Compressive Sensing Theory for Improving the Robustness and the Security of the Discrete Wavelet Transform-Singular Value Decomposition Watermarking Scheme

: Digital watermarking has been suggested as a solution to protect copyright and ownership of digital images. The effective watermarking scheme should satisfy imperceptibility, robustness, capacity and security requirements. In this research, the security issue and the lack of reconstruction quality of the watermark logo of the integration of Discrete Wavelet Transform (DWT), Singular Value Decomposition (SVD) and Compressive Sensing (CS) watermark technique are addressed. Wilkinson measurement matrix is proposed to increase the robustness and improve the reconstruction quality of the watermark logo in the DWT and SVD based on the CS watermarking scheme. Additionally, the security issue of the SVD is solved by sending a scrambled compressed logo then extracting U, V matrices from this compressed logo at the decoder. The proposed method not only increases security but also reduces the size of the transmitted watermark logo. The robustness of the proposed algorithm is evaluated in the presence of different attacks like signal processing, compression, geometrical and noise attacks. The simulation results demonstrate that the proposed technique is more robust than the conventional techniques for all attacks, as well as its robust security. Moreover, the quality of the watermarked image is measured and the results show that the imaging performance of the proposed technique is approximately 0.7 to 4 dB better than that of the conventional methods. Consequently, the proposed watermarking scheme proves its ability to achieve the demands of security, imperceptibility and robustness against different attacks.


Introduction
Although the internet is considered a beneficial communication tool that recently can make people's lives much more manageable, it also has terrible effects. One of these harmful effects is that the owner's right is threatened by the circulation of illegal and unauthorized digital content. Every day vast amounts of digital information are exposed to different types of attacks via the internet as data theft, spying, data corruption and denial of service attacks. This leads to the risk of violating owner copyright and hampering digital content's authenticity; therefore, the owners of digital content need to provide a form of proof for their owners to preserve their ownership rights. Many researchers introduced various techniques to protect ownership rights by embedding some extra information called watermark into digital content (Potdar et al., 2005) and extracting it later for copyright protection and integrity authentication. Strong security of copyright content with sufficient robustness against different attacks is required to design an excellent watermarking scheme. Watermarking schemes are divided according to working domains into spatial or transform domain techniques.
Spatial domain watermarking techniques slightly modify the images' pixels by simple addition or replacement of bits in the selected region according to the watermark. However, these techniques have less robustness against various attacks. Alternatively, the transform domain techniques are proposed in different literature to overcome these spatial domain technique limitations. In transform domain techniques, the watermark embedding method is applied to the coefficients of the transform domain such as Discrete Cosine Transform (DCT) (Xu et al., 2011), Discrete Wavelet Transform (DWT) (Li et al., 2008) and Singular Value Decomposition (SVD) (Bahadur and Chandra, 1959;Liu and Tan, 2002;Chang et al., 2005) of the image. The advantages of these techniques compared to spatial domain techniques are that they provide more robustness against many types of attacks, especially the SVD technique. SVD is characterized by its good noise immunity of the singular values of an image against any disturbance. Therefore, various research combine between SVD and different transform domain techniques to strengthen the watermarking scheme versus different attacks (Jane and Elbaşi, 2014;Hu and Hsu, 2015;Fazli and Moeini, 2016;Singh and Singh, 2017;Thakkar and Srivastava, 2017;Zhou et al., 2018). Although the SVD technique is usually used because of its robustness and stability to various attacks, it has a security issue called the False Positive Problem (FPP) (Makbol et al., 2018). It was discovered that the orthogonal matrices U and V keep the geometric properties of an image as they illustrate the Eigenvectors of the corresponding singular values. Therefore, when inverse SVD is applied and any different singular matrix is used with U and V, a correlated output is generated instead of the actual output. This issue is considered a security threat because an attacker may replace his/her own U, V set with the original one during watermark extraction and claims false ownership. This problem is solved in (Gupta and Raval, 2012) by adding a signature based on U and V matrix in the LL band. At the receiver, the extracted signature is compared to the generated signature from the sent U and V to prove the ownership. This solution improves the security problem of the SVD but degrades the quality of the watermarked image.
Compressive Sensing (CS) theory is presented as an alternative way of signal sampling that differs from the standard Shannon-Nyquist theorem in the last ten decades. The compressed sensing paradigm integrates the sensing and compression processes for the sparse signals in a straightforward linear measurement step. Compressive sensing gets randomly projected measurements that containing the content features of the image. The image is reconstructed from the compressed measurements based on the sensing measurement matrix's knowledge (Candès, 2006;Candès and Wakin, 2008). The scrambling sensing method of CS draws the researchers' attention in the security research area. Thus, CS is used in the watermarking schemes to improve the security of the conventional watermarking technique and provide copyright protection.
Different researches have integrated the CS theory and the watermarking scheme in various methods. One of these methods is to apply the CS to the host image by embedding the watermark logo into the host image's measurements (Huang and Chang, 2014;Chi and Feng, 2014). However, this method's disadvantage is that any alteration in the host image can lead to a modification in the measurements. Thus, embedding the watermark directly into the measurements is not robust to frequent attacks. The other method is to apply the CS technique on the watermark logo to represent it by a small number of measurements. Then these measurements are embedded directly into the host image. This method aims to enhance the security of the watermark and provide more payload capacity.
Various researches integrate between the CS, SVD and DWT to get the merits of all of them and overcome the security issue of SVD (Chen et al., 2013;Wang et al., 2017). However, it was discovered in other researches that this integration suffers from some degradation in the quality of the watermark logo and the watermarked image due to the use of the random measurement matrices in CS theory. These conventional random matrices usually don't provide good reconstruction quality and need high computational times.
In this proposed paper, the integration between CS, DWT and SVD watermarking scheme is addressed to overcome its previously mentioned problem. First, the watermark images are compressed sensing using the Wilkinson matrix instead of the conventional random matrices to improve the reconstructed watermark logo's quality. Then, the compressed sensing watermark logo's singular values are embedded in the HH band of the DWT of the host image. The FPP problem of SVD is solved by sending a compressed watermark as a secret key to the receiver. Therefore, if the attacker replaces his watermark with the original compressed watermark, the compressive sensing reconstruction process is applied to his watermark and a distorted image is produced. Therefore, the CS technique improves the quality of the watermark logo, overcomes the FPP security issue of SVD and reduces the size of the sent watermark logo.
The rest of this paper is prearranged as follows: Section 2 briefly illustrates the CS theory, measurement matrix, OMP and Singular value decomposition method. In section 3, the embedding and the extracting watermark schemes are explained. The evaluation metrics used to evaluate the watermarked and watermark images are introduced in section 4. In section 5, the simulation results are presented and discussed. Finally, section 6 concludes the idea and the results of this study.

