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Article

A Hybrid Domain Color Image Watermarking Scheme Based on Hyperchaotic Mapping

by
Yumin Dong
1,*,
Rui Yan
1,
Qiong Zhang
2 and
Xuesong Wu
1
1
College of Computer and Information Science, Chongqing Normal University, Chongqing 401331, China
2
College of Geography and Tourism, Chongqing Normal University, Chongqing 401331, China
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(12), 1859; https://doi.org/10.3390/math12121859
Submission received: 14 May 2024 / Revised: 5 June 2024 / Accepted: 11 June 2024 / Published: 14 June 2024
(This article belongs to the Topic Computer Vision and Image Processing, 2nd Edition)
Browse Figures
Figure 1
<p>Attractor projection of Lorenz hyperchaotic system. (<b>a</b>) x-y plane, (<b>b</b>) x-z plane, (<b>c</b>) x-w plane, (<b>d</b>) y-z plane, (<b>e</b>) y-w plane, (<b>f</b>) z-w plane.</p> "> Figure 2
<p>Lorenz chaotic map encryption effect, (<b>d</b>–<b>f</b>) respectively (<b>a</b>–<b>c</b>), corresponding to the histogram.</p> "> Figure 3
<p>Example of discrete wavelet transform: (<b>a</b>) original image; (<b>b</b>) first-order wavelet transform; and (<b>c</b>) second-order wavelet transform.</p> "> Figure 4
<p>Watermark embedding process.</p> "> Figure 5
<p>Watermark extraction process.</p> "> Figure 6
<p>Cover image and watermark image.</p> "> Figure 7
<p>Watermarked image and corresponding extracted watermark image.</p> "> Figure 8
<p>Histogram comparison between cover image and embedded watermark image.</p> "> Figure 9
<p>Attacked cover image.</p> "> Figure 10
<p>Watermark images extracted from different attacks.</p> ">
Versions Notes

Abstract

:
In the field of image watermarking technology, it is very important to balance imperceptibility, robustness and embedding capacity. In order to solve this key problem, this paper proposes a new color image adaptive watermarking scheme based on discrete wavelet transform (DWT), discrete cosine transform (DCT) and singular value decomposition (SVD). In order to improve the security of the watermark, we use Lorenz hyperchaotic mapping to encrypt the watermark image. We adaptively determine the embedding factor by calculating the Bhattacharyya distance between the cover image and the watermark image, and combine the Alpha blending technique to embed the watermark image into the Y component of the YCbCr color space to enhance the imperceptibility of the algorithm. The experimental results show that the average PSNR of our scheme is 45.9382 dB, and the SSIM is 0.9986. Through a large number of experimental results and comparative analysis, it shows that the scheme has good imperceptibility and robustness, indicating that we have achieved a good balance between imperceptibility, robustness and embedding capacity.

