Application of Empirical Wavelet Transform in Digital Image Watermarking

Until now, fixed filters were used for multimedia models in each image. This procedure makes it impossible to display the optimal filters. A better way is to produce filters based on image features.

Huang et al. [15] proposed a new method for signal analysis based on empirical mode analysis. The purpose of this method is to find oscillatory narrow band components called intrinsic state functions. The advantage of this method is associated with the issue that it does not require any prior assumptions. This method was used for watermarking process.

However, there were two major problems dealing with applying the empirical mode decomposition method. First, it lacked mathematical theory, and second, it did not guarantee that different input images would be adapted to each other. This has led to production of images that are different from input image and bring artifacts.

As we know, discrete wavelet transform is also used for image decomposition. In this way, when the image is transmitted into transform domain, the pre-prescribed design is broken down into approximations and details layers, which in turn affect the signal analysis results. However, their wavelet base sets are fixed and make it impossible to optimize the filter design for displaying image. So it is the best alternative to design a set of filters based on the processed signals. This has led to emergence of experimental wavelet transforms that do not have the above problems [7]. The experimental wavelet transform method had strong theoretical basis and was theoretically better than the empirical mode decomposition and discrete wavelet transform methods.

It is due to this has the ability to build empirical wavelet sets based on adaptive filter banks. Experimental wavelet transform is concerned with multi-resolution decomposition of a signal that decomposes the image hierarchically.

A new method called Experimental Wavelet transform method is presented [16]. This method decomposes images into approximation and details layers, which are consisted of low and high frequency information, respectively. The approximation layer retains low frequency information such as critical information from images such as intensity levels and structures. Whereas, minor layers retain high frequency information such as edges and textures.

Therefore, great performance of this method in the process of image fusion [17] motivated us to apply experimental wavelet transforms in watermarking. So far, this method has not been used for watermarking. This paper is aimed to employ this method for watermarking. In the next section, a full description is represented for two-dimensional experimental wavelet transform algorithm [18].

2.1 Two-dimensional empirical wavelet transform

In the real world, 2D signals need to be processed. Therefore, it is important to develop a two-dimensional empirical wavelet transform method. An alternative is to convert one-dimensional empirical wavelet transform to two-dimensional empirical wavelet transform. Then, the approximation layer and empirical layer are mapped to 2D space. However, we may lose some spatial information of adjacent pixels in this process. For this reason, two-dimensional Littlewoods-Paley wavelet transform is used among different wavelet transforms methods.

In the two-dimensional empirical wavelet transform, $\mathrm{f}_{2}(\mathrm{t})$ shows the two-dimensional input signal (input image) and Fourier transform of $\mathrm{f}_{2}(\mathrm{t})$ as $\mathfrak{F}_{t}\left(f_{2}\right)(\omega)$. Therefore, the approximation layer and details layers of Eq. (1) and Eq. (2) can be obtained [16]:

$\mathrm{W}\left(f_{2}\right)(0, \mathrm{t})=\mathfrak{F}_{\omega}^{-1}\left(\mathfrak{F}_{t}\left(f_{2}\right)(\omega) \overline{\mathfrak{F}_{t}\left(\phi_{1}\right)(\omega)}\right)$                  (1)

$\mathrm{W}\left(f_{2}\right)(\mathrm{m}, \mathrm{t})=\mathfrak{F}_{\omega}^{-1}\left(\mathfrak{F}_{t}\left(f_{2}\right)(\omega) \overline{\mathfrak{F}_{t}\left(\varphi_{\mathrm{m}}\right)(\omega)}\right)$            (2)

The inverse Fourier transform with $\mathfrak{F}_{\omega}^{-1}$ is shown in the above equations. As can be seen, the approximation layer and details layers in Eq. (1) and Eq. (2) are obtained by internal multiplication of the Fourier transform of the image in conjugate with the decomposed layer of approximation and details layers [17].

Finally, inverse of the two-dimensional empirical wavelet transform is shown in Eq. (3):

$f_{2}(\mathrm{t})=\mathrm{W}\left(f_{2}\right)(0, \mathrm{t}) \circ\left(\phi_{\mathrm{t}}\right)+\sum_{\mathrm{m}=1}^{\mathrm{M}-1} \mathrm{~W}\left(f_{2}\right)(\mathrm{m}, \mathrm{t}) \diamond \varphi_{\mathrm{m}}(\mathrm{t})$           (3)

In the Eq. (3), sign ⋄ is related to the convolution operator. In other words, integer multiplication of the input Fourier transform is approximated for the input images through sum of the Fourier transform of the input images with respect to internal multiplication of the details layers. Therefore, the two-dimensional empirical wavelet transform algorithm has been introduced. Next section revolves around watermarking methods.

To implement and test the proposed method, a computer system was used with the following specifications: Windows 8, 64-bit, Intel Core i7-4720 CPU @ 2.60 GHz, 6 GB memory. MATLAB R2019a software package was used for analysis process. To compare the robustness of the proposed method with the other methods, PSNR (Peak Signal to Noise Ratio) and NCC (Normalized Cross Correlation) criteria are used as shown in Eq. (5) and (6).

$\mathrm{PSNR}=10 \times \log _{10} \frac{255^{2}}{\sum_{i=1}^{m} \sum_{j=1}^{n}\left(P_{i j}-q_{i j}\right)^{2}}(\mathrm{db})$           (5)

where, $P_{i j}$ represents the values for pixels in rows i and j of the host image, and $P_{i j}$ denotes the pixel values in rows i and j of the embedded image.

$\mathrm{NCC}=\frac{\sum_{i=1}^{m} \sum_{j=1}^{n} P_{i j} q_{i j}}{\sum_{i=1}^{m} \sum_{j=1}^{n} P_{i j}^{2}}$          (6)

where, $P_{i j}$ represents the values for pixels in rows i and j of the host image, and $P_{i j}$ denotes the pixel values in rows i and j of the embedded image.