Compressive Sensing Theory
Shannon-Nyquist theorem is the most important theorem that sets a limit to the sampling rate, guaranteeing recovery of the signal. This theorem states that the sampling rate of the signal should be at least twice of its bandwidth. Unfortunately, in many important and emerging applications, the resulting Nyquist rate is so high that we end up with far too many samples. CS theory overcomes this Nyquist theorem problem. CS theory states that if the signal has a sparse representation in a spatial or in another domain; it can be recovered exactly from a far fewer linear, non-adaptive measurements (Candès, 2006;Romberg, 2008). In compressed sensing, the signal to be sampled is generally represented as a vector, say xR N that has at most k (k << N) nonzero components. The compressed signal y with the length of M (M<N) is gotten through the linear transformation, y = x where R MN (Candès and Wakin, 2008). The compressive sensing goal is to acquire the signal x with no or insignificant loss of information using fewer than N measurements to accurately reconstruct x from y with M<N. Therefore, the design of the measurement matrix is an important issue in compressive sensing since data recovery depends on how well the limited measurements provide information about the structure of the signal. To guarantee that the measurement matrix gets the largest amount of information from the signal, it should satisfy certain property which is the mutual coherence property. The less coherent the columns of  are, the better the reconstruction works because any two closely related columns will carry the same information about the signal and therefore, they may mislead any reconstruction technique. Therefore in compressive sensing, the measurement matrix should have very low coherence. Many researches introduce measurement matrices that possess these properties one of these matrices is the Random Gaussian matrix. These random matrices have the useful property of being almost maximally incoherent with almost all sparsifying bases. Thus, they are a universal choice for measurement Qaisar et al., 2013). However, these random matrices suffer from very high computational complexity and large storage capacity due to their inherent random structure. Also, these matrices reduce the quality of the reconstructed image from CS.
Reconstructing the original signal x from the compressed signal requires solving an under-determined set of linear equation y = x where the number of unknowns (N) is much larger than the number of measurements (M). Thus, the estimation of x is considered an ill-posed problem.
A wide variety of approaches exist to recover the sparse signal from a small number of linear measurements. The two major CS recovering approaches are L1-minimization and greedy algorithms Because the greedy algorithms are faster and simpler to implement compared to the L1 (Needell et al., 2008), one of the greedy algorithm which is the Orthogonal Matching Pursuit (OMP) will be used in this study for signal reconstruction (Tropp and Gilbert, 2007).
The Orthogonal Matching Pursuit (OMP) main idea is to select columns of the measurement matrix that contribute to greedily generating measurements y. The measurement matrix  and the measured vector y is used as inputs for the OMP and an estimate x of the original signal x is provided as an output. The OMP constructs an approximation by going through an iteration process. During each iteration, one column of  is chosen, which is most strongly correlated with the residual vector r. then, the contribution of this column is removed to compute a new residual. The initial condition of the residual vector is chosen equal to the vector that is required to be approximated i.e., r = y. The OMP algorithm can be described as follows (Meenakshi, 2015): Compute the least square problem in order to obtain the new estimate of the sparse signal: At the end of the iterations, the non-zero elements of xt will be at the locations corresponding to the respective columns of the dictionary matrix Φ at the index set Vt and the rest elements will be zeros, making it a sparse vector of size N × 1 (Rabah et al., 2014). The OMP will require more iterations to reach the halting criteria if the measurement matrix and the sparsity transform are not incoherent together or the measurement matrix is a dense matrix. This will produce less representative measurement vector y and lead the OMP to select the inappropriate columns during the reconstruction process, resulting in a lack of reconstruction quality.