1. Introduction

With the rapid spread of the Internet, the dissemination of information is easier than ever before, which has led to serious copyright infringement issues, such as unauthorized copying [1], distribution [2] and illegal tampering with digital works [3]. Therefore, in order to improve the security of Internet information, copyright protection becomes very important. The most common infringement problem is mainly the infringement of images. Image infringement will have a serious impact on the copyright party, especially important medical or military-related images. If they are tampered with or illegally used by criminals, they will have very important consequences. Therefore, in order to ensure the safe development of the information society, the security protection of digital images is urgent. As one of the widely used protection technologies, the watermarking method has been applied in many fields of multimedia copyright protection. As a common information embedding technology, it plays an important role in protecting image, video and audio information.
The embedding methods of image watermarking can be divided into many different types. According to the embedding process of watermark, the watermark can be divided into spatial domain watermark [4] and frequency domain watermark [5]. Spatial domain watermark is mainly embedded by directly changing the pixels of the cover image, such as the least significant bit (LSB) algorithm. The spatial domain watermark has low computational complexity and high operation efficiency, but the robustness is poor. Frequency domain watermarking is mainly realized by some commonly used image transformations. The pixels of the image are transformed into frequency coefficients by transformation, and then the watermark information is embedded in the frequency coefficients. The frequency domain algorithm greatly improves the robustness of the watermark, but the computational complexity is usually higher and the operation time is longer. Fourier transform (DFT), discrete wavelet transform (DWT), discrete cosine transform (DCT), singular value decomposition (SVD) and so on are commonly used frequency domain watermarking schemes.
Robustness, imperceptibility, embedding capacity and security are the four core indicators to evaluate the performance of image watermarking algorithms. There is often a trade-off relationship between these indicators, that is, in practical applications, it is difficult to optimize these indicators at the same time. Alpha hybrid technology [6] can be used to solve this problem. By finely adjusting the embedding strength, a relatively balanced state can be found between these four evaluation indicators, so as to ensure certain robustness and security while minimizing the imperceptibility of the watermark. By adaptively calculating the embedding strength, the robustness of the watermark image can be significantly enhanced. In contrast, a fixed embedding strength may cause the watermark to be more easily damaged. Therefore, adaptive computing provides a more effective way to improve the performance of watermarking [7]. In the face of geometric attacks, the watermarking method based on wavelet transform shows a certain degree of vulnerability [8], but the extraction of image geometric features by matrix decomposition can overcome this limitation. The combination of wavelet transform and matrix decomposition is often used for watermark embedding, which aims to prevent image tampering and resist geometric attacks. In addition, due to the concentration of DCT coefficients in the low-frequency part of the image, the image can still maintain good stability when experiencing compression, clipping, rotation and other operations [9]. In addition, color space also plays an important role in image watermarking technology. In particular, the YCbCr color space separates the brightness information (Y component) from the chromaticity information (Cb and Cr components). Among them, the Y component expresses the brightness of the image, while the Cb and Cr components carry the color information of the image. This separation makes the watermark embedding process more targeted; in particular, the brightness information to hide the watermark is more effective so as to ensure that the watermark is embedded at the same time and as far as possible to reduce the overall image quality of the potential impact [10].
Hajjaji et al. [11] proposed an efficient hardware-based image watermarking scheme based on Haar discrete wavelet transform (DWT). The system uses DWT technology to subtly embed key information fragments into digital content with excellent robustness, and also designs an ultra-large-scale integrated architecture based on FPGA. This architecture is specifically optimized for watermarking algorithms to accelerate the media authentication process. Hamidi et al. [12] proposed an innovative hybrid robust blind image watermarking technology, which combines the advantages of DFT-DCT and Arnold transform, and significantly enhances the imperceptibility and robustness of the watermark. In the embedding process, the technology makes specific adjustments to the DCT coefficients of the intermediate band through the key. In order to ensure the security of the watermark, Arnold transform is used to encrypt the watermark before embedding. This method enables the watermark to maintain a high degree of invisibility in the face of multiple attacks, thereby effectively protecting the image content.
Li et al. [13] proposed a reversible watermarking scheme based on wavelet transform. The cover image is divided into blocks, each block is transformed by wavelet transform as the carrier signal, and two different chaotic maps are used for encryption, which improves the security of the watermark image. Hsu et al. [14] proposed a new adaptive blind watermarking technique for deep forgery detection. This technique embeds the deep forgery detection information into the image, extracts the image without additional information and verifies the authenticity of the image. The scheme uses a combination of mixed modulation and partial symbol change mean to embed a set of coefficients. These coefficients not only maintain the image quality but also enhance the robustness against attacks. Dey et al. [15] designed a blind watermarking method for medical image authentication. This method embeds the patient’s information into the computer’s scanned copy in digital form, and applies the SHA-512 hash algorithm to process the watermark. The generated hash value is embedded into the digital copy of the scanned image to ensure the authenticity and integrity of the scanned copy.
Bhinder et al. [16] proposed an adaptive image watermarking technique that combines discrete wavelet transform (DWT) and fast Walsh–Hadamard transform (FWHT). In this method, the cover image is first divided into blocks, and then the entropy value of each block is calculated. The watermark embedding factor is determined by the statistical characteristics of the transform coefficients in the high entropy block (such as mean, standard deviation, entropy and kurtosis), so as to realize the effective embedding of the watermark. Hua et al. [17] proposed an adaptive watermarking scheme for remote sensing images based on information complexity. The scheme first converts the remote sensing image from RGB color space to YCbCr color space, and performs two-stage discrete wavelet transform (DWT) on the brightness component. Subsequently, the scheme selects the high-frequency coefficients of the low-frequency component as the embedding area of the watermark. At the same time, in order to determine the best watermark embedding position, the image is further divided into blocks, and the entropy value of each sub-block is calculated. Finally, the sub-block with the largest entropy value is selected as the watermark embedding target. Su et al. [18] proposed a robust color image watermarking method combining spatial domain and LU decomposition depth fusion. This method embeds the color watermark image into the color cover image by adjusting the high-correlation elements in the matrix L, and can realize the correction function of the geometric attack. Yuan et al. [19] designed a robust feature watermarking algorithm for encrypted medical images, which combines multi-level discrete wavelet transform (DWT), chrysanthemum descriptor and discrete cosine transform (DCT). In order to enhance the security of the image, logistic mapping is also used to encrypt the original medical image.
The main contributions of this paper are as follows:
  • A large-capacity robust image watermarking scheme based on DWT-DCT-SVD is proposed;
  • The security of the watermarking algorithm is improved by encrypting Lorenz hyperchaotic map;
  • In the YCbCr color space, the Bhattacharyya distance between the cover image and the watermark image is used to adaptively calculate the embedding factor, which solves the balance between the robustness and imperceptibility of the watermark.
The rest of this article is structured as follows. Section 2 introduces some techniques used in this paper, including Lorenz hyperchaotic system, discrete wavelet transform, discrete cosine transform and singular value decomposition. Section 3 introduces the watermarking scheme proposed in this paper. Section 4 is the experimental results of this scheme. Section 5 summarizes the work of this paper.

