A Domain-independent Dual-image based Robust Reversible Watermarking

Abstract

Robust reversible watermarking has attracted widespread attention in the field of information hiding in recent years. It should not only have robustness against attacks in transmission but also meet the reversibility of distortion-free transmission. According to our best knowledge, the most recent robust reversible watermarking methods adopt a single image as the carrier, which might lead to low efficiency in terms of carrier utilization. To address the issue, a novel dual-image robust reversible watermarking framework is proposed in this paper to effectively utilize the correlation between both carriers (namely dual images) and thus improve the efficiency of carrier utilization. In the dual-image robust reversible watermarking framework, a two-layer robust watermarking mechanism is designed to further improve the algorithm performances, i.e., embedding capacity and robustness. In addition, an optimization model is built to determine the parameters. Finally, the proposed framework is applied in different domains (namely domain-independent), i.e., Slantlet Transform and Singular Value Decomposition domain, and Zernike moments, respectively to demonstrate its effectiveness and generality. Experimental results demonstrate the superiority of the proposed dual-image robust reversible watermarking framework.

1. Introduction

To protect the security of multimedia information such as information integrity, copyright protection, etc., digital watermarking is designed and widely utilized. In general, traditional digital watermarking might cause irreversible distortion to the cover image during the secret data hiding process. To solve this issue, reversible watermarking (RW) has been proposed [1], which could not only extract the hidden secret data but recover the cover image, which is significant for some specific fields such as medical and military fields.

So far, many RW methods have been proposed on a single image as the carrier and could be mainly divided into four categories, namely lossless compression based [2-3], histogram shifting based [4-9], difference expansion based [10-12] and prediction error expansion based [13-18]. Recently, in order to improve the utilization of carriers, dual-image RW is proposed. In dual-image RW schemes [19-25], the sender first duplicates the cover image to make two identical versions. Different secret information is then hidden into the above-mentioned identical versions to generate the corresponding two stego-images. The receiver could extract those secret data from both received stego-images and restore the cover image in the case of no attack is encountered.

However, most traditional RW methods [2-29] belong to fragile watermarking, which means the hidden secret data could not be completely extracted when some normal attacks are encountered, i.e., JPEG compression, Gaussian noise, and even malicious attacks. To solve this issue, robust reversible watermarking (RRW) is proposed, and its framework is shown in Fig. 1. When no attack occurs, RRW enables the receiver to extract the watermark and to recover the cover image. Otherwise, the hidden secret data could be fully extracted even if the reversibility is lost.

Fig. 1. RRW general framework

In recent years, several RRW schemes are proposed [30-42] based on a single image as the carrier, which could mainly be classified into three categories: (1) histogram rotation (HR) based schemes [30-31], (2) schemes based on the modifications of traditional HS [32-33], (3) two-stage RRW framework based schemes [34-42]. The HR based methods [30,31] accomplish robust lossless watermarking by slightly rotating the centroid vectors of two random regions in the non-overlapping blocks. It was reported that these methods are robust to resist JPEG compression but are sensitive to “salt-and-pepper” noise. To solve the problem, schemes [32-33] based on a large modification of traditional HS can achieve good robustness. The method proposed by Zeng et al. in [32] divides the cover image into blocks and then exploits the arithmetic difference of each block to create the histogram. Gao et al. extended Zeng’s scheme by using a more efficient statistical quantity histogram [33]. For the third category, a two-stage general framework for RRW is proposed by Coltuc et al. [35-36]. The watermark is embedded into the cover image robustly in the first stage to generate an intermediate image. In the second stage, the information for recovery is embedded into the intermediate image by a reversible watermarking method. An et al. proposed a method based on a large modification of traditional HS in the wavelet domain [38]. In [38], the cover image would be divided into non-overlapping blocks and transformed into another domain to collect the coefficients. Then the watermark would be embedded into the coefficients by using histogram shifting. To improve the robustness, many two-stage RRW framework based schemes are developed in some transform domains and other domains such as the geometrical moment. Some discrete cosine transform domain (DCT) based RRW methods [35]-[36] are proposed to embed the secret data by modifying their frequency coefficients. In addition, the Slantlet transform domain (SLT) is used for RRW since it provides an optimal trade-off between time localization and smoothness characteristics [41]. Recently, Liu et al. proposed an RRW scheme based on SLT-SVD [34]. The method could further increase the robustness of the watermark by combining the SLT and the singular values decomposition (SVD). To improve the robustness for resisting most geometric attacks (e.g., rotation and scaling), Hu et al. proposed novel RRW schemes [37,39] to achieve the robustness to resist the geometric attacks by employing the polar harmonic transform (PHT) and Zernike moments (ZMs).

