# Robust Watermarking Algorithm against the Geometric Attacks based on Non-Subsampled Shearlet Transform and Harris-Laplace Detector

Authors: Ling Zhou; Mei-Juan Zuo; Hao Shi; Ye Zhang; Li-Hua Gong
Journal: Security and Communication Networks
 (2022)

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## Abstract

With the rapid spread of network information, the information maintenance has become the focus of information security on networks. Digital watermarking is one of the effective methods to protect information security, achieve anticounterfeiting traceability, and protect copyright, and it is an important branch of information hiding technology. However, one of the most challenging questions of digital watermarking is how to present strong robustness in geometric attacks. Nowadays, most watermarking algorithms are relatively weak robustness against geometric attacks. A robust watermarking algorithm against geometric attacks based on the non-subsampled shearlet transform and the Harris-Laplace detector is proposed. The host image is decomposed into subbands with different directions by the shearlet transform, and the Harris-Laplace detector is utilized to obtain the feature regions. Then, the nonoverlapping regions with strong robustness are selected to embed watermark by the fuzzy c-means cluster algorithm. The experimental results indicate that the proposed watermarking scheme can well resist geometric attacks.


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## Body

## 1. Introduction

As we all know, with the rapid development of network technology, digital watermarking has become one of the key technologies for ensuring data integrity and intellectual property rights [1–4]. On the premise of ensuring a certain visual quality of digital information, digital watermarking is directly embedded in the media content. When the media content needs to be certified, the watermark can be extracted to identify whether it is true and complete, and as a media authenticity and integrity protection. Therefore, it is necessary to propose secure watermarking schemes with authentication. And to protect the ownership of the digital products, a blind watermarking scheme was proposed by Luo et al. [5]. In the past several decades, digital watermarking algorithms based on discrete wavelet transform have been widely used [6, 7]. Wavelet transform can approach the one-dimensional signal with nonlinear sparsity well, and it can also capture the point singularity of the one-dimensional signal effectively [8, 9]. However, when the wavelet transform is utilized to represent high-dimensional signals, it not only lacks directional sensitivity, but also cannot capture the linear singularity of high-dimensional signals effectively [10]. To solve this problem, some multiscale directional transformation tools have been discovered, such as ridgelets [11], curvelets [12, 13], contourlets [14], and shearlets [15, 16]. Ridgelet transform maps the linear singularities of images into singularities of the Radon domain by the Radon transform, and then singularities in the frequency domain are processed by wavelet transform [17]. However, ridgelet analysis is not so efficient for natural images whose edge lines are dominated by curves, and the redundancy of analysis is very large. To settle the problem of curve analysis, the curvelet transform, which is a combination of a special filtering process with multiscale ridgelet transform, was proposed by Starck et al. [18]. Subsequently, the contourlet transform inherits the benefits from the ridgelet transform and the curvelet transform. It is decomposed by multiscale and multidirectional filtering, while the contourlet transform has no translation invariance [19].As previously explained, some complex structures of curves, edges, and textural regions are not easy to be captured by the ridgelet transform; however, the shearlet transform has the excellent ability to capture image features precisely. More and more researchers have proposed a large number of image watermarking schemes based on the shearlet transform [20–22]. Mardanpour and Chahooki put forward an image watermarking method based on the discrete shearlet transform (DST) and the bidiagonal singular value decomposition (BSVD) factorization [23]. They took advantage of shearlet transform to achieve higher imperceptibility and made use of diagonal singular value decomposition to enhance robustness and security; however, the watermarking algorithm could not well resist the geometric attacks.At present, there are still many problems to be solved in the copyright protection of digital watermarking. However, one of the most important questions is how to overcome the attack problem encountered in the transmission of digital products. With the development of the information technology, a variety of digital watermarking algorithms in resistance to attacks have emerged endlessly. Ma et al. proposed a novel watermarking algorithm resistant to the geometric attacks based on accurate polar harmonic Fourier moments (PHFMs) and chaotic mapping [24]. The robustness of this algorithm against the geometric attacks has been improved by relying on the geometric invariance of an accurate PHFMs. Hu and Xiang took advantage of the polarity harmonic transformation of reversible and robust watermarking techniques to achieve the lossless robust watermarking [25]. However, in the method of polar harmonic invariant domain, since the interpolation caused by the invariant domain increases the synchronization error, the watermark embedding and the watermark detection are not aligned. Therefore, the image is easily affected by the shear.Hence, these watermarking algorithms based on image feature points with better robustness to resist the geometric attacks were discovered, such as Harris detection [26, 27], Harris-Laplace detection [28], scale-invariant feature transformation (SIFT) [29], and affine scale-invariant feature transformation (ASIFT) [30]. During the detection process, the watermark is less prone to synchronization error. These watermarking algorithms based on the image feature points, also called the second generation of digital watermarking, are to select the appropriate watermark embedding area with the use of image characteristics, such as corner and texture, which are efficient to resist the RST and cutting attacks. Among them, point information has a good resistance to rotation and translation, and it is insensitive to noise, which can well express image features. And more and more scholars have been studying the second generation of digital watermarking [31–33]. Wang et al. introduced a robust image watermarking algorithm combining the DWT with the ASIFT [30]. The ASIFT is applied to obtain feature points invariant to geometric attacks. Gong et al. presented a robust watermarking algorithm for medical images based on the Harris-SURF-DCT [27]. The algorithm based on the Harris-SURF-DCT has strong robustness to conventional attacks and geometric attacks. And Feng et al. proposed an image watermarking method in resistance to affine attack by investigating characteristics of the ASIFT feature points and Delaunay triangles [34]. However, there are still a lot of deficiencies when to face the general image attacks. Deng et al. proposed an image watermarking scheme based on the local histogram to solve the serious distortion by the interpolation error and the shift problem [35]. The extraction of feature points and the construction of local circular regions are conducted with Harris-Laplace detector. To increase the readability of the section, the pros and the cons associated with different watermarking techniques are presented in Table 1.Table 1
The analysis of some references.
ReferencesWatermarking schemesAdvantagesDisadvantages[6–10]Based on the DWTThe DWT can approach the one-dimensional signal wellThe DWT lacks of directional sensitivity[11, 17]Based on ridgeletRidgelet can capture the singularity of high dimensional signals effectivelyThe redundancy of analysis is very large[12, 13, 18]Based on curveletsIt is so efficient for natural images whose edge lines are dominated by curvesCurvelets have high redundancy[14, 19]Based on contourletsContourlets inherits the benefits from the ridgelet transform and the curvelet transformContourlets have no translation invariance[15, 16, 20–23]Based on shearletShearlet possesses the characteristics of multi-scale and the ability to capture the geometry of multidimensional dataThe discrete shearlet transform may produce the pseudo-Gibbs phenomenon[24, 25]Based on PHFMsPHFMs are image rotation invariantIt would lead to higher numerical errors when processing smaller images[26, 27]Based on Harris detectionHarris feature points have good rotation invarianceHarris feature points are short of the scale invariance and affine invariance[28]Based on Harris-Laplace detectionHarris-Laplace feature points have scale invariance and affine invarianceIt would lead to the overlapping of feature regions[29]Based on SIFTThe SIFT feature points are invariant to the geometric attacksThe SIFT feature points do not have affine invariance[30, 34]Based on ASIFTThe ASIFT feature points have affine invarianceIt can not well resist rotation attackTo enhance the security of copyright protection, a blind watermarking scheme of the NSST and the Harris-Laplace detector is proposed. First, the host image is decomposed by the NSST with different directional subbands. And the feature point area is selected with the Harris-Laplace detector and the fuzzy c-means cluster algorithm. By the human visual characteristics, the low frequency subband is selected to structure the optimal embedding regions. The designed watermarking scheme is performed by the Harris-Laplace detection directly on the attacked image without the participation of original image and original watermark during the detection process.The remainder of the paper is organized in the following manner. The theoretical context is presented in Section2. The watermark embedding and the extracting schemes are reviewed in Section 3. Simulation results and analyses are described in Section 4. A comparison with existing work is provided in Section 5. Finally, it is summed up in Section 6.