Singular Value Decomposition
SVD is a linear algebra scheme, which is used to decompose a matrix into three matrices. For a M×N matrix A, its SVD representation is defined as:  M and N×N respectively. If A is an image, the matrix U represents the horizontal and V represents the vertical detail of this image (Thakkar and Srivastava, 2017). Rotation invariant and translation invariant are the properties of SVD, which make it Robust to many kinds of image processing attacks. Also, SVD is characterized by the immunity of its singular values to noise. A little variation in singular values when a small perturbation is added to an image intensity values does not affect the visual quality of this image. These properties of SVD make it the right candidate for the watermarking application (Rani et al., 2015).

The Proposed Watermarking Scheme
In this study, the watermarking scheme based on CS, DWT and SVD techniques is addressed to overcome its previous scheme drawbacks and improve its security and robustness features. First, the proposed scheme provides the system with three different security keys in various stages to guarantee that the scheme is very secure. Second, the security issue of the SVD technique is solved by sending the compressed sensing logo and extracting the U and V matrices at the decoding side to guarantee that an attacker cannot replace the U, V of the original watermark logo with his watermark. Moreover, even if the watermark is replaced by the attacker's watermark, the compressive sensing reconstruction process is applied to it and produces a distorted image. Third, the proposed scheme improves the quality of the reconstructed watermark logo and watermarked images by replacing the conventional random matrices in the CS scheme with the Wilkinson matrix.
The Wilkinson matrix, the proposed watermark embedding and the extraction scheme will be explained in detail in the next subsection.

Wilkinson Matrix
The Wilkinson matrix is asymmetric tri-diagonal matrix with 1's on the off-diagonals its structure shown in Fig. 1.
The advantage of this matrix is when re-orthognalize only M rows from its rows, different columns with zero value are generated at the end of the matrix as shown in Fig. 2 (Shoitan et al., 2018a). This advantage is useful if the DCT is the sparsity transform. The low-frequency components of an image in the DCT domain, which is considered the significant DCT elements, are found on the top left.
Therefore, the compressed vector y is a linear multiplication of the most vital coefficients of the DCT and the columns of non-zero values in the Wilkinson matrix, while the columns with zero value are multiplied by the DCT coefficients which contain less information. Thus, the compressed vector y is constructed from the most significant information in an image. At the receiver, OMP search for the columns in Φ (Wilkinson matrix), which strongly correlated with the measured vector y, then removes its contribution from y.   Then, search for the second highly correlated column from the residual and repeat this step until the DCT elements of x are recreated according to the OMP algorithm's steps. Thus, when the OMP search for the Wilkinson matrix columns most correlated with y, it will find all the columns with non-zero coefficients. However, if the vector y is constructed from a linear multiplication of most of the columns of the Gaussian measurement matrix Φ and most of the DCT coefficients, this misdirects the OMP to find the proper columns from the Gaussian matrix (Shoitan et al., 2018b). Figure 3 presents the structure of the proposed watermark embedding scheme. At the transmitter, the watermark is embedded using two main processes. The first process is applying CS to the sparse watermark logo in the DCT domain. Then the Wilkinson measurement matrix is applied to the produced sparse watermark logo and then these measurements are sent by the owner as keys. The second process is applying DWT to the host image. Because the HH band contains the finer details and contributes insignificantly to the image energy, it is selected to replace their singular values with the compressed watermark's singular values. The watermark logo cannot be reconstructed at the receiver without inversing the produced sparse coefficients using IDCT. Thus the measurement matrix and the sparsity transform are considered as secret keys at the receiver to restore the Watermark logo from the host image. The algorithm steps for watermark embedding are given below: The Proposed Watermark Extraction Scheme Figure 4 shows the block diagram for the proposed extraction procedure. In the proposed watermark extraction scheme, the Wilkinson measurement matrix, the compressed logo and the DCT sparsity transform are considered secret keys that should be given as an input for the extraction algorithm to extract the watermark logo the watermarked image. So if someone wants to extract the watermark, all the keys must be obtained, which increases the proposed system's security. The algorithms steps of the proposed watermark extraction process are given below:

Evaluation Metrics
In this study, the perceptual quality and the robustness of the proposed watermarking scheme are evaluated using the following parameter.

Peak Signal to Noise Ratio (PSNR)
PSNR metric is used to validate the perceptual similarity between host image and watermarked image. The PSNR represents the ratio between the maximum power of the image and the power of distorting noise that affects the quality of its representation. While the maximum power is expressed as MAX 2 and MAX is the maximum pixel intensity. All images here are expressed using 8-bit intensity values per pixel; thus, the maximum pixel power MAX 2 is 255 2 . The mathematical representation of PSNR is given as follows: The PSNR approach infinity as MSE approaches zero; this shows that a higher PSNR value provides a highquality image (Mrak, 2004).

Correlation Coefficient (CC)
The correlation coefficient metric is used to measure the robustness of the proposed watermark scheme against various attacks. The correlation coefficient measures the closeness or similarity between the original and the extracted watermark logo.
The CC values vary between -1 to +1. The two images are very similar when the CC closes to +1.
However, CC value close to -1 indicates that the two images are highly dissimilar. The formula to compute

Experimental Results
In this section, the proposed watermarking scheme's performance is numerically evaluated and compared to the Conventional DWT+SVD and the DWT+SVD using the signature technique (Gupta and Raval, 2012). Furthermore, the impact of the Wilkinson matrix on the performance of the proposed method is compared to the impact of the Gaussian matrix by comparing the proposed algorithm to DWT+SVD+CS based on Gaussian matrix algorithm.
In this experiment, the measurement rate in the proposed method and the CS+SVD+DWT based on Gaussian matrix is adopted at M/N = 0.5. The proposed method's block size is empirically chosen as 6464, as mentioned (Shoitan et al., 2018b). The sparsity transform in the CS+DWT+SVD based on the Gaussian matrix is the DCT and OMP is the reconstruction algorithm.
Two different sets of host images are employed for performance evaluation. These test images are the Barbara and the Lena images of 512512 pixels shown in Fig. 5. Lena is a smooth image in which most of its content is low-frequency components, while Barbara is a detailed image in which most of its content are highfrequency components. Copyrights logo of size 512512 pixels is used as a watermark logo.
The performance of the proposed and the conventional watermarking techniques are evaluated by applying various standard watermarking attacks. Geometric attacks such as shear, rotation and cropping; Signal processing attacks as mean filter, median filter and Gaussian low pass filter. Noise attacks such as Gaussian noise, Salt and Pepper noise, JPEG compression and histogram equalization attacks. PSNR measures watermarked image quality while CC measures the reconstructed watermark logo quality. To evaluate the perceptual quality of the proposed technique's watermarked image relative to the other techniques, the PSNR is calculated and presented in Table 1.
From the results given in this table, it can be realized that the quality of the watermarked Lena image increased by approximately 4d B using the proposed technique over the conventional DWT+SVD with signature technique. This result reflects that the signature used to solve the security issue of SVD in the conventional DWT+SVD degrades the quality of the watermarked image.
The proposed technique enhances the quality of Lena images over the rest of the conventional methods by approximately 0.1 to 0.5 dB. For Barbara image, the proposed technique is slightly better than the conventional techniques by approximately 0.04 to 0.7 dB.  For evaluating the FPP security issue of the proposed technique, the watermark extraction program is run many times by inserting wrong inputs.
First, if the attacker adds a different compressed logo to the proposed technique, a distorted watermark logo is obtained, as shown in Fig. 6a. Second, if the attacker inserts the whole logo instead of the compressed logo, a distorted watermark logo is obtained, as shown in Fig. 6b. On the contrary, if the attacker inserts his logo instead of the required logo to the conventional DWT+SVD, the attacker logo is extracted instead of the owner logo. This result confirms that the program is very secure compared to the DWT+SVD technique.