2. Background

2.1. Lorenz Hyperchaotic System

Chaotic phenomena are common in nonlinear systems that are sensitive to initial conditions and control parameters, and their orbits exhibit randomness and unpredictability. This feature makes chaotic systems have significant advantages in the field of security, such as randomness, complexity and reliability [20]. Therefore, chaotic systems are often used as encryption methods. As a classical chaotic model, the Lorenz hyperchaotic system is defined as follows:
d x d t = a ( y − x ) + w , d y d t = c x − y − x z , d z d t = x y − b z , d w d t = − y z + r w .
The hyperchaotic system needs to meet the following conditions. First, its phase space is at least four-dimensional. Secondly, it has at least two positive Lyapunov exponents [21,22].
In Equation (1), a, b, c and r are control parameters. When a = 10 , b = 8 / 3 , c = 28 , and − 1.52 ≤ r ≤ − 0.06 , the system is in a hyperchaotic state. When r = − 1 , the four Lyapunov exponents of the above formula are λ 1 = 0.3381 , λ 2 = 0.1586 , λ 3 = 0 , and λ 3 = − 15.1752 . The initial value range of the Lorenz hyperchaotic system is x 0 ∈ ( − 40 , 40 ) , y 0 ∈ ( − 40 , 40 ) , z 0 ∈ ( 1 , 81 ) , w 0 ∈ ( − 250 , 250 ) . Figure 1 shows six strange attractors of the Lorenz hyperchaotic system, where the phase diagram of Figure 1c resembles a butterfly, and is also known as the butterfly attractor.
Figure 2 shows the encryption of a 256 × 256 grayscale image using the chaotic sequence generated by the Lorenz hyperchaotic system in the case of initial values x 0 = 3.5739 , y 0 = 3.7979 , z 0 = 4.2980 , w 0 = 3.9282 .
From Figure 2, it can be found that the histogram of the original image and the decrypted image is almost the same, indicating that the encryption process has little effect on the image quality. However, the encrypted image histogram is significantly different from the original image, and its distribution is more uniform. As a visual display of image pixel frequency distribution, histogram is an important parameter to measure the anti-statistical analysis ability of the image encryption algorithm. In general, the higher the uniformity of the encrypted image histogram, the stronger its ability to resist statistical analysis attacks. The histogram of the original image often has specific rules and characteristics, while the histogram of the encrypted image tends to be flat or have uniform distribution, which is significantly different from the original image. This reflects that the image encryption algorithm based on the Lorenz hyperchaotic system significantly reduces the statistical correlation between pixels, making it difficult to reveal the obvious rules of the image through histogram analysis, thereby improving the security of image encryption.

2.2. Discrete Wavelet Transform

Discrete wavelet transform (DWT) is a key technology in the field of signal processing and image processing, which realizes the multi-resolution analysis of signals or images [23]. In image processing, DWT can decompose the image into four sub-bands: LL, LH, HL and HH. The LL sub-band is rich in the main low-frequency information of the image, and the energy is concentrated and stable. Therefore, the LL sub-band is usually rotated during watermark embedding. The LH sub-band combines the characteristics of horizontal low frequency and vertical high frequency, which are suitable for edge detection. The HL sub-band is used for image enhancement because it carries horizontal high-frequency and vertical low-frequency information. HH sub-bands are sensitive to the texture and edge details of the image, but these sub-bands are also relatively vulnerable to attack. In order to further improve the security of watermark embedding, the LL sub-band can be further decomposed by a double DWT. Figure 3 shows an example of image discrete wavelet transform.

2.3. Discrete Cosine Transform

Discrete cosine transform (DCT) is a key technology in image processing, which converts images from the spatial domain to the frequency domain [24]. DCT decomposes the image into components of different frequencies, where the low-frequency component reflects the overall structure of the image, while the high-frequency component contains the details of the image. DCT is widely used in the fields of image compression, coding and feature extraction. Its advantages are energy concentration, strong robustness, independent transform coefficients and wide adaptability. DCT also plays an important role in image watermarking processing. Through frequency domain conversion, DCT decomposes image information into components of different frequencies, making watermark embedding more efficient. Since the watermark can be hidden in the high-frequency component and is highly similar to the original image, it is not easy to be detected. In addition, the characteristics of DCT make the watermark have certain robustness. Even if the image is compressed, rotated or scaled, the effective extraction of the watermark can be guaranteed by selecting the appropriate DCT coefficients for watermark embedding. This is because the energy of DCT is mainly concentrated in a few low-frequency coefficients, which are more important to the overall characteristics of the image, so it can resist the influence of image processing operations.