Because the embedding phases of the two stages are performed on the same domain, the watermark would be influenced by the reversible embedding stage. In [40], Wang et al. proposed a method to address the above-mentioned problem by using an independent domain.

However, it is observed that most robust reversible watermarking methods are performed on a single image. Inspired by dual-image RW [25], a novel dual-image RRW scheme is proposed in this paper to effectively exploit the correlation of dual images to achieve better algorithm performances, i.e., high visual quality of stego-image and large embedding capability. The main contributions of the proposed scheme are summarized as follows.

1. A novel dual-image RRW (denoted as DIRRW) framework is proposed.

2. A two-layer robust watermarking mechanism is designed in the proposed DIRRW framework to further improve the algorithm performances.

3. An optimization model is developed by optimizing the parameter.

4. Proposed framework, namely DIRRW is applied in different domains, i.e., SLT-SVD and Zernike domains, respectively to demonstrate its effectiveness and generality.

The rest of this paper is organized as follows. Section 2 briefs the related works for readers to better understand the proposed framework. The proposed method is presented in Section 3. Experimental results compared with some state-of-the-art RRW methods are reported in Section 4. Finally, we conclude this paper in the last section.

2. Related Works

In this section, we briefly introduce some related works in the following subsections. First, a two-stage RRW framework isintroduced, which is the basis of our proposed framework. Then, a robust quantization index modulation watermarking algorithm [43] is briefly described.

2.1 Two-stage RRW framework

The two-stage general framework proposed by Coltuc et al in [35] and [40] is widely used in RRW methods. As shown in Fig. 2, the framework consists of two stages, namely the robust embedding stage and the reversible embedding stage. In the first stage, a watermark is embedded into the cover image by using a robust embedding method to create the robust image (also called the intermediate image).

Fig. 2. Two-stage RRW framework

After that, in the second stage, the distortion produced by the robust embedding stage will be collected and then reversibly hidden into the robust image to generate the final stego-image.

2.2 A classical quantization index modulation watermarking algorithm

A classical robust watermarking algorithm, namely Quantization Index Modulation (QIM) watermarking algorithm, is introduced in [43]. Specifically, the secret data embedding phase and extraction operation are calculated as follows.

1) Secret data embedding:

\(\begin{aligned}e^{\prime}= \begin{cases}P * \operatorname{round}(e / P)+P / 2, & \text { if } \bmod (\operatorname{round}(e / P), 2)=w \\ P * \operatorname{round}(e / P)-P / 2, & \text { if } \bmod (\operatorname{round}(e / P), 2) \neq w \end{cases} \end{aligned}\)       (1)

where quantization step length 𝑃 is the threshold to control the robustness, round(∙) is the rounded operation, and mod(∙) indicates the modular operation. 𝑒 indicates the original value and 𝑒′ is the value embedded with the secret data 𝑤.

2) Secret data extraction:

The secret data 𝑤_𝑖 could be extracted in the decoder by (2)

𝑤_𝑖 = mod(⌊𝐸_𝑖⁄𝑃⌋, 2), 𝑖 = 1,2       (2)

where ⌊∙⌋ indicates the floor function.

Then, an example is taken to further illustrate the algorithm [43] as shown in Fig. 3.

Fig. 3. Example of the embedding and extraction process of the QIM in [43]

Embedding phase: The quantization step length 𝑃 is set to be 10, and the low-frequency coefficient is considered as 𝑒 ∈ (55,65) or (65,75). As shown in Fig. 3, when 1-bit secret data 𝑤 ∈ {0,1} is embedded, 𝑒 would be changed to 𝑒′. Specifically, for 𝑒 ∈ (55,65), 𝑒′ = 55 is got when 𝑤 = 1, while 𝑒′ = 65 is obtained when 𝑤 = 0. If 𝑒 is in the range (65,75), 𝑒′ = 75 is acquired when 𝑤 = 1, while 𝑒′ = 65 could be obtained when 𝑤 = 0.

Extraction phase: After the embedding phase, we could easily extract the embedded secret data 𝑤 according to (2). If the received 𝑒′ is not distorted, 𝑤 = 1 could be extracted from 𝑒′ = 55 or 75, while 𝑤 = 0 could be extracted from 𝑒′ = 65. It is noted that, even the distorted version of 𝑒′ (denoted as 𝑒̃ ′) is received , the secret data could be correctly extracted. For example, 𝑤 = 0 could be correctly extracted when 𝑒̃ ′ ∈ (60,70). Obviously, the length of the range (60,70) is equal to 𝑃, which means that the length robustness range to perfectly extract secret data is determined by 𝑃.