## 2. Theoretical Background

### 2.1. Non-Subsampled Shearlet Transform

Shearlet transform is an advanced multiscale geometric analysis tool, which owns the characteristics of translation invariability, multidirection, and low computational complexity. When the dimensionn is 2, the affine system is the collection of the form(1)MABψ=ψj,l,kx=detAj/2ψBlAjx−k|j,l∈z,k∈z2,where ψ is ψ∈L2R2, and j, l, k are the scale parameter, the shear parameter, and the translation parameter, respectively. A, B are the anisotropic matrix and the shear matrix, respectively [16]. If MABψ can form a Parseval frame, the elements of the affine system are called composite wavelets. Aj is related to the scale transform, and Bl is associated with the geometric transform. Assuming(2)A=a00a,B=1s01.Ma,s,tψ is called the shear wave system if it satisfies the following affine system for a,s∈R+×R.(3)Ma,s,tψ=ψa,s,tx=a−3/4ψA−1B−1x−t,a∈R+,s∈R,t∈R2.The non-subsampled shearlet transform is mainly divided into two processes: the non-subsampled Laplacian pyramid decomposition and the directional filter. The process of the non-subsampled Laplacian pyramid decomposition is to ensure the multiscale characteristics of the non-subsampled shearlet transform. In other words, the image can be decomposed into different scales. The process of the directional filter is that the image on each scale is divided into different directions to keep the multidirectional characteristics of the transform [16]. The decomposition process of the non-subsampled shearlet transform is shown in Figure 1. Combining the non-subsampled tower-type transform with the non-subsampled filter is to make the size of the transformed subband image be consistent with the original image and to improve the redundancy of image.Figure 1
The process of the shearlet transform.


### 2.2. Harris-Laplace Detector

Harris detector has good rotation invariance; however, it is short of the scale invariance and affine invariance. To obtain the scale invariance, one of the most intuitive methods is to establish multiscale space. Therefore, it is necessary to introduce multiscale space in the conventional methods and append many feature points to other scale space on the original feature point space [36]. The added feature points correspond to the images of different scale space, which increases the robustness of the target scale. In this way, a scale autocorrelation matrix Mx,y can be expressed as(4)Mx,y=σD2gσIIx2x,y,σDIxIyx,y,σDIxIyx,y,σDIy2x,y,σD,where gσI is the Gaussian convolution kernel of σ=σI, σI is the integral scale, and σD is the differential scale. x is the image coordinate, and Ix,y are the derivatives of Gaussian smooth image at x and y directions. The characteristic intensity of pixels x and y can be determined by σI and σD.However, for the spatial extreme values of each scale, the extreme points of the scale space all represent the local characteristics of the image. Then, the local feature points with invariant scale features can be extracted with the Laplacian of Gaussian (LoG), and the Gaussian filtering and Laplace edge detection are combined by the LoG operation.(5)LoG=σD2Gxxx,y,σD+Gyyx,y,σDIx,y,Gx,y,σD=12πσDexp−x4+y42σ.And the characteristic intensity can be calculated as(6)RH=DetM−k(TrM)2,where the range of k is from 0.04 to 0.06, Det⋅ represents the determinant of the matrix, and Tr⋅ is the trace of the matrix. In addition, the value of RH is directly related to the robustness of the feature points. The response value of the characteristic intensity is greater, and the robustness of the feature points is better.


### 2.3. Fuzzy c-Means Cluster

The algorithm of fuzzy c-means (FCM) is a cluster method, which permits one piece of data to belong to two or more clusters. It is based on the minimization of the following objective function:(7)Jm=∑i=1N∑j=1Cμijmxi−cj2,where m is any real number greater than 1, μij is the degree of membership of sample i in the cluster j, xi is a sample with d-dimensional characteristics of the measured data, cj is the center of the cluster j, and ⋅ is the expression between the measured data and the center.Actually, the fuzzy c-means cluster algorithm is a process of iterating continuously to calculate membership degree and cluster center [37]. The membership degree μij and the cluster center cj are updated by the iterative optimization of the following objective function.(8)μij=xi−cj−2/m−1∑k=1Cxi−ck−2/m−1,cj=∑i=1Nμijm∗xi∑i=1Nμijm.However, whether the optimal cluster center and membership degree will be found in the end can be determined by(9)maxijμijk+1−μijk<ε,where k is the iteration step, and ε is the error threshold.The implementation of the FCM cluster process is illustrated in Figure2.Figure 2
Process of the FCM cluster algorithm.The detailed process of FCM cluster algorithm is also described asStep. 1 The input matrix and the cluster centercj are initialized randomly.Step. 2 The data points are classified into the nearest cluster centercj according to the nearest distance principle.Step. 3 The location of cluster centercj is calculated, and the cluster center matrix is updated.Step. 4 According to the iteration termination condition, if the membership degreeμij satisfies equation (9), the cluster results can be obtained, and the process is stopped. Otherwise, Steps 2 and 3 are repeated until the results of the cluster center are no longer changed.


## 2.1. Non-Subsampled Shearlet Transform

Shearlet transform is an advanced multiscale geometric analysis tool, which owns the characteristics of translation invariability, multidirection, and low computational complexity. When the dimensionn is 2, the affine system is the collection of the form(1)MABψ=ψj,l,kx=detAj/2ψBlAjx−k|j,l∈z,k∈z2,where ψ is ψ∈L2R2, and j, l, k are the scale parameter, the shear parameter, and the translation parameter, respectively. A, B are the anisotropic matrix and the shear matrix, respectively [16]. If MABψ can form a Parseval frame, the elements of the affine system are called composite wavelets. Aj is related to the scale transform, and Bl is associated with the geometric transform. Assuming(2)A=a00a,B=1s01.Ma,s,tψ is called the shear wave system if it satisfies the following affine system for a,s∈R+×R.(3)Ma,s,tψ=ψa,s,tx=a−3/4ψA−1B−1x−t,a∈R+,s∈R,t∈R2.The non-subsampled shearlet transform is mainly divided into two processes: the non-subsampled Laplacian pyramid decomposition and the directional filter. The process of the non-subsampled Laplacian pyramid decomposition is to ensure the multiscale characteristics of the non-subsampled shearlet transform. In other words, the image can be decomposed into different scales. The process of the directional filter is that the image on each scale is divided into different directions to keep the multidirectional characteristics of the transform [16]. The decomposition process of the non-subsampled shearlet transform is shown in Figure 1. Combining the non-subsampled tower-type transform with the non-subsampled filter is to make the size of the transformed subband image be consistent with the original image and to improve the redundancy of image.Figure 1
The process of the shearlet transform.