2.4. Singular Value Decomposition

Singular value decomposition (SVD) is a powerful linear algebraic tool [24] for decomposing matrices into three key parts: two orthogonal matrices and a diagonal matrix. In the field of image processing, SVD is widely used in dimensionality reduction, denoising and compression scenarios. Its core idea is to identify and extract the dominant features in the matrix, which are measured by singular values, so that the matrix can be decomposed into multiple feature subspaces. Matrix A ∈ R m × n can represent an image, and the size of the matrix is m × n ; then,
U = [ U 1 , U 2 , U 3 , ⋯ , U m ]
V = [ V 1 , V 2 , V 3 , ⋯ , V m ]
which enable the following:
U T A V = σ 1 σ 2 ⋱ σ m = Σ
where V ∈ R m × n and U ∈ R m × n , both of which are orthogonal matrices, and Σ ∈ R m × n denotes a diagonal matrix whose values on the off-diagonal are all 0, and whose values on the diagonal are satisfied:
σ 1 ≥ σ 2 ⋯ ≥ σ r ≥ σ r + 1 = ⋯ = σ m = 0
where r denotes the rank of A whose size is the number of non-zero real numbers on the diagonal, and σ i ( i = 1 , 2 , ⋯ , m ) is the singular value of the matrix.
From the perspective of image processing, the singular values of the image matrix are closely related to the key features of the image itself, which makes the singular values have excellent stability. Even if the image is slightly disturbed or attacked, its singular value can remain relatively stable and will not change significantly. It is worth mentioning that the singular values of images are invariant to geometric transformations (such as rotation, scaling and translation), which makes them resistant to various image processing operations. Therefore, the introduction of singular value decomposition in digital-image watermarking technology can greatly enhance the robustness of the algorithm and make the watermark more difficult to be destroyed or tampered with. This method not only enhances the security of the watermark, but also ensures the stable extraction of the watermark in the image processing process.

2.5. Bhattacharyya Distance

Bhattacharyya distance [25] is a method to evaluate the similarity between two probability distributions. It calculates the distance between two distributions based on comparing the similarity between them. For two discrete probability distributions p and q, the distance provides a quantitative metric, which is defined as follows:
D b ( p , q ) = − ln ( B C ( p , q ) )
where B C ( p , q ) represents the Bhattacharyya coefficient. In the case of discrete distribution and continuous distribution, the Bhattacharyya coefficient is defined as
B C ( p , q ) = ∑ p ( x ) q ( x )
B C ( p , q ) = ∫ p ( x ) q ( x ) d x
For two normal distributions p i = N ( μ i , σ i 2 ) , where μ i and σ i denote the expectation and standard deviation of the distribution, respectively, the Bhattacharyya distance is defined as follows [25]:
D b = 1 8 ( μ 1 − μ 2 ) T σ − 1 ( μ 1 − μ 2 ) + 1 2 ln | σ | | σ 1 | | σ 2 |
where σ = ( σ 1 + σ 2 ) 2 .

3. The Proposed Watermarking Scheme

3.1. Adaptive Embedding Factor

The determination of the watermark embedding factor is a trade-off problem, which needs to be adjusted according to specific applications and image attributes. In the actual operation, it may need to be tested and adjusted many times to find the best embedding factor, so as to ensure a good balance between the watermark effect and the image quality. In digital images, since the probability density function usually conforms to the normal distribution characteristics, the embedding factor is determined by evaluating the similarity between images, and the watermark is embedded into the cover image accordingly, making it difficult to be detected.
The non-adaptive calculation of the embedding factor is easy to make the watermark image fragile, and it is cumbersome to calculate these values. Therefore, adaptive computing becomes the preferred solution. We use the Bhattacharyya distance between the cover image and the watermark image to adaptively calculate the embedding factor, and normalize the distance by an exponential function to obtain the embedding factor. The specific calculation formula is as follows:
α = λ × 1 1 + e d
where λ is the parameter to be adjusted, the value range is 0 to 1, and d is the Bhattacharyya distance between the cover image and the watermark image.

3.2. Watermark Embedding

In this paper, the cover image is a color image, and the watermark image is a gray image. In order to improve the security of the watermarking algorithm, we use Lorenz hyperchaotic mapping to encrypt the watermark image. Then, the singular value decomposition operation is performed on the encrypted image. In order to enhance the imperceptibility of the watermarking algorithm, we convert the cover image to the YCbCr color space, and select the Y component for discrete wavelet decomposition. The selection of the YCbCr color space is designed to improve the visual transparency of the watermarking algorithm, and the Y component is selected because the Y component has stronger robustness when embedding watermarks than the Cb and Cr components. Then, we perform discrete wavelet decomposition on the Y component and perform discrete cosine transform on the obtained LL sub-band. Finally, we perform singular value decomposition on the image after discrete cosine transform. The embedding factor is adaptively calculated by the Bhattacharyya distance of the cover image and the watermark image to perform Alpha hybrid technology to embed the watermark. The steps of watermark embedding are shown in Figure 4.
The specific steps of watermark embedding are as follows:
Step 1: Read the gray watermark image and encrypt it with Lorenz hyperchaotic map. The control parameters of the chaotic system are a = 10 , b = 8 / 3 , c = 28 and r = 1 . The initial values of the chaotic system are x 0 = 3.5739 , y 0 = 3.7979 , z 0 = 4.2980 and w 0 = 3.9282 .
W E = L o g r e n z ( W I )
Step 2: Using singular value decomposition, the image W E is decomposed into three matrices U w , S w and V w :
[ U w , S w , V w ] = S V D ( W E )
Step 3: Read the color cover image and convert it to the YCbCr color space, and then select the Y component:
[ H Y , H C b , H C r ] = R G B 2 Y C b C r ( H R G B )
Step 4: Use Equation (10) to adaptively calculate the embedding factor a.
Step 5: The Y component ( H Y ) of the cover image is decomposed by the first-order Haar wavelet, and it is decomposed into four sub-bands: LL, HL, LH and HH:
[ L L , H L , L H , H H ] = D W T ( H Y )
Step 6: Perform discrete cosine transform on LL to obtain
L L D = D C T ( L L )
Step 7: The singular value decomposition of the image after discrete cosine transform is performed to obtain three matrices U H , S H and V H :
[ U H , S H , V H ] = S V D ( L L D )
Step 8: Watermark embedding is performed by Alpha mixing the result S w of the singular value decomposition in step 2 with the result S H of singular value decomposition in step 7:
S H ′ = α × S w + ( 1 − α ) × S H
Step 9: According to the matrix U H , S H ′ and V H , the L L D part is reconstructed by inverse singular value decomposition:
L L D ′ = U H × S H ′ × V H T
Step 10: The L L ′ sub-band is reconstructed by inverse discrete cosine transform:
L L ′ = I D C T ( L L D ′ )
Step 11: According to the four sub-bands of L L ′ , HL, LH and HH, the inverse discrete wavelet transform is performed to obtain the Y channel image Y ′ embedded with watermark:
Y ′ = I D W T [ L L ′ , H L , L H , H H ]
Step 12: Use Cb, Cr and Y ′ to convert the image into RGB color space to obtain the image after embedding the watermark:
H R G B ′ = Y C b C r 2 R G B [ H Y , H C b , H C r ]