3. Proposed Scheme

A novel dual-image RRW framework, denoted as DIRRW, is proposed in this section. The framework is established in Subsection 3.1. The algorithm for two-layer robust watermarking is then presented in Subsection 3.2. In Subsection 3.3, a theoretical analysis is offered by comparing the proposed scheme and classical schemes with a single image as the carrier. Finally, implementation details for the proposed scheme are mentioned in Subsection 3.4.

3.1 Proposed DIRRW framework

The proposed DIRRW framework is illustrated in Fig. 4. Two main parts are contained in the framework, i.e., robust embedding stage and reversible embedding stage.

Fig. 4. Proposed dual-image RRW framework

Firstly, two duplicate images 𝐼₁ and 𝐼₂ are copied from the cover image 𝐼 at the sender. Both 𝐼₁ and 𝐼₂ are then transformed into a certain domain, i.e., SLT-SVD or Zernike moments to collect the low-frequency coefficients for two channels. 𝐸₁ = {𝑒₁¹, 𝑒₁², ⋯ , 𝑒₁^𝐿/3 } would be collected from the channel 1 (the channel where image 𝐼₁ is located). Similarly, 𝐸₂ = {𝑒₂¹, 𝑒₂², ⋯ , 𝑒₂^𝐿/3} would be collected from the channel 2.

Then, the 𝐿 -length watermark 𝑊 would be embedded into 𝐸₁ and 𝐸₂ by using the proposed two-layer robust watermarking algorithm, where 𝑊 are divided into 𝑊_𝑖 = {𝑤_𝑖¹, 𝑤_𝑖², . . . , 𝑤_𝑖^𝐿/3|𝑖 = 1,2,3}. In detail, 𝑊₁ and 𝑊₂ would be embedded into 𝐸₁ and 𝐸₂ in the first layer. After that, 𝐸₁′ = {𝑒₁′¹, 𝑒₁′², ⋯ , 𝑒₁′^𝐿/3} and 𝐸₂′ = {𝑒₂′¹, 𝑒₂′², ⋯, 𝑒₂′ ^𝐿/3} could be obtained, which would be modified to embed 𝑊₃ in the second layer by using their correlation. Last, 𝐸₁′′ = {𝑒₁′′¹, 𝑒₁′′², ⋯, 𝑒₁′′^𝐿/3 } and 𝐸₂′′ = {𝑒₂′′¹, 𝑒₂′′², ⋯, 𝑒₂′′^𝐿/3} would be generated.

Later, the low-frequency coefficients sets 𝐸₁′′ and 𝐸₂′′ embedded with the secret data 𝑊 would be utilized to replace the original sets 𝐸₁and 𝐸₂. Then, two intermediate images 𝐼_𝑖1 and 𝐼_𝑖2 would be reconstructed by inversing the domain. Considering that the pixels in the constructed transition images and the final stego-images should be integers, 𝐼_𝑡1 and 𝐼_𝑡2 would be rounded to generate two robust images 𝐼_𝑟1 and 𝐼_𝑟2.

To ensure the reversibility, the difference between the robust images and the cover image should be recorded as the auxiliary information 𝐻 by referring to [37]. The auxiliary information consists of two parts, i.e., quantization errors occurred in the robust embedding phase (denoted as 𝑑_𝑞) and rounded error occurred in the above-mentioned image rounded operation (denoted as 𝑑_𝑟).

Finally, 𝐻 would be hidden into the robust images 𝐼_𝑟1 and 𝐼_𝑟2 by employing the dual-image reversible data hiding method [25] to generate the final stego-images 𝐼_𝑤1 and 𝐼_𝑤2.

In the case of no attack in the transmission, the cover image could be recovered and the watermark could be extracted at the decoder. In general, the auxiliary information 𝐻 could be extracted from the stego-images 𝐼_𝑤1 and 𝐼_𝑤2 by referring to [25]. Meanwhile, the robust images 𝐼_𝑟1 and 𝐼_𝑟2 could be recovered. Later, the watermark 𝑊 could be extracted from 𝐼_𝑟1 and 𝐼_𝑟2 and the cover image could be recovered from 𝐼_𝑟1 by using auxiliary information 𝐻. In the case of some attacks in the transmission, the reversibility is lost due to the vulnerability of the reversible embedding stage. However, the watermark 𝑊 could be extracted directly from the distorted version of 𝐼_𝑤1 and 𝐼_𝑤2 due to the robustness of the two-layer robust watermarking algorithm in the proposed framework. The two-layer robust watermarking algorithm would be described in detail in the following subsection.