## 2.2. Harris-Laplace Detector

Harris detector has good rotation invariance; however, it is short of the scale invariance and affine invariance. To obtain the scale invariance, one of the most intuitive methods is to establish multiscale space. Therefore, it is necessary to introduce multiscale space in the conventional methods and append many feature points to other scale space on the original feature point space [36]. The added feature points correspond to the images of different scale space, which increases the robustness of the target scale. In this way, a scale autocorrelation matrix Mx,y can be expressed as(4)Mx,y=σD2gσIIx2x,y,σDIxIyx,y,σDIxIyx,y,σDIy2x,y,σD,where gσI is the Gaussian convolution kernel of σ=σI, σI is the integral scale, and σD is the differential scale. x is the image coordinate, and Ix,y are the derivatives of Gaussian smooth image at x and y directions. The characteristic intensity of pixels x and y can be determined by σI and σD.However, for the spatial extreme values of each scale, the extreme points of the scale space all represent the local characteristics of the image. Then, the local feature points with invariant scale features can be extracted with the Laplacian of Gaussian (LoG), and the Gaussian filtering and Laplace edge detection are combined by the LoG operation.(5)LoG=σD2Gxxx,y,σD+Gyyx,y,σDIx,y,Gx,y,σD=12πσDexp−x4+y42σ.And the characteristic intensity can be calculated as(6)RH=DetM−k(TrM)2,where the range of k is from 0.04 to 0.06, Det⋅ represents the determinant of the matrix, and Tr⋅ is the trace of the matrix. In addition, the value of RH is directly related to the robustness of the feature points. The response value of the characteristic intensity is greater, and the robustness of the feature points is better.


## 2.3. Fuzzy c-Means Cluster

The algorithm of fuzzy c-means (FCM) is a cluster method, which permits one piece of data to belong to two or more clusters. It is based on the minimization of the following objective function:(7)Jm=∑i=1N∑j=1Cμijmxi−cj2,where m is any real number greater than 1, μij is the degree of membership of sample i in the cluster j, xi is a sample with d-dimensional characteristics of the measured data, cj is the center of the cluster j, and ⋅ is the expression between the measured data and the center.Actually, the fuzzy c-means cluster algorithm is a process of iterating continuously to calculate membership degree and cluster center [37]. The membership degree μij and the cluster center cj are updated by the iterative optimization of the following objective function.(8)μij=xi−cj−2/m−1∑k=1Cxi−ck−2/m−1,cj=∑i=1Nμijm∗xi∑i=1Nμijm.However, whether the optimal cluster center and membership degree will be found in the end can be determined by(9)maxijμijk+1−μijk<ε,where k is the iteration step, and ε is the error threshold.The implementation of the FCM cluster process is illustrated in Figure2.Figure 2
Process of the FCM cluster algorithm.The detailed process of FCM cluster algorithm is also described asStep. 1 The input matrix and the cluster centercj are initialized randomly.Step. 2 The data points are classified into the nearest cluster centercj according to the nearest distance principle.Step. 3 The location of cluster centercj is calculated, and the cluster center matrix is updated.Step. 4 According to the iteration termination condition, if the membership degreeμij satisfies equation (9), the cluster results can be obtained, and the process is stopped. Otherwise, Steps 2 and 3 are repeated until the results of the cluster center are no longer changed.


## 3. Watermark Embedding and Extraction Scheme

### 3.1. Structure of the Characteristic Area

To ensure that the extracted feature points have good robustness against the geometric attacks, it is important to select the higher quality feature points as the watermark embedding region. As evidenced by Figure3, the different feature regions may contain the same feature points or intersect with each other, which would lead to the overlapping of feature regions and the mutual influence of watermark information with the improper construction method. Consequently, the feature region is constructed with the fuzzy c-means cluster algorithm, so that the feature points of the constructed region achieve stability and do not overlap one another.Figure 3
Results of Harris-Laplace detector: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)The specific process of constructing the algorithm is as follows:Step. 1 The Harris-Laplace detector method is utilized to extract feature points from the host image. Firstly, a scale space is established, and the extreme points of the region in the image are detected by Harris points as candidate points. Then, the Laplacian of Gaussian (LoG) values of the candidate points in all scales are calculated by an iterative method.Step. 2 A matrixP containing the location and the feature ratio of the feature points is obtained. Figure 3 shows the results for different host images after the Harris-Laplace detector. Then, the appropriate value of the cluster center is selected with the fuzzy c-means cluster algorithm.Step. 3 The feature points detected by the Harris-Laplace detector are divided into three categories; in other words, the value of the cluster center is defined as three, as shown in Figure4.Step. 4 A new feature point matrixP1 containing the location of pixel value is obtained by the fuzzy C-means cluster algorithm. Then, the location of pixel value, which is the closest to the cluster center, is selected as the feature region to embed watermark. The nonoverlapping region of the embedding watermark is shown in Figure 5.Figure 4
Results of fuzzy c-means cluster: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)Figure 5
The selected region of the watermark embedding processed by the Harris-Laplace detector and the fuzzy c-means cluster scheme: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)


### 3.2. Watermark Embedding

The watermark embedding algorithm illustrated in Figure6 includes the following steps.Step 1. The feature points of the host image are extracted by the Harris-Laplace operator, and then the extracted feature points are categorized by the fuzzy c-means cluster algorithm.Step 2. The host image is decomposed by the non-subsampled shearlet transform to acquire the directional subbands. Then, these points closest to cluster centers are selected from the initially extracted feature points, respectively. And in the lower frequency subband, the selected feature points are taken as the center to intercept the same size region to embed watermark.Step 3. The watermark image is encrypted with chaotic sequence and Arnold transform. Afterwards, the encrypted watermark is divided into blocks of size4×4, and DCT is executed on each block, and the lower-left coefficient of each block is selected to form a new matrix Jw.Step 4. The SVD is performed onJw and three feature point regions J to obtain the diagonal matrix Sw and Si respectively. Singular values represent that the intrinsic algebraic image properties and the singular values of an image have good stability. If a small perturbation is added into an image, its singular values will not be changed significantly.(10)Uw,Sw,Vw=SVDJw,Ui,Si,Vi=SVDJ,1≤i≤C,whereJ is a cell array of size 1×3, and every cell is of size 16×16. The three blocks are decomposed by the SVD, respectively.Step 5.Sw of the watermark image is embedded into the diagonal matrices of three blocks as(11)Si∗=Si+kSw,wherek is the embedding factor. Therefore, singular value of the watermark image is embedded into the three feature regions.Step 6. The inverse SVD is carried out onSi∗ and the new matrix J∗ can be acquired,(12)J∗=UiSi∗ViT.Similarly,J∗ is also a cell array of size 1×3, and every cell is of size 16×16.Step 7. The matrices embedded with watermark information are reconstructed. And the image containing watermark information is obtained by the inverse non-subsampled shearlet transform with other high frequency subbands.Figure 6
Watermark embedding.The embedding factork, the singular value matrix Si of the host image, and the left and the right singular vectors Uw and Vw of the watermark image are all kept as keys for watermark extraction.


