark, as demonstrated in Equation (11): Apply k times inverse Arnold transform, inverse NSST and inverse DWT on Subband mark to obtain I Ymark . Merge I Ymark with I Cb , I Cr , then transform YCbCr to RGB to obtain the watermarked color image, i.e., WatermarkedImage, as demonstrated in Equation (12) [31]:

The Watermark Extracting Scheme
The watermark extracting process is presented in Figure 5. It contains three steps.

The Watermark Extracting Scheme
The watermark extracting process is presented in Figure 5. It contains three steps. Step 1 The watermarked image is transformed RGB to YCbCr color space to extract luminance component ′. Apply DWT and NSST on ′ to obtain the directional sub-band containing the watermark denoted as ′. Perform the same transformations on the original image to obtain the directional sub-band .
Step 2 Apply times Arnold transform on directional sub-bands ′ and , and then apply SVD on ′ and to obtain the singular matrices ′ and . Extract the embedding information as in Equation (13): Step 3 Extract the principle components of the watermark as in Equation (14): Then extract the watermark using the side information using the following Equation (15):

Optimization of Embedding Strength
Most of the algorithms presented in the literature consider the fixed value of embedding strength. However, different images have different characteristics in terms of color, texture, and some other characteristics. The fixed embedding strength is unable to realize the good imperceptibility and robustness on various kinds of images. S-AES considers the characteristics of the host image and the watermark to optimize the embedding strength using ABC.
Normalized correlation (NC) is used to evaluate the robustness, and peak signal-to-noise ratio Step 1 The watermarked image WatermarkedImage is transformed RGB to YCbCr color space to extract luminance component I Ymark . Apply DWT and NSST on I Ymark to obtain the directional sub-band containing the watermark denoted as Subband mark . Perform the same transformations on the original image to obtain the directional sub-band Subband.
Step 2 Apply k times Arnold transform on directional sub-bands Subband mark and Subband, and then apply SVD on Subband mark and Subband to obtain the singular matrices S mark and S. Extract the embedding information as in Equation (13): Step 3 Extract the principle components of the watermark as in Equation (14): Then extract the watermark using the side information using the following Equation (15):

Optimization of Embedding Strength
Most of the algorithms presented in the literature consider the fixed value of embedding strength. However, different images have different characteristics in terms of color, texture, and some other characteristics. The fixed embedding strength is unable to realize the good imperceptibility and robustness on various kinds of images. S-AES considers the characteristics of the host image and the watermark to optimize the embedding strength using ABC.
Normalized correlation (NC) is used to evaluate the robustness, and peak signal-to-noise ratio (PSNR) is used to evaluate imperceptibility, as demonstrated in Equations (16) and (17) [33]: PSNR(I, I ) = 10 log 10 255 2 where W and W represent the original watermark and the extracted watermark, respectively. I and I represent the host image and the watermarked image, respectively.
It makes sense to satisfy imperceptibility for the watermarking algorithms. The main idea of S-AES is to achieve higher robustness for multiple attacks under the premise of satisfying imperceptibility. N times different types and intensity attacks are applied to the watermarked image when the PSNR is above the threshold. The robustness of S-AES is maximized by maximizing the proposed objective function presented in Equation (18), where W is the original watermark, and W represents the extracted watermark.
The embedding strength optimization process is presented in Figure 6. The values of parameters are presented in Table 1.
where and ′ represent the original watermark and the extracted watermark, respectively. and ′ represent the host image and the watermarked image, respectively.
It makes sense to satisfy imperceptibility for the watermarking algorithms. The main idea of S-AES is to achieve higher robustness for multiple attacks under the premise of satisfying imperceptibility. N times different types and intensity attacks are applied to the watermarked image when the PSNR is above the threshold. The robustness of S-AES is maximized by maximizing the proposed objective function presented in Equation (18), where W is the original watermark, and W' represents the extracted watermark.
The embedding strength optimization process is presented in Figure 6. The values of parameters are presented in Table 1.

Results and Analysis
The experiments of S-AES are performed in MATLABR2017b, and they involve 10 host images and 2 watermark images that were selected from the standard image data-base. The size of each host image is 512 × 512, while that of the watermarked images is 128 × 128, as presented in Figure 7. The threshold value is 38 dB in this experiment. Figure 8 illustrates the convergence graph for the 10 host images embedding with the "UPC" watermark in Figure 7.