3.2 The two-layer robust watermarking algorithm

The proposed two-layer robust watermarking algorithm mentioned above would be utilized to embed the watermark 𝑊. After the embedding phases, the low-frequency coefficients 𝐸₁ and 𝐸₂ would be modified to 𝐸₁′′ and 𝐸₂′′. The embedding and extraction processes would be introduced in detail as follows.

Embedding phase

In the first layer, the proposed algorithm is formed by mixing two advanced methods. As mentioned in Subsection 2.2, one is the QIM watermarking algorithm [43], which provides robustness for the embedding of the watermark. The other is the decimal preservation operation [37], which could reduce the cost of collecting auxiliary information before the reversible embedding stage.

Specifically, 1-bit secret data 𝑤₁ and 𝑤₂ would be embedded into a pair of 𝑒₁ and 𝑒₂ utilizing the first-layer embedding algorithm by (3).

\(\begin{aligned}e_{i}^{\prime}=\begin{cases}P * \operatorname{round}\left(e_{i} / P\right)+P / 2+\Delta_{i}, & \text { if } \bmod \left(\operatorname{round}\left(e_{i} / P\right), 2\right)=w_{i} \\ P * \operatorname{round}\left(e_{i} / P\right)-P / 2+\Delta_{i}, & \text { if } \bmod \left(\operatorname{round}\left(e_{i} / P\right), 2\right) \neq w_{i} \end{cases} \end{aligned}\)       (3)

where 𝑃 is the quantization step length as mentioned in Subsection 2.2, ∆_𝑖 = 𝑒_𝑖 − ⌊𝑒_𝑖⌋ is the decimal part of 𝑒_𝑖 . After the first layer embedding phase, 𝑒₁′ and 𝑒₂′ would be obtained. Meanwhile, 𝑑_𝑞 would be recorded via the first channel in our proposed scheme as follows:

𝑑_𝑞 = 𝑒₁′ − 𝑒₁       (4)

In the second layer, the correlation between 𝑒₁′ and 𝑒₂′ could be exploited to provide a new space for further watermark embedding. In detail, two correlation coefficients are computed by (5).

\(\begin{aligned}\begin{cases} D=e_{1}^{\prime}-e_{2}^{\prime} \\ A=e_{1}^{\prime}+e_{2}^{\prime}\end{cases} \end{aligned} \)       (5)

where 𝐷 and 𝐴 represent the difference and the sum of 𝑒₁′ and 𝑒₂′ , respectively. Then, 𝑤₃ would be embedded by (6).

\(\begin{aligned}e_{i}^{\prime \prime}=\begin{cases} e_{i}^{\prime}+(-1)^{i} \frac{P}{4}, \quad \text { if } w_{3}=\bmod \left(\operatorname{round}\left(\frac{D}{P}\right), 2\right) \\ e_{i}^{\prime}-(-1)^{i} \frac{P}{4}, \quad \text { if } w_{3} \neq \bmod \left(\operatorname{round}\left(\frac{D}{P}\right), 2\right) \end{cases} \end{aligned}\)       (6)

After all of the watermark 𝑊 are embedded, 𝐸₁′′ and 𝐸₂′′ would be obtained and be utilized to generate two robust images 𝐼_𝑟1 and 𝐼_𝑟2.

Extraction and Recovery

If there is no attack during the transmission, the authorized users would receive two stego-images 𝐼_𝑤1 and 𝐼_𝑤2 . As mentioned in Subsection 3.1, auxiliary information 𝐻 = {𝑑_𝑞 𝑑_𝑟} could be extracted at first, then the robust images 𝐼_r1 and 𝐼_r2 could be obtained. Then, 𝐸₁′′ and 𝐸₂′′ could be calculated. For each pair of 𝑒₁′′ and 𝑒₂′′, 𝐷′ and 𝐴′ could be calculated as follows:

\(\begin{aligned}\begin{cases} D^{\prime}=e_{1}^{\prime \prime}-e_{2}^{\prime \prime} \\ A^{\prime}=e_{1}^{\prime \prime}+e_{2}^{\prime \prime} \end{cases} \end{aligned}\)       (7)

Then, the secret data 𝑤₃ could be extracted by (8).