### 3.3. Watermark Extraction

The watermark extraction algorithm shown in Figure7 consists of the following steps.Step 1. Similarly, the feature points of the watermarked image are initially extracted by the Harris-Laplace detector without involving original image and original watermark, and then all the extracted feature points are classified by the fuzzy c-means cluster algorithm.Step 2. Like Step 2 in the watermark embedding, the Harris-Laplace detector and the fuzzy c-means cluster algorithm are utilized to select the appropriate extraction area for the watermarked image.Step 3. The SVD is performed onJ′ to acquire the diagonal matrix Si′,(13)Ui′,Si′,Vi′=SVDJ′.Step 4. With the singular value matrixSi of the host image and the embedding factor k, the diagonal matrix Swi of the scrambling watermark image can be obtained:(14)Swi=Si′−Sik.Step 5. The inverse SVD is, respectively, executed on three diagonal matricesSwi and the new matrices Wi can be achieved,(15)Wi=UwSwiVwT.SinceC is three, three matrices will be obtained.Step 6. After the reconstruction of these matrices, the watermark images of different robustness are obtained by the inverse DCT and the inverse scrambling of chaos sequence and Arnold transform. The watermark image of the best robustness will be selected as the final extracted watermark image.Figure 7
Watermark extraction.


## 3.1. Structure of the Characteristic Area

To ensure that the extracted feature points have good robustness against the geometric attacks, it is important to select the higher quality feature points as the watermark embedding region. As evidenced by Figure3, the different feature regions may contain the same feature points or intersect with each other, which would lead to the overlapping of feature regions and the mutual influence of watermark information with the improper construction method. Consequently, the feature region is constructed with the fuzzy c-means cluster algorithm, so that the feature points of the constructed region achieve stability and do not overlap one another.Figure 3
Results of Harris-Laplace detector: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)The specific process of constructing the algorithm is as follows:Step. 1 The Harris-Laplace detector method is utilized to extract feature points from the host image. Firstly, a scale space is established, and the extreme points of the region in the image are detected by Harris points as candidate points. Then, the Laplacian of Gaussian (LoG) values of the candidate points in all scales are calculated by an iterative method.Step. 2 A matrixP containing the location and the feature ratio of the feature points is obtained. Figure 3 shows the results for different host images after the Harris-Laplace detector. Then, the appropriate value of the cluster center is selected with the fuzzy c-means cluster algorithm.Step. 3 The feature points detected by the Harris-Laplace detector are divided into three categories; in other words, the value of the cluster center is defined as three, as shown in Figure4.Step. 4 A new feature point matrixP1 containing the location of pixel value is obtained by the fuzzy C-means cluster algorithm. Then, the location of pixel value, which is the closest to the cluster center, is selected as the feature region to embed watermark. The nonoverlapping region of the embedding watermark is shown in Figure 5.Figure 4
Results of fuzzy c-means cluster: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)Figure 5
The selected region of the watermark embedding processed by the Harris-Laplace detector and the fuzzy c-means cluster scheme: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)


## 3.2. Watermark Embedding

The watermark embedding algorithm illustrated in Figure6 includes the following steps.Step 1. The feature points of the host image are extracted by the Harris-Laplace operator, and then the extracted feature points are categorized by the fuzzy c-means cluster algorithm.Step 2. The host image is decomposed by the non-subsampled shearlet transform to acquire the directional subbands. Then, these points closest to cluster centers are selected from the initially extracted feature points, respectively. And in the lower frequency subband, the selected feature points are taken as the center to intercept the same size region to embed watermark.Step 3. The watermark image is encrypted with chaotic sequence and Arnold transform. Afterwards, the encrypted watermark is divided into blocks of size4×4, and DCT is executed on each block, and the lower-left coefficient of each block is selected to form a new matrix Jw.Step 4. The SVD is performed onJw and three feature point regions J to obtain the diagonal matrix Sw and Si respectively. Singular values represent that the intrinsic algebraic image properties and the singular values of an image have good stability. If a small perturbation is added into an image, its singular values will not be changed significantly.(10)Uw,Sw,Vw=SVDJw,Ui,Si,Vi=SVDJ,1≤i≤C,whereJ is a cell array of size 1×3, and every cell is of size 16×16. The three blocks are decomposed by the SVD, respectively.Step 5.Sw of the watermark image is embedded into the diagonal matrices of three blocks as(11)Si∗=Si+kSw,wherek is the embedding factor. Therefore, singular value of the watermark image is embedded into the three feature regions.Step 6. The inverse SVD is carried out onSi∗ and the new matrix J∗ can be acquired,(12)J∗=UiSi∗ViT.Similarly,J∗ is also a cell array of size 1×3, and every cell is of size 16×16.Step 7. The matrices embedded with watermark information are reconstructed. And the image containing watermark information is obtained by the inverse non-subsampled shearlet transform with other high frequency subbands.Figure 6
Watermark embedding.The embedding factork, the singular value matrix Si of the host image, and the left and the right singular vectors Uw and Vw of the watermark image are all kept as keys for watermark extraction.


## 3.3. Watermark Extraction

The watermark extraction algorithm shown in Figure7 consists of the following steps.Step 1. Similarly, the feature points of the watermarked image are initially extracted by the Harris-Laplace detector without involving original image and original watermark, and then all the extracted feature points are classified by the fuzzy c-means cluster algorithm.Step 2. Like Step 2 in the watermark embedding, the Harris-Laplace detector and the fuzzy c-means cluster algorithm are utilized to select the appropriate extraction area for the watermarked image.Step 3. The SVD is performed onJ′ to acquire the diagonal matrix Si′,(13)Ui′,Si′,Vi′=SVDJ′.Step 4. With the singular value matrixSi of the host image and the embedding factor k, the diagonal matrix Swi of the scrambling watermark image can be obtained:(14)Swi=Si′−Sik.Step 5. The inverse SVD is, respectively, executed on three diagonal matricesSwi and the new matrices Wi can be achieved,(15)Wi=UwSwiVwT.SinceC is three, three matrices will be obtained.Step 6. After the reconstruction of these matrices, the watermark images of different robustness are obtained by the inverse DCT and the inverse scrambling of chaos sequence and Arnold transform. The watermark image of the best robustness will be selected as the final extracted watermark image.Figure 7
Watermark extraction.