Results and Analysis
The experiments of S-AES are performed in MATLABR2017b, and they involve 10 host images and 2 watermark images that were selected from the standard image data-base. The size of each host image is 512 × 512, while that of the watermarked images is 128 × 128, as presented in Figure 7. The threshold value is 38 dB in this experiment. Figure 8 illustrates the convergence graph for the 10 host images embedding with the "UPC" watermark in Figure 7.

Imperceptibility Analysis
The imperceptibility results are presented in Table 2.

Results and Analysis
The experiments of S-AES are performed in MATLABR2017b, and they involve 10 host images and 2 watermark images that were selected from the standard image data-base. The size of each host image is 512 × 512, while that of the watermarked images is 128 × 128, as presented in Figure 7. The threshold value is 38 dB in this experiment. Figure 8 illustrates the convergence graph for the 10 host images embedding with the "UPC" watermark in Figure 7.

Imperceptibility Analysis
The imperceptibility results are presented in Table 2.

Imperceptibility Analysis
The imperceptibility results are presented in Table 2. According to the human visual system [34], the watermark has good imperceptibility when PSNR > 35 dB. As presented in Table 2, the PSNR values are about 38 dB, indicating that S-AES has good imperceptibility.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].
where B is the number of erroneous bits in the extracted watermark, and P × Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images "Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3.
In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. The BERs of the extracted watermark under different types and parameter attacks are shown in Table 5. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].

= ×
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].

= ×
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].

= ×
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Robustness Analysis
NC and the bit error ratio (BER) are used to evaluate the robustness of the proposed algorithm. The larger value of NC results in better robustness. The BER is an indicator to measure the difference between the extracted watermark and the original watermark. Its mathematical definition is presented in Equation (19) [35].

= ×
where B is the number of erroneous bits in the extracted watermark, and P×Q represents the total number of bits in the extracted watermark. The lower value of the BER shows greater robustness of the proposed algorithm.

Robustness Results
The extracted "UPC" and "Logo" watermarks from the host images '"Lena" and "Pepper" under crop, filer, and rotation attacks are presented in Table 3. Table 3. Extracted "UPC" and "Logo" watermarks from "Lena" and "Pepper" under different attacks.

Centre crop 256 × 256
Average filter (3,3) In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4. In Table 3, we can see that the extracted watermarks are slightly different from the original watermarks, and can represent the copyright information clearly. This indicates that S-AES has strong robustness under different attacks.
To illustrate the robustness of the algorithm accurately, the NC values are shown in Table 4.    According to the information presented in Tables 4 and 5, the NC value of the extracted watermark decreases and the BER increases with the increase in the intensity of the attack. The robustness of "Lena" is stronger than "Pepper" in most cases under the same attack. The NC values of the watermarks are mostly around 0.99 and the BERs are mostly around 0.01. The NC value is above 0.95 and the BER is below 0.05 even under high intensity attacks. The NC values of S-AES are always above 0.99 under crop, Gaussian filter, compression, and speckle noise attacks. However, they drop to 0.95 under an average filter attack with the parameter of (5,5). This is because S-AES uses NSST to capture the texture and directional features of the image. However, the watermark can be extracted successfully, indicating that S-AES has the capability to resist average filter attacks.

Optimization of Embedding Strength Comparison
We embed "UPC" into "Lena" using the embedding strength optimized by the proposed method and the method presented in [9]. We then perform eight types of attacks on watermarked images. The comparison of NC values is presented in Figure 9.
It is evident from Figure 9 that the robustness is stronger using the embedding strength optimized by the proposed objective function than [9]. Notably, the NC values decrease more slightly as compared to [9] when the attack intensity increases. Therefore, the proposed algorithm is more robust.

Optimization of Embedding Strength Comparison
We embed "UPC" into "Lena" using the embedding strength optimized by the proposed method and the method presented in [9]. We then perform eight types of attacks on watermarked images. The comparison of NC values is presented in Figure 9. It is evident from Figure 9 that the robustness is stronger using the embedding strength optimized by the proposed objective function than [9]. Notably, the NC values decrease more slightly as compared to [9] when the attack intensity increases. Therefore, the proposed algorithm is more robust.