𝑤₃ = mod(round(⌈𝐷′⁄𝑃⌉),2)        (8)

where ⌈∙⌉ means the rounded-up function. After that, 𝑒₁′ and 𝑒₂′ could be recovered as follows:

\(\begin{aligned}e_{i}^{\prime}= \begin{cases} e_{i}^{\prime \prime}+(-1)^{i} \frac{P}{4}, & \text { if } w_{3}=\bmod \left(\left|\frac{A^{\prime}}{P}\right|, 2\right) \\ e_{i}^{\prime \prime}-(-1)^{i} \frac{P}{4}, & \text { if } w_{3} \neq \bmod \left(\left|\frac{A^{\prime}}{P}\right|, 2\right)\end{cases} \end{aligned}\)       (9)

Later, 𝑤₁ and 𝑤₂ could be extracted by (2), and 𝑒₁ and 𝑒₂ could be recovered by (10).

e₁ = e₁' - d_q       (10)

As mentioned in Subsection 3.1, the cover image 𝐼 could be recovered by using 𝐸₁and 𝑑_𝑟 in the end.

If the stego-images are distorted during transmission, the reversibility of the proposed method would be lost. However, the watermark 𝑊 could be extracted correctly even if stego-images are slightly distorted due to the robustness of the proposed two-layer robust watermarking.

An example is shown in Fig. 5 to further illustrate the embedding and the extraction procedures. Specifically, if 𝑒₁ and 𝑒₂ are both set to 60, both 𝑤₁ and 𝑤₂ are set to 0, and thus both 𝑒₁′ and 𝑒₂′ would be equal to 65 after the first layer embedding. Thus, 𝐷 would be acquired as 0. If 𝑤₃ is equal to 1 in the second layer embedding process, 𝑒₁′′ would be changed to 67.5, and 𝑒₂′′ is changed to 62.5. Then, 𝐷′ could be acquired to 5 and 𝑤₃ could be extracted to 1 at the receiver. After that, 𝑒_𝑖′ could be recovered to 65 by (9) and both 𝑤₁ and 𝑤₂ could be extracted to 0 by (2). It is noted that 𝑤₃ could be correctly extracted as long as 𝐷′ is in the range of 0 to 10, which means that the robustness is similar to that of the first layer.

Fig. 5. Example of the embedding and extraction of the second layer

3.3 Theoretical analysis

In this subsection, the proposed scheme is analyzed to demonstrate its advantages from three aspects: embedding distortion (𝐸𝐷), embedding capacity (𝐸𝐶), and robustness.

On the one hand, the EDs of two typical QIM based robust watermarking algorithms are calculated. The ED of the QIM based watermarking algorithm [43] utilizing in the first layer of our proposed scheme is denoted as Dis₍₁₎ and calculated by (11).

\(\begin{aligned}D i s_{(1)}=\frac{\frac{1}{2} \int_{a}^{a+P}(x-a)^{2}+(x-a-P)^{2} d x}{P}=\frac{1}{3} P^{2}\end{aligned}\)       (11)

The ED of the QIM based [37] is expressed as Dis₍₂₎ and calculated as follows:

\(\begin{aligned}D_{(2)}=\frac{\frac{1}{2} \int_{a-\frac{P}{2}}^{a+\frac{P}{2}}(x-a)^{2}+(x-a-P)^{2} d x}{P}=\frac{7}{12} P^{2}\end{aligned}\)       (12)

It is noted that Dis₍₁₎ is smaller than Dis₍₂₎ under the same value of 𝑃 so that the less ED could be acquired in the former algorithm. On the other hand, the auxiliary information of an image would be compressed and then embedded into the robust image by using the method in [37]. While in our proposed DIRRW scheme, the auxiliary information 𝐻 would be similarly collected from an image but embedded into two robust images by employing the method [25]. In other words, the auxiliary information embedded in each robust image in our proposed scheme is less than half of that in [37]. In summary, the better visual quality could be reached in our proposed scheme.

For EC, the EC_s of the watermarking methods in [37] and [43] are equal to 2𝐿/3 due to their single-channel property, while the EC of the proposed scheme could reach to 𝐿.

In addition, the robustness of our proposed two-layer embedding algorithm is equal to 𝑃 as mentioned in Subsection 3.2. Noted that the large EC and the less ED could be accomplished in our proposed method. Thus, stronger robustness could be reached in our proposed scheme when the dual-image visual qualities are adjusted to be similar to that in the single-image RRW methods [37] and [43].

3.4 Implementation details

The encoding and decoding processes in the proposed scheme are designed. In addition, 𝑃 would be optimized according to the user's needs to better control the robustness and visual effects.

Encoding Steps

Step1: The sender copies the original cover image 𝐼 to generate two cover images 𝐼₁ and 𝐼₂. 𝐼₁ and 𝐼₂ are transformed into another domain (SLT-SVD or ZMs adopted in this paper). The methods for selecting low-frequency coefficients in the SLT-SVD domain and Zernike moments can be referred to [34] and [37]. Then, two sets of low-frequency coefficients 𝐸₁ and 𝐸₂ are collected for the robust watermarking.