## 4. Results and Analyses

### 4.1. Theoretical Analysis

The proposed watermarking scheme is primarily a combination of the non-subsampled shearlet transform, the Harris-Laplace detector, and the fuzzy c-means cluster algorithm. The multiscale geometric analysis overcomes the disadvantage that wavelet transform cannot effectively describe high-dimensional signals. As a multiscale analysis function is constructed in recent years, the shearlet transform has been widely used in digital watermarking due to its strong directional decomposition as shown in Figure8. However, if the shearlet transform is used alone in the watermarking algorithm, it cannot resist all kinds of attacks well and may produce the pseudo-Gibbs phenomenon. Therefore, the non-subsampled shearlet transform and the Harris-Laplace detector are combined to enhance the robustness of the watermarking algorithm.Figure 8
Directional subbands diagram after shearlet decomposition.
(a)(b)(c)(d)(e)(f)The feature points extracted by the traditional Harris operator are sensitive to scale attacks, while the Harris-Laplace detector overcomes this drawback due to the adaptive scale characteristics to some extent. However, the feature points generated by the Harris-Laplace detector would overlap, which could result in the interlacing of the watermarking information. Therefore, the feature regions are selected by the fuzzy c-means cluster algorithm to prevent the feature regions from overlapping. To further enhance the robustness against the geometric attacks, an image watermarking algorithm based on the non-subsampled shearlet transform, the Harris-Laplace detector, and the fuzzy c-means cluster is proposed.


### 4.2. Experiment Results

Watermarking algorithms are generally assessed by two important parameters: invisibility and robustness. The invisibility is that the watermarked image neither distorts nor diminishes the perceived quality of the watermarked image in comparison to the original image. Robustness means that the watermark is still extracted clearly from the watermarked image after the attacks. To evaluate two performances, the peak signal-to-noise ratio (PSNR) and the normalized correlation (NC) are adopted. PSNR shows the imperceptibility of the watermark, while NC evaluates the robustness of the extracted watermark. For a gray-scale image of the pixel value 256, PSNR value can be calculated as(16)PSNR=10lg2552MSE,where MSE means the mean square error between a watermarked image fyi,j and its original image fxi,j,(17)MSE=1m1n1∑i=1m1∑j=1n1fyi,j−fxi,j2,where fyi,j and fxi,j denote the pixel values at position i,j of the original image and watermarked image, and m1 and n1 are the image dimensions.Normalized correlation (NC) coefficient, the similarity of original watermarks and extracted ones, is expressed as(18)NC=∑i=1m2∑j=1n2Ioi,jIwi,j∑i=1m2∑j=1n2Ioi,j2∑i=1m2∑j=1n2Iwi,j2,where m1 and n1 denote the image dimensions, and Ioi,j and Iwi,j are the pixel values at position i,j of the original watermark and the extracted watermark.In experiment, the gray-scale images “Cameraman,” “Peppers,” and “Couple” with512×512 pixels are regarded as the host images, and the gray-scale image “Hand” with 64×64 pixels is considered as watermark. The feature point region of watermark extraction is displayed in Figure 9. It can be concluded that the extracted regions do not change due to embedding watermark with reference to Figure 5. The results of test images without attacks are displayed in Figure 10. Table 2 shows that the NC values of the extracted watermark are all close to 1.0. And the PSNR values for different images can also reach 66 dB. In general, if the PSNR value exceeds 40 dB, the imperceptibility can be guaranteed.Figure 9
The selected region of the watermark extraction processed by the Harris-Laplace detector and the fuzzy c-means cluster scheme: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)Figure 10
Test images: original images: (a) cameraman, (b) peppers, (c) couple; original watermark: (a1), (b1), (c1); watermarked images: (d), (e), (f); the extracted watermark: (d1), (e1), (f1).Table 2
PSNR (dB) and NC values of the watermarked images.
Cover imageCameramanPeppersCouplePSNR66.009066.011365.9934NC0.98490.98900.9905The low-frequency component of the image carries the vast majority of the information about the original image, which also contains the most important visual perception part of the image. If the DCT low-frequency component is selected as the watermark embedding region, the visual quality of the host image is significantly reduced, thus reducing the invisibility of the watermarking algorithm. The high-frequency part of the image is easily lost during lossy data compression. Considering these factors, it is more appropriate to embed the watermark into the DCT mid-frequency component. And as can be seen, the PSNR values are high to demonstrate a good degree of imperceptibility of the proposed method.


### 4.3. Attack Tests

#### 4.3.1. Rotation Attack

The rotation of a digital image actually moves every pixel of the image along a circular path. Rotation attack is one of the numerous attack methods. Although the rotation attack does not remove the watermark information from the image, it makes the detection and embedding of watermark lose synchronization, leading to the failed detection of the watermark.Table3 displays the extracted results of applying rotation attack to some watermarked images. The watermarked images are rotated by 45∘, 90∘, and 135∘ in the counterclockwise direction, respectively. In general, the scale of the image would change due to the discreteness of the digital image after rotation of an image. To keep the rotated image with the same size as the original image, a part of the edge information of the rotated image is cut off. However, since the Harris-Laplace detector has the benefits of rotation invariance and angle invariance, the extracted watermarks can still be recognized by human eyes of Figure 11. Table 3 also shows that the NC values of the extracted watermark from different angles are as high as 0.95.Table 3
NC values of the watermarked images under rotation attack.
Cover imageCameramanPeppersCouple45∘0.99010.98660.985190∘0.95910.97580.9807135∘0.96600.95980.9823Figure 11
Rotation attack on different watermarked images with45∘, 90∘, 135∘: Cameraman (a), (c), (e), extracted watermark (b), (d), (f); Peppers (a1), (c1), (e1), extracted watermark (b1), (d1), (f1); Couple (a2), (c2), (e2), extracted watermark (b2), (d2), (f2).


#### 4.3.2. Cutting Attack

The cutting attack means that an attacker cuts out an area of an image containing watermark and discards it. A good watermarking algorithm should be able to extract the watermark effectively on the premise of keeping the information from the main image, while the image is attacked by cutting.Simulation results are performed to test the robustness of the proposed approach against cutting attack on Figure12. The extracted results from a cutting version of the watermarked images with different cutting areas, respectively, are presented in Table 4. Since it is impossible to retrieve a part of the image contents discarded by the attacker during the watermark detection, the damage caused by the cutting attack to the image is irreversible. When the cutting area is quite large, the robustness of the extracted watermark may be not well. However, the contour of the extracted watermark images could be still identified by human eyes. The experimental results demonstrate that the proposed algorithm based on non-subsampled shearlet transform and Harris-Laplace detector has more visually recognizable information about copyright property, and the quality of the watermarked images is still in a good situation, even under the large cutting area. The extracted watermarks and the original watermarks are strongly correlated.Figure 12
Cutting attack on different images with quarter, half, irregular: cameraman (a), (c), (e), extracted watermark (b), (d), (f); peppers (a1), (c1), (e1), extracted watermark (b1), (d1), (f1); couple (a2), (c2), (e2), extracted watermark (b2), (d2), (f2).Table 4
NC values of the watermarked images under cutting attack.
Cover imageCameramanPeppersCoupleQuarter cutting0.99050.98340.9890Half cutting0.96500.98900.9780Irregular cutting0.98710.98870.9903