To evaluate the robustness of the proposed method versus different attacks, the CC between the original and extracted watermark is measured and tabulated in Table 2 to 4. Table 2 summarizes the proposed technique's CC results and the conventional techniques in signal processing and compression attacks. It can be noticed from Table 2 that the proposed technique is more robust to signal processing and compression attacks compared to the conventional techniques, especially if the logo is embedded in the Barbara image.         The CC results of the proposed technique and the conventional techniques against noise attacks are given in Table 3. The results in this table reflect that the proposed method improves the robustness in noise attacks compared to the conventional techniques, especially the DWT+SVD+CS technique based on the Gaussian matrix. Also, concerning the geometric attacks, the CC is measured and given in Table 4. It can be observed that the performance of the proposed method is more robust than conventional techniques against the geometric attack.
Therefore, it can be concluded from the tables' results that the proposed DWT+SVD+CS based on the Wilkinson matrix technique outperforms the other techniques in robustness. However, it can be noticed that the proposed technique has a strong ability to resist signal processing, compression and geometric attack better than the noise attack.
To compare visually between the extracted watermark logo obtained from the proposed technique and the conventional techniques Figs. 7 and 8 present the extracted watermark logo from those techniques. Figure 7 shows the extracted watermark logo after applying signal processing and compression attack. It can be perceived that the extracted watermark logo from DWT+SVD and DWT+SVD with signature and the proposed technique is clear except the logo extracted after the mean filter attack, the images suffer from some darkness. The extracted watermark logo from CS+DWT+SVD based on the Gaussian matrix suffers from the darkness for mean attack and suffers from distortion for JPEG and histogram equalization attacks. Figure 8 shows the extracted watermark logo for noise attacks and geometric attacks after applying these attacks for all the mentioned techniques. It can be noticed from row 1 and row 2 that the extracted watermark logo from CS+DWT+SVD based on the Gaussian matrix suffers from a high distortion for all the noise attacks. However, the DWT+SVD, DWT+SVD with signature and the proposed techniques suffer from a little distortion.
For geometric attack, the extracted watermark logo from CS+DWT+SVD based on the Gaussian matrix suffers from high distortion and darkness for the geometric attack except for rotate 90 the distortion is smaller than the other geometric attacks as shown in row 3,4 and 5 in Fig. 8. On the contrary, the extracted watermark logo from the proposed technique and the DWT+SVD and DWT+SVD with signature techniques suffer from a bit of distortion compared to DWT+SVD+CS based on Gaussian matrix for rotate 45 and crop attacks. This visual analysis is matched with the numerical results in the tables.

Conclusion
In this research, the idea of scramble compressed sensed watermark logo based on Wilkinson matrix is proposed. The objective of the proposed work is to overcome the lack of reconstruction quality of the watermark logo, increase the robustness and improve the traditional DWT+SVD+CS scheme's security. Signal processing, compression, geometric and noise attacks are applied to the traditional and the proposed technique to evaluate their robustness. PSNR and CC are used as metrics to evaluate the watermarked image and watermark logo's quality, respectively. First, the PSNR results illustrate that the proposed technique improves the watermarked image quality compared to all the conventional techniques, especially for a smooth image. Second, the CC results demonstrated that the proposed techniques resist all the attacks better than conventional techniques, especially for Signal processing, compression and geometric attacks and these results are confirmed through visual analysis. Third, the security is tested by using an unauthorized watermark logo for the extraction program. The proposed technique did not extract the watermark logo; however, DWT+SVD extract the attacker logo.
Moreover, sending the watermark logo in a compressed form increases the security and is considered an advantage for limited resource applications that need to save power and reduce the required bandwidth to transmit the whole watermark logo. However, the proposed technique has a limitation on the property of the selected watermark. It should be sparse on time or frequency domains to apply the Compressive sensing to it and improve the performance.
Applying the proposed method to the color images using different color systems will be addressed in future research using different measurement matrix and reconstruction algorithms in compressive sensing theory.