Step2: In the proposed two-layer robust watermarking algorithm, secret data 𝑤₁ and 𝑤₂ are individually embedded into 𝑒₁ and 𝑒₂ during each channel by (3) in the first layer. Then, 𝑒₁′ and 𝑒₂′ are obtained.

Step3: The correlation between 𝑒₁′ and 𝑒₂′ is exploited and then modified to embed 𝑤₃ by (6) in the second layer. Later, 𝑒₁′′ and 𝑒₂′′ are obtained.

Step4: After the watermark 𝑊 is fully embedded, 𝐸₁′′ and 𝐸₂′′ are utilized to reconstruct the intermediate images 𝐼_𝑖1 and 𝐼_𝑖2 as mentioned in Subsection 3.1. Subsequently, the robust images 𝐼_𝑟1 and 𝐼_𝑟2 are generated by rounding 𝐼_𝑖1 and 𝐼_𝑖2 to UINT8 data type.

Step5: Before the reversible embedding in this paper, 𝑑_𝑞 is collected by (4) and 𝑑_𝑟 = 𝐼 − 𝐼̂₁ is collected by referring to [37]. Afterwards, the auxiliary information 𝐻 = {𝑑_𝑞 𝑑_𝑟} would be compressed and then embedded into the robust images 𝐼_𝑟1 and 𝐼_𝑟2 by using the dual-image reversible data hiding method [25]. In the end, the stego-images 𝐼_𝑤1 and 𝐼_𝑤2 are generated.

Optimization process: To optimize 𝑃, the robust embedding distortion (denoted as Dis) is devised in this paper as follows:

\(\begin{aligned}Dis=L \log _{10} \sqrt{\frac{1}{4} \int_{3 P / 2}^{5 P / 2}\left(x-\frac{5 P}{4}\right)^{2}+\left(x-\frac{7 P}{4}\right)^{2}+\left(x-\frac{9 P}{4}\right)^{2}+\left(x-\frac{11 P}{4}\right)^{2} d x}\end{aligned}\)       (13)

where 𝐿 represents the length of the watermark. The optimization model of the parameter 𝑃 could be expressed as follows:

\(\begin{aligned} \begin{cases} \text { Max } P \\ s . t . \\ \begin{cases} L=\text { selected number } \\ \text { Dis } \leq k \end{cases} \end{cases} \end{aligned}\)       (14)

Among them, 𝑘 is a parameter determined by users. In this paper, 𝐿 is set to 128 and 𝑃 is optimized by exhaustive search in the set {1 < 𝑃 < 50} when the ZMs domain is selected. In general, the lowest peak signal-to-noise (PSNR) of the stego-image in a RRW method should be greater than 38dB to guarantee the visual quality. In this case, the experimental data k is set to 388. The optimized 𝑃 = 14 is obtained by (14) when ZMs is chosen as the transform domain. Similarly, the optimized 𝑃 = 260 is obtained when the SLT-SVD domain is selected.

Decoding Steps

1) Undistorted:

Step1: Extract the auxiliary information 𝐻 = {𝑑_𝑞 𝑑_𝑟} and the robust images 𝐼_𝑟1 and 𝐼_𝑟2 could be recovered according to [25] in the meantime.

Step2: Transform 𝐼_𝑟1 and 𝐼_𝑟2 into the same domain which is selected in the encoding process. After that, the modified low-frequency coefficients sets 𝐸₁′′ and 𝐸₂′′ are obtained.

Step3: The secret data 𝑤₃ could be extracted according to (8) and both 𝑒₁′ and 𝑒₂′ could be recovered by (9).

Step4: The secret data 𝑤₁ and 𝑤₂ could be extracted according to (2). Meanwhile, 𝑒₁ could be restored with the quantized error 𝑑_𝑞 by (10).

Step5: Repeat the steps 3-4 of the decoding process until the watermark 𝑊 is extracted completely. Then, the image 𝐼̂₁ could be reconstructed by reversing 𝐸₁ into pixel domain. Finally, the original cover image 𝐼 would be recovered by formula as follows:

I = 𝐼̂₁ + d_r       (15)

2) Distorted:

The distorted stego-images 𝐼̃_𝑤1 and 𝐼̃_𝑤2 would be received at the decoder. The decoding steps in the case of distortion are similar to those in the undistorted case except- ing recover the original cover image. Although the reversibility is lost, the watermark 𝑊 could be extracted due to the robustness of the proposed DIRRW scheme.