#### 4.3.3. Scaling Attack

The scaling of an image can be divided into two situations, equal scaling and unequal scaling. An equal scaling signifies that the image is scaled in the same proportion in the horizontal and vertical directions. The unequal scaling is that the image is scaled in different proportions. And the content of the image would be distorted with the unequal scaling.The results of the watermarked images and the corresponding extracted watermarks under different scaling factors on different watermarked images are given in Figure13. The contents of the image do not change when an image is attacked by scaling; however, there is an interpolation operation in the process of scaling. Table 5 presents the NC values for different scaling factors on different watermarked images. Since the scaling attack on the image is an irreversible operation, the loss part of the original information would have a great impact on the watermarking system when the scaling factor is higher than 1. However, the extracted watermarks can still be recognized. The robustness of the watermarking algorithm based on the shearlet transform is well overall.Figure 13
Scaling attack on different images with different scaling factors 1.5 and 0.75, respectively: cameraman (a), (a1), extracted watermark (b), (b1); peppers (c), (c1), extracted watermark (d), (d1); couple (e), (e1), extracted watermark (f), (f1).Table 5
NC values of the watermarked images under the different scaling factors.
Cover imageCameramanPeppersCouple1.50.94670.94420.95110.750.98230.98440.9769


#### 4.3.4. Other Common Attacks

In addition to rotation attack, cutting attack, and scaling attack, the proposed watermarking algorithm based on non-subsampled shearlet transform and Harris-Laplace detector also displays good robustness against common attacks such as Gaussian low pass filtering, noise, and compression, as shown in Table6.Table 6
NC values of the proposed scheme on different watermarked image with various attacks.
AttacksImagePeppersCoupleCameramanJPEG compression (75%)0.98830.98420.9889Gaussian low-pass filtering (0.1)0.99010.98900.9890Gaussian noise (0.1)0.97830.93620.9843Salt and pepper noise (0.03)0.97050.98710.9891Scaling (0.7) & rotation (10∘)0.92810.99070.9674Since the feature points typically own the geometric invariance, embedding watermark in feature points or embedding watermark with feature points as a reference can make watermark resist the geometric attacks. And the non-subsampled shearlet transform abandons the operation of down sampling in the process of the standard shearlet transform and overcomes the phenomenon of the spectrum aliasing after transform. Therefore, the watermarking performance can be further improved by combining the directional sensitivity of non-subsampled shearlet transform with the geometric invariance of the Harris-Laplace detector.


## 4.1. Theoretical Analysis

The proposed watermarking scheme is primarily a combination of the non-subsampled shearlet transform, the Harris-Laplace detector, and the fuzzy c-means cluster algorithm. The multiscale geometric analysis overcomes the disadvantage that wavelet transform cannot effectively describe high-dimensional signals. As a multiscale analysis function is constructed in recent years, the shearlet transform has been widely used in digital watermarking due to its strong directional decomposition as shown in Figure8. However, if the shearlet transform is used alone in the watermarking algorithm, it cannot resist all kinds of attacks well and may produce the pseudo-Gibbs phenomenon. Therefore, the non-subsampled shearlet transform and the Harris-Laplace detector are combined to enhance the robustness of the watermarking algorithm.Figure 8
Directional subbands diagram after shearlet decomposition.
(a)(b)(c)(d)(e)(f)The feature points extracted by the traditional Harris operator are sensitive to scale attacks, while the Harris-Laplace detector overcomes this drawback due to the adaptive scale characteristics to some extent. However, the feature points generated by the Harris-Laplace detector would overlap, which could result in the interlacing of the watermarking information. Therefore, the feature regions are selected by the fuzzy c-means cluster algorithm to prevent the feature regions from overlapping. To further enhance the robustness against the geometric attacks, an image watermarking algorithm based on the non-subsampled shearlet transform, the Harris-Laplace detector, and the fuzzy c-means cluster is proposed.


## 4.2. Experiment Results

Watermarking algorithms are generally assessed by two important parameters: invisibility and robustness. The invisibility is that the watermarked image neither distorts nor diminishes the perceived quality of the watermarked image in comparison to the original image. Robustness means that the watermark is still extracted clearly from the watermarked image after the attacks. To evaluate two performances, the peak signal-to-noise ratio (PSNR) and the normalized correlation (NC) are adopted. PSNR shows the imperceptibility of the watermark, while NC evaluates the robustness of the extracted watermark. For a gray-scale image of the pixel value 256, PSNR value can be calculated as(16)PSNR=10lg2552MSE,where MSE means the mean square error between a watermarked image fyi,j and its original image fxi,j,(17)MSE=1m1n1∑i=1m1∑j=1n1fyi,j−fxi,j2,where fyi,j and fxi,j denote the pixel values at position i,j of the original image and watermarked image, and m1 and n1 are the image dimensions.Normalized correlation (NC) coefficient, the similarity of original watermarks and extracted ones, is expressed as(18)NC=∑i=1m2∑j=1n2Ioi,jIwi,j∑i=1m2∑j=1n2Ioi,j2∑i=1m2∑j=1n2Iwi,j2,where m1 and n1 denote the image dimensions, and Ioi,j and Iwi,j are the pixel values at position i,j of the original watermark and the extracted watermark.In experiment, the gray-scale images “Cameraman,” “Peppers,” and “Couple” with512×512 pixels are regarded as the host images, and the gray-scale image “Hand” with 64×64 pixels is considered as watermark. The feature point region of watermark extraction is displayed in Figure 9. It can be concluded that the extracted regions do not change due to embedding watermark with reference to Figure 5. The results of test images without attacks are displayed in Figure 10. Table 2 shows that the NC values of the extracted watermark are all close to 1.0. And the PSNR values for different images can also reach 66 dB. In general, if the PSNR value exceeds 40 dB, the imperceptibility can be guaranteed.Figure 9
The selected region of the watermark extraction processed by the Harris-Laplace detector and the fuzzy c-means cluster scheme: (a) cameraman, (b) peppers, (c) couple.
(a)(b)(c)Figure 10
Test images: original images: (a) cameraman, (b) peppers, (c) couple; original watermark: (a1), (b1), (c1); watermarked images: (d), (e), (f); the extracted watermark: (d1), (e1), (f1).Table 2
PSNR (dB) and NC values of the watermarked images.
Cover imageCameramanPeppersCouplePSNR66.009066.011365.9934NC0.98490.98900.9905The low-frequency component of the image carries the vast majority of the information about the original image, which also contains the most important visual perception part of the image. If the DCT low-frequency component is selected as the watermark embedding region, the visual quality of the host image is significantly reduced, thus reducing the invisibility of the watermarking algorithm. The high-frequency part of the image is easily lost during lossy data compression. Considering these factors, it is more appropriate to embed the watermark into the DCT mid-frequency component. And as can be seen, the PSNR values are high to demonstrate a good degree of imperceptibility of the proposed method.