3.3. Watermark Extraction

In the watermark extraction process, the watermarked image is first converted to the YCbCr color space, and then the Y component is subjected to discrete wavelet decomposition. The LL sub-band is selected for discrete cosine transform and singular value decomposition. Then, the singular value matrix of the watermark is extracted by inverse Alpha mixing. Finally, the encrypted watermark image is obtained by inverse singular value decomposition, and then the watermark image is restored by the decryption process. The steps of watermark extraction are shown in Figure 5.
The specific steps of watermark extraction are as follows:
Step 1: Read the watermarked color image and convert it to the YCbCr color space:
[ W M Y , W M C b , W M C r = R G B 2 Y C b C r ( W M R G B ) ]
Step 2: Select the Y component and perform first-order Haar wavelet decomposition on it:
[ L L W M , H L W M , L H W M , H H W M ] = D W T ( W M Y )
Step 3: Discrete wavelet transform is performed on the L L W M sub-band to obtain D L L :
D L L = D C T ( L L W M )
Step 4: Perform the singular value decomposition of D L L , which is decomposed into three matrices U W M , S W M and V W M :
[ U W M , S W M , V W M ] = S V D ( D L L )
Step 5: The singular value matrix of the encrypted watermark can be obtained by embedding the factor α and the singular value matrix S H :
S W ′ = S W M − ( 1 − α ) × S H α
Step 6: Recover the encrypted watermark by performing inverse singular value decomposition on the matrices U W , S W ′ and V W :
W E ′ = U W × S W ′ × V W T
Step 7: Use Lorenz hyperchaotic map to decrypt the image and finally obtain the watermark image.

4. Experimental Results and Analysis

In this section, six color cover images with a resolution of 512 × 512 and a gray watermark with a resolution of 256 × 256 are used to verify the performance of the proposed scheme. Figure 6 shows the cover image and watermark image used in this paper. All the image data used in this paper are from USC-SIPI: Signal and Image Processing Institute, University of Southern California (https://sipi.usc.edu/database/) (accessed on 1 June 2024). The experiment is carried out in MATLAB (R2020b) environment on Intel (R) Core (TM) i5-12400F CPU 2.50 GHz, 16.0 GB RAM computer. The peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are used to measure the imperceptibility of the proposed scheme, while the normalized correlation coefficient (NCC) and bit error rate (BER) are used to measure the robustness of the proposed scheme. The embedding factor is calculated by using Equation (10). In this paper, the value of λ is 0.1. Table 1 shows the embedding factor of each cover image.

4.1. Invisibility Analysis

Invisibility means that the similarity between the watermarked image and the original cover image is difficult to identify. This slight change is almost invisible to the human eye. In this study, two key indicators, PSNR and SSIM, were selected to evaluate the watermarking methods applied on six different cover images. Among them, PSNR is used to measure the transparency of the watermark image and the original cover image in visual perception, while SSIM is responsible for evaluating the contrast, brightness and structural similarity of the image. When the PSNR value exceeds 30 dB, it means that the watermark image has an acceptable visual quality and maintains a high degree of similarity with the original image. When the SSIM value is close to 1, it indicates that the watermark image has reached a high level of perceptual quality [26]. Figure 7 shows these cover images with watermarks and the successfully extracted watermarks.
The PSNR, MSE and SSIM of different cover images are shown in Table 2. The average PSNR and SSIM of the scheme are 45.9382 dB and 0.9986, respectively. It can be seen that the scheme has good imperceptibility.
Image histogram is a tool similar to PSNR, which is widely used to measure the difference between the cover image and the watermarked image to evaluate the influence of the watermark on the original cover image. The image histogram intuitively shows the distribution of different pixel values in the image, thus revealing the core features of the image. By comparing the histograms of the two, we can more reliably judge the similarity between images, which also shows that the watermark embedding method does not significantly change the original characteristics of the cover image. Figure 8 intuitively shows the histogram comparison between the original cover image and the watermarked image. It can be observed from the figure that there is almost no significant change in the histogram before and after watermark embedding, which fully shows the excellent performance of our scheme in imperceptibility.