4. Experimental Results

In the section, the proposed method is evaluated by comparing with other three state-of-the-art methods, i.e., Liu et al.’s [34], Hu et al.’s [37] and Wang et al.’s [40], in terms of robustness and visual quality. In this paper, 100 typical 512 × 512 × 8 bits grayscale images with different textures from the CVG-UGR and DIP databases [44,45] are employed as test images, where some standard images are shown in Fig. 6. The methods in [34], [37] and [40] are all based on a single image as the carrier. Liu et al.’s method is based on the SLT-SVD domain. Hu et al.’s method is implemented in the Zernike moments (ZMs). Wang et al.’s method is a typical RRW that is based on an independent domain. To perform a fair comparison, the length 𝐿 of the watermark in [34], [37] and [40] is set to 128. While the total embedding capacity is 256 bits in our proposed dual-image RRW method. Specifically, the size of non-overlapping blocks in Liu’s method and the proposed SLT-SVD based dual-image RRW (denoted as SS-DIRRW) are set to 8 × 8. The parameters 𝑇 =1000 and 𝑁 =31 are set in Hu’s method and the proposed Zernike moments dual-image RRW (denoted as ZMs-DIRRW). In [40], the block size is set to 64 × 16. 𝑃 in the proposed SS-DIRRW and ZMs-DIRRW is the threshold to adjust the tradeoff between the robustness and perception for the stego-image, which means the larger 𝑃 results in better robustness but worse visual quality for the stego-image, and vice versa. Similarly, the parameter ∆ in [34] and [37], and 𝜁 in [40] play the same role as 𝑃. The visual quality of the proposed scheme would be tested at first to indicate the visual difference between the dual images. PSNR is employed to measure the visual quality of the images, which is computed as follows:

Fig. 6. Four standard test images

\(\begin{aligned}P S N R=10 \log _{10}\left(\frac{255^{2} \times r \times c}{\sum_{i=1}^{r} \sum_{j=1}^{c}\left(I(x, y)-I_{w}(x, y)\right)^{2}}\right)\end{aligned}\)       (16)

where r, c demotes the length and the width of the image, respectively. And x, y is the pixel coordinate of the image, respectively. 𝐼 is the cover image and 𝐼_𝑤 is the stego-image.

By using the optimization model, 𝑃 is optimized in the SLT-SVD and ZMs, respectively. Experientially, 𝑃 is acquired to 260 in the proposed SS-DIRRW and is acquired to 14 in the proposed ZMs-RRW. The PSNRs of the dual stego-images (i.e., 𝐼_𝑤1 and 𝐼_𝑤2) are listed in Table 1, in which 𝑃 = 260 in the proposed SS-DIRRW and in the proposed ZMs-DIRRW with 𝑃 = 14. It is observed that the PSNRs of the dual stego-images for the proposed methods are close to each other, whether in SLT-SVD or ZMs. Then, the smaller PSNRs of the dual stego-images in the proposed methods would be exploited in the following experiments of visual quality and robustness.

Table 1. PSNRs of the dual stego-images in the proposed SS-DIRRW with 𝑃 = 260 and in the proposed ZMs-DIRRW with 𝑃 = 14

4.1 Visual quality

In this subsection, PSNRs of [34], [37], the proposed SS-DIRRW and the proposed ZMs-DIRRW are under the same level of the robust thresholds and are listed in Table 2. In order to reach the same level of the robust thresholds, 𝛥 in [34] and 𝑃 in the proposed SS-DIRRW are set to 260, 𝛥 in [37] is set to 28 and 𝑃 in the proposed ZMs-DIRRW is set to 14. Under this condition, the proposed schemes (SS-DIRRW and ZMs-DIRRW) achieve better visual quality than the other traditional single-image RRW methods.

Table 2. PSNRs of [34], [37], the proposed SS-DIRRW and the proposed ZMs-DIRRW under the same level of the robust thresholds

4.2 Robustness

The bit error rate (BER) is exploited to indicate the robustness of given schemes as follows:

\(\begin{aligned}BER=\frac{L_e}{L_o}\end{aligned}\)      (17)

where 𝐿_𝑒 refers to the number of error secret data bits and 𝐿_𝑜 denotes the number of original secret data bits. The less BER is, the better robustness is achieved.

Geometric deformations (e.g., rotation and scaling) and common attacks (e.g., JPEG compression and Gaussian noise) would occur from time to time during the network transmission of the images. To testify the robustness performance, the experiments are performed between the proposed scheme and the other three state-of-the-art works, i.e., [34], [37] and [40], based on the above-mentioned test images in Fig. 6. In order to guarantee the visual quality of the stego-images, PSNRs of all the contract methods are adjusted to no less than 38. Specifically, Δ=180 is set in [34], 𝛥 =18 is set in [37] and 𝜁 = 2.4 is set in [40]. Meanwhile, 𝑃 =240 is set in our proposed SS-DIRRW and 𝑃 =14 is set in the proposed ZMs-DIRRW which are optimized by our designed optimization model as mentioned in Section 3.