## 4.3. Attack Tests

### 4.3.1. Rotation Attack

The rotation of a digital image actually moves every pixel of the image along a circular path. Rotation attack is one of the numerous attack methods. Although the rotation attack does not remove the watermark information from the image, it makes the detection and embedding of watermark lose synchronization, leading to the failed detection of the watermark.Table3 displays the extracted results of applying rotation attack to some watermarked images. The watermarked images are rotated by 45∘, 90∘, and 135∘ in the counterclockwise direction, respectively. In general, the scale of the image would change due to the discreteness of the digital image after rotation of an image. To keep the rotated image with the same size as the original image, a part of the edge information of the rotated image is cut off. However, since the Harris-Laplace detector has the benefits of rotation invariance and angle invariance, the extracted watermarks can still be recognized by human eyes of Figure 11. Table 3 also shows that the NC values of the extracted watermark from different angles are as high as 0.95.Table 3
NC values of the watermarked images under rotation attack.
Cover imageCameramanPeppersCouple45∘0.99010.98660.985190∘0.95910.97580.9807135∘0.96600.95980.9823Figure 11
Rotation attack on different watermarked images with45∘, 90∘, 135∘: Cameraman (a), (c), (e), extracted watermark (b), (d), (f); Peppers (a1), (c1), (e1), extracted watermark (b1), (d1), (f1); Couple (a2), (c2), (e2), extracted watermark (b2), (d2), (f2).


### 4.3.2. Cutting Attack

The cutting attack means that an attacker cuts out an area of an image containing watermark and discards it. A good watermarking algorithm should be able to extract the watermark effectively on the premise of keeping the information from the main image, while the image is attacked by cutting.Simulation results are performed to test the robustness of the proposed approach against cutting attack on Figure12. The extracted results from a cutting version of the watermarked images with different cutting areas, respectively, are presented in Table 4. Since it is impossible to retrieve a part of the image contents discarded by the attacker during the watermark detection, the damage caused by the cutting attack to the image is irreversible. When the cutting area is quite large, the robustness of the extracted watermark may be not well. However, the contour of the extracted watermark images could be still identified by human eyes. The experimental results demonstrate that the proposed algorithm based on non-subsampled shearlet transform and Harris-Laplace detector has more visually recognizable information about copyright property, and the quality of the watermarked images is still in a good situation, even under the large cutting area. The extracted watermarks and the original watermarks are strongly correlated.Figure 12
Cutting attack on different images with quarter, half, irregular: cameraman (a), (c), (e), extracted watermark (b), (d), (f); peppers (a1), (c1), (e1), extracted watermark (b1), (d1), (f1); couple (a2), (c2), (e2), extracted watermark (b2), (d2), (f2).Table 4
NC values of the watermarked images under cutting attack.
Cover imageCameramanPeppersCoupleQuarter cutting0.99050.98340.9890Half cutting0.96500.98900.9780Irregular cutting0.98710.98870.9903


### 4.3.3. Scaling Attack

The scaling of an image can be divided into two situations, equal scaling and unequal scaling. An equal scaling signifies that the image is scaled in the same proportion in the horizontal and vertical directions. The unequal scaling is that the image is scaled in different proportions. And the content of the image would be distorted with the unequal scaling.The results of the watermarked images and the corresponding extracted watermarks under different scaling factors on different watermarked images are given in Figure13. The contents of the image do not change when an image is attacked by scaling; however, there is an interpolation operation in the process of scaling. Table 5 presents the NC values for different scaling factors on different watermarked images. Since the scaling attack on the image is an irreversible operation, the loss part of the original information would have a great impact on the watermarking system when the scaling factor is higher than 1. However, the extracted watermarks can still be recognized. The robustness of the watermarking algorithm based on the shearlet transform is well overall.Figure 13
Scaling attack on different images with different scaling factors 1.5 and 0.75, respectively: cameraman (a), (a1), extracted watermark (b), (b1); peppers (c), (c1), extracted watermark (d), (d1); couple (e), (e1), extracted watermark (f), (f1).Table 5
NC values of the watermarked images under the different scaling factors.
Cover imageCameramanPeppersCouple1.50.94670.94420.95110.750.98230.98440.9769


### 4.3.4. Other Common Attacks

In addition to rotation attack, cutting attack, and scaling attack, the proposed watermarking algorithm based on non-subsampled shearlet transform and Harris-Laplace detector also displays good robustness against common attacks such as Gaussian low pass filtering, noise, and compression, as shown in Table6.Table 6
NC values of the proposed scheme on different watermarked image with various attacks.
AttacksImagePeppersCoupleCameramanJPEG compression (75%)0.98830.98420.9889Gaussian low-pass filtering (0.1)0.99010.98900.9890Gaussian noise (0.1)0.97830.93620.9843Salt and pepper noise (0.03)0.97050.98710.9891Scaling (0.7) & rotation (10∘)0.92810.99070.9674Since the feature points typically own the geometric invariance, embedding watermark in feature points or embedding watermark with feature points as a reference can make watermark resist the geometric attacks. And the non-subsampled shearlet transform abandons the operation of down sampling in the process of the standard shearlet transform and overcomes the phenomenon of the spectrum aliasing after transform. Therefore, the watermarking performance can be further improved by combining the directional sensitivity of non-subsampled shearlet transform with the geometric invariance of the Harris-Laplace detector.


## 4.3.1. Rotation Attack

The rotation of a digital image actually moves every pixel of the image along a circular path. Rotation attack is one of the numerous attack methods. Although the rotation attack does not remove the watermark information from the image, it makes the detection and embedding of watermark lose synchronization, leading to the failed detection of the watermark.Table3 displays the extracted results of applying rotation attack to some watermarked images. The watermarked images are rotated by 45∘, 90∘, and 135∘ in the counterclockwise direction, respectively. In general, the scale of the image would change due to the discreteness of the digital image after rotation of an image. To keep the rotated image with the same size as the original image, a part of the edge information of the rotated image is cut off. However, since the Harris-Laplace detector has the benefits of rotation invariance and angle invariance, the extracted watermarks can still be recognized by human eyes of Figure 11. Table 3 also shows that the NC values of the extracted watermark from different angles are as high as 0.95.Table 3
NC values of the watermarked images under rotation attack.
Cover imageCameramanPeppersCouple45∘0.99010.98660.985190∘0.95910.97580.9807135∘0.96600.95980.9823Figure 11
Rotation attack on different watermarked images with45∘, 90∘, 135∘: Cameraman (a), (c), (e), extracted watermark (b), (d), (f); Peppers (a1), (c1), (e1), extracted watermark (b1), (d1), (f1); Couple (a2), (c2), (e2), extracted watermark (b2), (d2), (f2).