4.2. Robustness Analysis

In this section, we focus on evaluating the robustness of the proposed scheme against various attacks. Based on this, we use three main metrics: PSNR, NCC and BER. These three indicators play a crucial role in evaluating the performance of the scheme. PSNR is used to show the perceptual transparency of the watermark image relative to the original cover image. The NCC value directly reflects the anti-attack ability of the scheme. The higher the value is, the stronger the resistance of the scheme to the attack is. The decrease in bit error rate means that the degree of watermark distortion is smaller, which is very important in maintaining the integrity of watermark information.
In this paper, various attacks are comprehensively tested. These attacks are mainly divided into four categories: noise attack, enhanced technology attack, geometric transformation attack and compression attack. In the experiment, we compare the attacked cover image and the successfully restored watermark. The comparison results are shown in Figure 9 and Figure 10, where [a–l] represent different types of attacks, respectively: no attack, salt and pepper noise (v = 0.05), Gaussian noise (m = 0, v = 0.05), speckle noise (v = 0.05), sharpening, Gaussian filtering, mean filtering, rotation 10°, vertical flipping, cropping 20%, JPEG compression (QF = 90), and JPEG 2000 compression (QF = 90). These attacks represent a variety of situations that may be encountered in the real world, and our scheme has been thoroughly tested to ensure robustness in all situations.
Through our experimental results, we can clearly see that our scheme can effectively resist other attacks, except rotation and cropping attacks, and the extracted watermark image has only slight distortion. This shows that our scheme has high robustness and can effectively deal with attacks in various situations in the real world, thus protecting the integrity and stability of watermark information.
Table 3 describes the PSNR of the attacked watermark cover image, Table 4 describes the PSNR of the watermark image extracted from different attacks, Table 5 describes the NCC of the watermark image extracted from different attacks, and Table 6 describes the BER of the watermark image extracted from different attacks. Under the condition of no attack, the average PSNR of this method is 45.6849 dB, the average NCC of this method is 1, and the average bit error rate is 0. And under various attacks, the average NCC and average BER can maintain a good level. Table 7 describes the robustness comparison results with other watermarking schemes. Taking the Baboon cover image as an example, it can be seen from Table 7 that the watermarking scheme in this paper can effectively extract the watermark image in the face of various attacks.
It can be seen from Table 3 that when the cover image is attacked by different attacks, there is a big difference between the PSNR of the attacked image and the original image, and the quality of the image is significantly reduced, but the watermark image extracted from it can still maintain high quality. The average PSNR of the extracted watermark can maintain 20 dB, indicating that the watermark scheme in this paper has good robustness. From the comparison with other schemes in Table 7, it can also be seen that most of the NCC values of the schemes in this paper are higher than other schemes, indicating that the scheme in this paper has higher robustness.

4.3. Embedding Capacity Analysis

The embedding capacity is a measure of the number of bits of information that can be inserted into the cover image [31]. In this scheme, a color cover image with a size of 512 × 512 is used, and a grayscale watermark image with a size of 256 × 256 is used. Therefore, the embedding capacity of the scheme is (256 × 256 × 8) / (512 × 512 × 3) = 0.6667 bpp. Most of the other watermarking schemes use 64 × 64 or 128 × 128 watermarks, and the embedding capacity is 0.0417 and 0.1667, respectively. The embedding capacity is low, and the scheme in this paper has high embedding capacity.

4.4. Security Analysis

The security of the image watermark is a key point that the watermark algorithm needs to consider. A robust algorithm needs to ensure that illegal personnel cannot extract the watermark image from the cover image. In this paper, we use Lorenz hyperchaotic map encryption to protect the watermark. Even if the attacker extracts the watermark, only the encrypted image is obtained. Lorenz chaotic map has good chaotic properties, including sensitivity dependence on initial conditions, long-term unpredictability, etc. A small change in initial conditions can cause a significant deviation from the system trajectory, making it form a complex trajectory in the phase space. These characteristics ensure that the generated chaotic sequence has a high degree of randomness and improve the strength of watermark encryption. In addition, the scheme has good robustness. In the face of some attacks, the watermark image can be completely extracted, which greatly improves the security of the watermark.