The robustness of the schemes is testified based on several types of attacks, i.e., rotation, scaling, JPEG compression, additive Gaussian noise and JPEG2000 compression. Fig. 7 illustrates the five versions of the image Lena after the above different types of attacks.

Fig. 7. The image Lena after being attacked

Since the stego-images 𝐼_𝑤1 and 𝐼_𝑤2 would be generated in our proposed methods (SS-DIRRW and ZMs-DIRRW), two cases are considered in the paper as follows: Case 1: The Same Attacks (𝐼_𝑤1 and 𝐼_𝑤2 encounter the same attacks under this case). Case 2: Different Attacks (𝐼_𝑤1 and 𝐼_𝑤2 suffer from different attacks).

Case 1: The Same Attacks

The comparison between our proposed method (SS-DIRRW and ZMS-DIRRW) and the other three state-of-the-art works (i.e., [34], [37] and [40]) are performed under the above-mentioned five attacks and they are illustrated from Fig. 8 to Fig. 11. As shown in Fig. 8, it is observed that the proposed ZMs-DIRRW achieves the same or even better robustness resistant to the geometric attacks especially the scaling attack when the factor is 0.6. This situation demonstrates the proposed DIRRW framework could significantly improve the robustness.

Fig. 8. Robustness to resist geometric attacks with [37]

It is observed that the BER of our proposed SS-DIRRW is less than which of [34] under the common attacks like JPEG compression, Gaussian noise and JPEG2000 compression. Similarly, the BER of the proposed ZMs-DIRRW is less than Hu’s method [37]. In Fig. 9, the proposed SS-DIRRW achieves the least BER (<5%) at Gaussian noise with a variance of 0.029. The proposed ZMs-DIRRW achieve the similar experimental results when JPEG2000 compression reaches 100:1 for four standard test images. In addition, the results of the proposed methods (ZMs-DIRRW and SS-DIRRW) are excellent and stable. Even when the quality factor is set to 10 in JPEG compression, all of the secret data are extracted without any errors in our proposed methods.

Fig. 9. Robustness to resist additive gaussian noise compared with [34], [37] and [40]

Fig. 10. Robustness to resist JPEG2000 compression compared with [34], [37] and [40]

It is obviously that the robustness of the proposed SS-DIRRW and ZMs-DIRRW is slightly inferior to Wang’s method [40] under the Gaussian noise with a variance of 0.029 for Lena and Baboon as shown in Fig. 9. However, for most cases, i.e., JPEG compression, JPEG2000 compression, rotation, and scaling, the robustness of the proposed schemes is superior to the other contrast schemes.

Besides the above-mentioned five attacks, some other common noise attacks such as medium and average filter operations are tested in the robustness experiment. The experimental results are shown in Fig. 12. The less BER of the proposed SS-DIRRW and ZMs-DIRRW further demonstrates the superiority of the proposed methods under the most common attacks.

Fig. 11. Robustness to resist JPEG compression compared with [34], [37]and [40]

Fig. 12. The BERs for compared methods under other common noise attacks

Case 2：Different Attacks.

The proposed ZMs-DIRRW is taken for robustness test under this condition. Set several different kinds of attacks as follows: Rotation with angle 100, scaling with factor 0.5, gaussian noise with mean 0 and variance 0.02 and JPEG2000 compression with ratio 70:1. Six different attacks are performed on stego-images 𝐼_𝑤1 and 𝐼_𝑤2 to test the robustness under Case 2 and the result is listed in Table 3. Evidently, the excellent and comparable BERs of the ZMs-DIRRW are obtained under different attacks, which means similar robustness could be achieved regardless of Case 1 (same attacks) or Case 2 (different attacks). The phenomenon is attributed to the high utilization of the generated dual stego-images.

Table 3. BERs of the proposed ZMs-DIRRW under the different attacks

5. Conclusion

In this paper, a novel dual-image based robust reversible watermarking framework is proposed. In addition, to further improve high performance, a two-layer robust quantization watermarking algorithm is designed. Meanwhile, both different domains are incorporated into the proposed framework to build two examples, i.e., SS-DIRRW and ZMs-DIRRW, to demonstrate its superiority. Experimental results demonstrate that the proposed scheme could achieve better performance in terms of robustness and visual quality. In the future, we will focus on extending the application field of the proposed framework, such as the utilization of our framework into audio and video carriers.