Comparison of Robustness
In order to test the stronger robustness of S-AES, the NC values of the extracted watermark are compared with [9] under attacks with the same parameters. In addition, we also perform comparison with [13,15,17,21,22]. The results are presented in Table 6.

Comparison of Robustness
In order to test the stronger robustness of S-AES, the NC values of the extracted watermark are compared with [9] under attacks with the same parameters. In addition, we also perform comparison with [13,15,17,21,22]. The results are presented in Table 6. In Table 6, we can see that the NC values of S-AES are larger than [9] under the same attacks. S-AES has stronger robustness than other algorithms for compression and geometric attacks, such as crop and rescale. In most cases, S-AES has stronger robustness than other algorithms under translation attacks. The NC values of S-AES are slightly lower than [15] under the filter attack a larger window size. This is because S-AES uses NSST to select the embedding position.
Under the attacks with the same parameters, we compare the BER of S-AES with methods presented in [13,14,18,35]. The results are presented in Table 7.
It is evident from the comparison of the BER presented in Table 7 that the S-AES algorithm has better robustness.
We compared the NC values of the extracted watermark with [15,24] under multiple attacks. The results are shown in Table 8. According to Table 8, most of the NC values of S-AES are larger than those in [15] and [24]. Therefore, S-AES has the ability to resist multiple attacks effectively.
The aforementioned experimental results show that S-AES has the capacity to resist not only single attacks but also multiple attacks. Furthermore, S-AES has stronger robustness compared with the existing algorithms.

False Positive Problem
In order to test the false positive problem, the watermarked image "Lena" is embedded with the "UPC" watermark used as a disputed image. "Logo" is used as a false watermark. The extracted results are presented in Figure 10.

False Positive Problem
In order to test the false positive problem, the watermarked image "Lena" is embedded with the "UPC" watermark used as a disputed image. "Logo" is used as a false watermark. The extracted results are presented in Figure 10.  Figure 10 shows that the watermark cannot be extracted successfully using side information of the false watermark. This indicates that S-AES overcomes the false positive problem.
According to the aforementioned experiments, S-AES has good imperceptibility and strong robustness for single attacks and multiple attacks. It also overcomes the false positive problem.

Conclusions
In this work, we propose an adaptive embedding strength watermarking algorithm based on shearlets' capture directional features (S-AES). The multiscale and multidirectional characteristics of NSST are used to improve the watermarking algorithms in the domain of DWT. This enables the algorithm to resist common attacks and geometric attacks effectively. S-AES fully considers the watermarking characteristics of imperceptibility and robustness and optimizes the embedding strength using the proposed objective function, achieving maximum robustness under the premise of satisfying imperceptibility. As compared with other algorithms, S-AES has a greater ability to resist high intensity attacks. We embedded the principle components enclosing the unique features of the watermark into the host image to overcome the false positive problem. Even if someone provides a wrong singular matrix obtained from a false watermark, it cannot extract the watermark successfully using side information of the false watermark. The robustness of S-AES decreases slightly under average and median filter attacks with a larger window size. This can be improved in the future. In addition, S-AES can also be extended to the field of video watermarking.    Figure 10 shows that the watermark cannot be extracted successfully using side information of the false watermark. This indicates that S-AES overcomes the false positive problem.
According to the aforementioned experiments, S-AES has good imperceptibility and strong robustness for single attacks and multiple attacks. It also overcomes the false positive problem.

Conclusions
In this work, we propose an adaptive embedding strength watermarking algorithm based on shearlets' capture directional features (S-AES). The multiscale and multidirectional characteristics of NSST are used to improve the watermarking algorithms in the domain of DWT. This enables the algorithm to resist common attacks and geometric attacks effectively. S-AES fully considers the watermarking characteristics of imperceptibility and robustness and optimizes the embedding strength using the proposed objective function, achieving maximum robustness under the premise of satisfying imperceptibility. As compared with other algorithms, S-AES has a greater ability to resist high intensity attacks. We embedded the principle components enclosing the unique features of the watermark into the host image to overcome the false positive problem. Even if someone provides a wrong singular matrix obtained from a false watermark, it cannot extract the watermark successfully using side information of the false watermark. The robustness of S-AES decreases slightly under average and median filter attacks with a larger window size. This can be improved in the future. In addition, S-AES can also be extended to the field of video watermarking.