5. Conclusions

In watermarking technology, it is necessary to make a trade-off between imperceptibility, robustness and embedding capacity, but many watermarking schemes ignore this problem. In order to solve this problem, we propose an adaptive color image watermarking scheme based on DWT-DCT-SVD. The embedding factor is adaptively determined by calculating the Bhattacharyya distance between the cover image and the watermark image. At the same time, in order to enhance the security of the watermark, we use Lorenz hyperchaotic mapping to encrypt the watermark image. Experimental results show that our scheme achieves a balance between imperceptibility, robustness and embedding capacity. The digital watermarking of color images is widely used, including digital copyright protection, security authentication, information hiding, digital media traceability and other fields. Therefore, the research and application of digital color image watermarking technology has become a very important topic in the field of digital information security. The future research direction of this study to improve on our findings mainly combines image watermarking with intelligent optimization algorithm or deep learning, which can search for better embedding factors through intelligent optimization. At the same time, the combination of deep learning and image watermarking will greatly improve the performance and application potential of watermarking.

Author Contributions

Conceptualization, R.Y.; methodology, R.Y.; validation, R.Y. and Q.Z.; formal analysis, X.W.; investigation, R.Y.; resources, R.Y.; data curation, R.Y.; writing—original draft preparation, R.Y. and Q.Z.; writing—review and editing, Y.D.; supervision, Y.D.; project administration, Y.D.; funding acquisition, Y.D. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by National Natural Science Foundation of China (No.61772295), Key Projects of Chongqing Natural Science Foundation Innovation Development Joint Fund (CSTB2023NSCQ-LZX0139), Open Fund of Advanced Cryptography and System Security Key Laboratory of Sichuan Province (Grant No. SKLACSS–202208) and Technology Research Program of Chongqing Municipal Education Commission (Grant no. KJZD-M202000501).

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:
DWTDiscrete wavelet transform
DCTDiscrete cosine transform
SVDSingular value decomposition
PSNRPeak signal-to-noise ratio
SSIMStructural similarity
MSEMean-squared error
NCCNonlinear correlation coefficient
EBRBit error rate