Two color image pairs are used to perform the watermarking process. In the first pair of images, the first color image is related to the image of a pepper and it is used as cover image (Figure 3). The second color image is relied on the image of an air rocket used as a cover image (Figure 4). This type of watermarking is the hidden purpose of military equipment. Low frequency content is considered for both images. As mentioned, the alpha compound is used. The size of both images is 256 by 256. The watermarking image lies within the cover image, and the value of A varies from 0.1 to 0.9.

3.png

Figure 3. Input images

4.png

Figure 4. Hidden images

The value of B is always constant. The best result is obtained when A=0.2, which makes the watermarking image process more clear and complete in the watermarking process. For the watermarking image extract process, the value of A varies from 0.9 to 0.1, depending on the watermarking algorithm. If A decreases to 0.2, the hidden image becomes darker and is completely invisible. In Figure 4, sections A and B show the hidden images with different values for A. Figure 5 a and b show the recovery of latent images based on different values for A.

5.png

Figure 5. Extract watermarking images

4.1 Extraction of hidden images using a variety of a and b values

In the proposed method, the results are cached and retrieved using two levels of analysis. Second level usually provides better results than first level decomposition due to more decomposition. At the second level, the approximate layer, LL1, is decomposed into four subgroups including LL2, LH2, HL2 and HH2. The results of the PSNR at the two breakdown levels are listed in Tables 1 and 2. The values obtained from the proposed method are plotted in both tables. It is done for both watermarking process and extract process.

Table 1. Image watermarking with different A and B values

Table 2. Extract image watermarking with different A and B values

Also, in the proposed method, considering that there is a time of decomposition and reconstruction processing, therefore, decomposition at different levels does not have such an effect on increasing the measurement criteria as the PSNR in the Figure 6.

6.png

Figure 6. Relationship of the PSNR metric with time

4.2 The robustness of the proposed method against attacks

When images are scattered on the Internet, they can be subject to various attacks. Therefore, the watermarking method should be adopted in a way that is resistant to attacks. Finally, we will provide some examples to test the images for detection and attacks below.

4.3 Strength to extract against white noise and salt & pepper attacks

In the proposed method, white noise watermarking or Gaussian noise with zero mean is added to the images. Figure 7 (a) shows the cover image after Gaussian noise attack. Figure 7 (b) depicts the watermarked image. Figure 7 (c) exhibits the extracted image after watermarking and noise removal process. The quality of the extracted image and related results of the algorithm in Table 3 indicate that this method is more robust against noise attacks.

Also, Table 3 shows the effect of Correlation by applying salt and peppers noise with different densities in the proposed method.

Table 3. Dependence of watermarking on images with different values of A and B

7a.jpg

(a) The cover image after from white noise attack

7b.jpg

(b) The image hidden after the white noise attack

7c.jpg

(c) The image extracted after denoising

Figure 7. The strength of the watermarking images

4.4 Extract strength against jpeg compression

Image compression involves in transferring important components. Thus, it tests the image watermarking versus JEPG image compression in which we paid. The compression results are shown in Table 4. The value of coefficient indicates the quality of the compressed image. The higher the quality factor, the higher the image quality, and the lower the quality if the quality factor is lower. The values range from 0 to 100. Dependency between the extracted image and original image is also considered for estimation of the watermarking. As such, if the correlation coefficients are closer to 1, the consistency between the original image and extracted image is better. The results of the Table 4 demonstrate that the algorithm proposed in this paper is resistant to JEPG compression.

Table 4. Watermarking of images against compression

4.5 Strength to extract against low pass filter

Figure 8 (a) shows the cover image after the Gaussian low-pass filter and Figure 8 (b) displays the extracted image for comparison. Figure 8 (c) indicates the extracted image after the latent filtering process and recovery after the low pass filter.

In this section, we compare the proposed method to other techniques including DWT [5], EMD [6], and DCT [8] methods. The comparison is performed on input images of a and b in Figure 3. Table 5 compares the values for PSNR and NC in the proposed method with other methods with respect to different attacks. Also, in Figures 8 and 9, we compare the proposed PNSR and NC with other methods.

Based on the results of the Table 5 and Figures 9 and 10, in the method of watermarking that performed by discrete wavelet transform, the components of the frequency bands are pre-prescribed at the time of decomposition. Thus, it is not desirable when reconstructing the image quality. In the DCT method, when parsing images, sometimes images get too much artifacts during reconstruction. It happens due to blocking, and thus it makes this algorithm less efficient.

Table 5. Comparison of the proposed algorithm with other methods

8a.jpg

(a) Image extracted

8b.jpg

(b) Cover image

8c.png

(c) Extracted image after applying low pass filter

Figure 8. Strength of images with low pass filter

9.png

Figure 9. PSNR performance comparison

10.png

Figure 10. NC performance comparison

EMD method is considered as algorithm-based approach and lacks mathematical theory. In this method, regeneration bands may not resemble the initial state sometimes, and this issue is taken into account as one of the weaknesses of this algorithm.

Finally, the proposed method is characterized as an experimental wavelet conversion method, because the coefficients are image-based and based on mathematical theory. They are highly desirable when reconstructing the image quality and quantitatively as shown in the evaluation section. The values are higher than other methods.

The disadvantage of the experimental wavelet transform method refers to the issue that when the input images are simultaneously split into several layers, the low frequency coefficients and high frequency coefficients are considered for both images in a similar range. For this reason, it leads to exactly match the coefficients of the images at the time of reconstruction. In the same way, if the coefficients overlap when the frequency bands are overlapped, they sometimes have trouble.