## 4.3.2. Cutting Attack

The cutting attack means that an attacker cuts out an area of an image containing watermark and discards it. A good watermarking algorithm should be able to extract the watermark effectively on the premise of keeping the information from the main image, while the image is attacked by cutting.Simulation results are performed to test the robustness of the proposed approach against cutting attack on Figure12. The extracted results from a cutting version of the watermarked images with different cutting areas, respectively, are presented in Table 4. Since it is impossible to retrieve a part of the image contents discarded by the attacker during the watermark detection, the damage caused by the cutting attack to the image is irreversible. When the cutting area is quite large, the robustness of the extracted watermark may be not well. However, the contour of the extracted watermark images could be still identified by human eyes. The experimental results demonstrate that the proposed algorithm based on non-subsampled shearlet transform and Harris-Laplace detector has more visually recognizable information about copyright property, and the quality of the watermarked images is still in a good situation, even under the large cutting area. The extracted watermarks and the original watermarks are strongly correlated.Figure 12
Cutting attack on different images with quarter, half, irregular: cameraman (a), (c), (e), extracted watermark (b), (d), (f); peppers (a1), (c1), (e1), extracted watermark (b1), (d1), (f1); couple (a2), (c2), (e2), extracted watermark (b2), (d2), (f2).Table 4
NC values of the watermarked images under cutting attack.
Cover imageCameramanPeppersCoupleQuarter cutting0.99050.98340.9890Half cutting0.96500.98900.9780Irregular cutting0.98710.98870.9903


## 4.3.3. Scaling Attack

The scaling of an image can be divided into two situations, equal scaling and unequal scaling. An equal scaling signifies that the image is scaled in the same proportion in the horizontal and vertical directions. The unequal scaling is that the image is scaled in different proportions. And the content of the image would be distorted with the unequal scaling.The results of the watermarked images and the corresponding extracted watermarks under different scaling factors on different watermarked images are given in Figure13. The contents of the image do not change when an image is attacked by scaling; however, there is an interpolation operation in the process of scaling. Table 5 presents the NC values for different scaling factors on different watermarked images. Since the scaling attack on the image is an irreversible operation, the loss part of the original information would have a great impact on the watermarking system when the scaling factor is higher than 1. However, the extracted watermarks can still be recognized. The robustness of the watermarking algorithm based on the shearlet transform is well overall.Figure 13
Scaling attack on different images with different scaling factors 1.5 and 0.75, respectively: cameraman (a), (a1), extracted watermark (b), (b1); peppers (c), (c1), extracted watermark (d), (d1); couple (e), (e1), extracted watermark (f), (f1).Table 5
NC values of the watermarked images under the different scaling factors.
Cover imageCameramanPeppersCouple1.50.94670.94420.95110.750.98230.98440.9769


## 4.3.4. Other Common Attacks

In addition to rotation attack, cutting attack, and scaling attack, the proposed watermarking algorithm based on non-subsampled shearlet transform and Harris-Laplace detector also displays good robustness against common attacks such as Gaussian low pass filtering, noise, and compression, as shown in Table6.Table 6
NC values of the proposed scheme on different watermarked image with various attacks.
AttacksImagePeppersCoupleCameramanJPEG compression (75%)0.98830.98420.9889Gaussian low-pass filtering (0.1)0.99010.98900.9890Gaussian noise (0.1)0.97830.93620.9843Salt and pepper noise (0.03)0.97050.98710.9891Scaling (0.7) & rotation (10∘)0.92810.99070.9674Since the feature points typically own the geometric invariance, embedding watermark in feature points or embedding watermark with feature points as a reference can make watermark resist the geometric attacks. And the non-subsampled shearlet transform abandons the operation of down sampling in the process of the standard shearlet transform and overcomes the phenomenon of the spectrum aliasing after transform. Therefore, the watermarking performance can be further improved by combining the directional sensitivity of non-subsampled shearlet transform with the geometric invariance of the Harris-Laplace detector.


## 5. Comparison with Existing Works

To further verify the performance of the proposed watermarking algorithm, it is compared with previous typical schemes [23, 27, 30]. Since the singular values of the BSVD are small variation when image is faced with a little distortion, and the DST possesses the multidirectional properties, the scheme of combining the DST with the BSVD is applicable to images with different textures [23]. However, the DST may produce the pseudo-Gibbs phenomenon. To overcome the problem of the destruction of watermark synchronization, an ASIFT algorithm based on the resynchronization was adopted to extract the feature points as reference points for matching establishment between watermarked and received images [30]. The scheme in [30] combining the DWT with the ASIFT can be used for distortion correction. However, the exhaustive method of the ASIFT involves heavy computations, and the physical significance of affine transform on the original image is not obvious. Afterwards, a robust watermarking algorithm for medical images based on the Harris-SURF-DCT was proposed to protect the patient information [27]. The Harris detector was utilized to extract the feature points of the medical image, and the extracted points were described through the speeded up robust features (SURF) algorithm to generate the feature descriptor matrix. Although the method describing feature points in the SURF has scale invariance, the Harris detector algorithm does not have scale invariance.Therefore, a watermarking scheme based on non-subsampled shearlet transform (NSST) and Harris-Laplace detector is proposed. The NSST is an improvement of shearlet transform, which inherits the advantages of the shearlet transform and avoids the pseudo-Gibbs phenomenon. The NSST is the shift-invariant version of the shearlet transform. The NSST differs from the shearlet transform in that the NSST eliminates the downsamplers and upsamplers. The NSST is a fully shift-invariant, multiscale, and multidirectional expansion. The NSST combined the nonsubsampled Laplacian pyramid transform (NSLP) with several different combinations of the shearing filters. Since the Harris-Laplace detector has scale invariance, the combination of the Harris-Laplace detector and the fuzzy c-means cluster can enhance the robustness of the watermarking scheme. The result of the NC values is shown in Table7. It can be seen from Table 7 that, for the conventional attacks, the overall performance of the proposed algorithm is not so good as the other algorithms. However, for geometric attacks, except for scaling attack, the performance of the proposed algorithm is significantly better than the other algorithms.Table 7
NC values of the existing schemes.
AttacksReference [23]Reference [30]Reference [27]ProposedDST & BSVDDWT & ASIFTHarris-SURF & DCTNSST & Harris-LaplaceJPEG compression (75%)0.99210.99711.00000.9827Gaussian low-pass filtering (0.5)0.98830.98530.95630.9866Gaussian noise (0.05)0.98370.92930.93740.9666Salt and pepper noise (0.05)0.98150.94520.88160.9620Quarter cutting0.90250.84610.87450.9841Rotation (45∘)0.98430.96080.85380.9870Scaling (0.8)0.95310.99630.93520.9860


## 6. Conclusion

Most image watermarking schemes are weak in the geometric distortions. A robust watermarking algorithm based on non-subsampled shearlet transform and Harris-Laplace detector against the geometric attacks is proposed. The stronger texture feature regions of the image are selected by Harris-Laplace detector and fuzzy c-means cluster scheme, and the trackless embedding of the watermark can be achieved with singular value decomposition. The robustness of the watermarking algorithm against the geometric attacks is improved by the geometric invariance of Harris-Laplace feature points. In conclusion, the proposed watermarking algorithm has the following advantages.(1)
The algorithm has strong robustness to conventional signal processing attacks and geometric attacks. Especially under geometric attacks, the algorithm performs better.(2)
The algorithm combines the chaotic encryption with the Arnold transform to ensure that the watermark information will not be easily leaked. It is more secure due to double secret keys.



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*Source: 7605595-2022-03-21.xml*