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Figure 1. Attractor projection of Lorenz hyperchaotic system. (a) x-y plane, (b) x-z plane, (c) x-w plane, (d) y-z plane, (e) y-w plane, (f) z-w plane.
Figure 1. Attractor projection of Lorenz hyperchaotic system. (a) x-y plane, (b) x-z plane, (c) x-w plane, (d) y-z plane, (e) y-w plane, (f) z-w plane.
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Figure 2. Lorenz chaotic map encryption effect, (d–f) respectively (a–c), corresponding to the histogram.
Figure 2. Lorenz chaotic map encryption effect, (d–f) respectively (a–c), corresponding to the histogram.
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Figure 3. Example of discrete wavelet transform: (a) original image; (b) first-order wavelet transform; and (c) second-order wavelet transform.
Figure 3. Example of discrete wavelet transform: (a) original image; (b) first-order wavelet transform; and (c) second-order wavelet transform.
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Figure 4. Watermark embedding process.
Figure 4. Watermark embedding process.
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Figure 5. Watermark extraction process.
Figure 5. Watermark extraction process.
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Figure 6. Cover image and watermark image.
Figure 6. Cover image and watermark image.
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Figure 7. Watermarked image and corresponding extracted watermark image.
Figure 7. Watermarked image and corresponding extracted watermark image.
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Figure 8. Histogram comparison between cover image and embedded watermark image.
Figure 8. Histogram comparison between cover image and embedded watermark image.
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Figure 9. Attacked cover image.
Figure 9. Attacked cover image.
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Figure 10. Watermark images extracted from different attacks.
Figure 10. Watermark images extracted from different attacks.
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Table 1. Embedding factors of cover image and watermark image.
Table 1. Embedding factors of cover image and watermark image.
Images α Images α
Airplane0.0214Baboon0.0307
Goldhill0.0381House0.0245
Lake0.0398Peppers0.02368
Table 2. The imperceptibility results of different watermark images.
Table 2. The imperceptibility results of different watermark images.
Cover ImagePSNRNSESSIM
Airplane45.39991.87540.9980
Baboon46.18301.56600.9994
Goldhill46.35811.50410.9998
House45.79421.71260.9985
Lake46.02331.62460.9986
Peppers45.87091.68260.9985
Table 3. PSNR of the attacked watermark cover image.
Table 3. PSNR of the attacked watermark cover image.
Type of AttackAirplaneBaboonGoldhillHouseLakePeppers
No attack45.399946.183046.358145.794246.023345.8709
Salt and pepper noise (m = 0, v = 0.05)24.192524.519924.283924.280824.065824.0156
Gaussian noise (m = 0, v = 0.05)21.960421.174321.347721.572021.684821.3546
Speckle noise (m = 0, v = 0.05)23.350124.268825.157623.462824.675924.1880
Sharpening22.758314.594221.546321.090820.083022.1108
Gaussian filter26.668823.838626.552526.276726.273526.7413
Mean filtering (3 × 3)26.334622.946826.197525.865425.905726.5179
Rotation (10°)11.722413.871413.642911.967711.875012.4293
Flip vertical13.315813.723611.601913.369810.680611.3423
Cropping (20%)17.946818.795820.915318.587318.605519.2897
JPEG (QF = 90)27.140727.234427.147727.140127.033727.1295
JPEG2000 (QF = 90)26.151022.592925.852125.387125.050826.2070
Table 4. PSNR of watermark image extracted from different attacks.
Table 4. PSNR of watermark image extracted from different attacks.
Type of AttackAirplaneBaboonGoldhillHouseLakePeppers
No attack45.123646.293745.964045.435245.602345.6903
Salt and pepper noise (m = 0, v = 0.05)21.993018.136719.501821.636222.598521.0464
Gaussian noise (m = 0, v = 0.05)16.498514.843615.189415.798115.261614.9989
Speckle noise (m = 0, v = 0.05)17.299115.623817.066518.883920.605618.9589
Sharpening17.813711.740413.739112.226714.380416.0217
Gaussian filter27.165517.429020.007823.235325.589927.6140
Mean filtering (3 × 3)25.498916.657819.307721.664323.966725.8653
Rotation (10°)12.568811.96378.92877.389510.93749.4181
Flip vertical45.123646.293745.964045.435245.602345.6903
Cropping (20%)15.180512.452916.882111.901110.419112.3544
JPEG (QF = 90)40.381729.152523.380835.202832.732137.1544
JPEG2000 (QF = 90)31.148718.661319.705025.829428.289130.7598
Table 5. NCC of watermark image extracted from different attacks.
Table 5. NCC of watermark image extracted from different attacks.
Type of AttackAirplaneBaboonGoldhillHouseLakePeppers
No attack111111
Salt and pepper noise (m = 0, v = 0.05)0.99180.99710.99180.99530.99620.9948
Gaussian noise (m = 0, v = 0.05)0.91460.92420.90600.91930.92460.9062
Speckle noise (m = 0, v = 0.05)0.98220.99470.99170.99150.99660.9976
Sharpening0.97450.89580.95620.95710.94020.9694
Gaussian filter0.99600.95750.99280.98930.98870.9959
Mean filtering (3 × 3)0.99440.94350.98980.99020.98390.9942
Rotation (10°)0.81820.87150.83790.83320.88440.9742
Flip vertical111111
Cropping (20%)0.88710.96360.98170.96130.90090.9848
JPEG (QF = 90)0.99860.99660.99920.99990.99950.9999
JPEG2000 (QF = 90)0.99830.96110.99510.99670.99180.9983
Table 6. BER of watermark images extracted from different attacks.
Table 6. BER of watermark images extracted from different attacks.
Type of AttackAirplaneBaboonGoldhillHouseLakePeppers
No attack000000
Salt and pepper noise (m = 0, v = 0.05)0.08910.09260.07220.07650.07040.0836
Gaussian noise (m = 0, v = 0.05)0.17130.17870.15980.15640.16270.1733
Speckle noise (m = 0, v = 0.05)0.13890.13000.07100.11080.09470.1128
Sharpening0.14730.25760.20590.18550.17540.1623
Gaussian filter0.07430.22800.11160.11210.08080.0640
Mean filtering (3 × 3)0.09070.26640.13560.13890.09940.0772
Rotation (10°)0.30670.29790.35380.33360.24710.2872
Flip vertical000000
Cropping (20%)0.31480.32780.16440.34710.29560.2659
JPEG (QF = 90)0.00750.06610.03710.01650.03470.0056
JPEG2000 (QF = 90)0.04070.12360.13810.07900.05490.0432
Table 7. Comparison of NCC with other schemes under different attacks.
Table 7. Comparison of NCC with other schemes under different attacks.
Type of AttackProposed[27][28][21][29][30]
No attack110.999210.99670.9976
Salt and pepper noise (m = 0, v = 0.05)0.99710.97800.9583-0.8693-
Gaussian noise (m = 0, v = 0.05)0.92420.90720.9294-0.79960.8339
Speckle noise (m = 0, v = 0.05)0.99470.90460.96250.91940.9276-
Sharpening0.8958-0.93850.9596-0.8560
Gaussian filter0.9575--1-0.8186
Mean filtering (3 × 3)0.94350.94200.9796-0.92690.8271
Rotation (10°)0.8715--0.9569--
Cropping (20%)0.96360.9997-0.9438--
JPEG (QF = 90)0.99660.9921-0.99790.99480.94779
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Dong, Y.; Yan, R.; Zhang, Q.; Wu, X. A Hybrid Domain Color Image Watermarking Scheme Based on Hyperchaotic Mapping. Mathematics 2024, 12, 1859. https://doi.org/10.3390/math12121859

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Dong Y, Yan R, Zhang Q, Wu X. A Hybrid Domain Color Image Watermarking Scheme Based on Hyperchaotic Mapping. Mathematics. 2024; 12(12):1859. https://doi.org/10.3390/math12121859

Chicago/Turabian Style

Dong, Yumin, Rui Yan, Qiong Zhang, and Xuesong Wu. 2024. "A Hybrid Domain Color Image Watermarking Scheme Based on Hyperchaotic Mapping" Mathematics 12, no. 12: 1859. https://doi.org/10.3390/math12121859

APA Style

Dong, Y., Yan, R., Zhang, Q., & Wu, X. (2024). A Hybrid Domain Color Image Watermarking Scheme Based on Hyperchaotic Mapping. Mathematics, 12(12), 1859. https://doi.org/10.3390/math12121859

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