# A Robust Image Watermarking Scheme Based on SVD in the Spatial Domain
## 5 Experimental Results and Performance Analysis  
In this section, some experiments are conducted to evaluate the invisibility and robustness of the proposed watermarking scheme. In the experiments, several grayscale images with a size of \(512\,\times\,512\) are adopted as the host images, and the binary logo of Shandong University with a size of \(64\,\times\,64\) is selected as the watermark. Figure 4 shows the original host images and the watermark image. To make full use of the conclusion mentioned in Section 3.2, the block size used in this paper is \(4\,\times\,4\). The secret key \(k\) of the Arnold transform is selected as \(12\). As we all know, the imperceptibility and robustness of the watermark is a pair of contradictions in digital watermarking schemes. From the watermark embedding process, it is noted that the larger the quantization step is, the greater the intensity of watermark embedding will be. At the same time, a higher watermark embedding intensity will bring about a greater influence on the quality of the watermarked image. To make a balance between imperceptibility and robustness, we first analyze the effect of different quantization steps on the watermarked images and extracted watermarks. As one of the evaluation indexes of watermark imperceptibility, the structural similarity (SSIM) index [25] is adopted to calculate the similarity between original image and its watermarked version, which is defined as:  
\[\text{SSIM}=l(\mathbf{I},\mathbf{I}_{\text{w}})c(\mathbf{I},\mathbf{I}_{\text{w}})s(\mathbf{I},\mathbf{ I}_{\text{w}}), \tag{15}\]  
\[l(\mathbf{I},\mathbf{I}_{\text{w}})=\frac{2\mu_{l}\mu_{\text{w}}+\,C_{1}}{\mu_{I}{}^{2 }+\,\mu_{\text{w}}{}^{2}+C_{1}},c(\mathbf{I},\mathbf{I}_{\text{w}})=\frac{2\sigma_{l} \sigma_{\mathbf{I}_{\text{w}}}+\,C_{2}}{\sigma_{\mathbf{I}}{}^{2}+\,C_{2}},\,s(\mathbf{I}, \mathbf{I}_{\text{w}})=\frac{\sigma_{\mathbf{I}_{\text{w}}}+\,C_{3}}{\sigma_{\mathbf{I}}{} \sigma_{\mathbf{I}_{\text{w}}}+\,C_{3}}, \tag{16}\]  
where \(l(\mathbf{I},\mathbf{I}_{\text{w}})\), \(c(\mathbf{I},\mathbf{I}_{\text{w}})\), and \(s(\mathbf{I},\mathbf{I}_{\text{w}})\) are three comparison functions for luminance, contrast, and structure, respectively; \(\mu_{I}\) and \(\sigma_{\mathbf{I}}\) are the average and variance of the host image \(\mathbf{I}\); \(\mu_{\text{w}}\) and \(\sigma_{\mathbf{I}_{\text{w}}}\) are the average and variance of the watermarked image \(\mathbf{I}_{\text{w}}\); \(\sigma_{\mathbf{I}_{\text{w}}}\) is the covariance between these two images; and \(C_{1}\), \(C_{2}\), and \(C_{3}\) are three parameters used to keep stability. To evaluate the robustness of the proposed method, the normalized correlation coefficient (NCC) is utilized to investigate the  
Figure 4: Host images and the original watermark: (**a**) Lena; (**b**) Barbara; (**c**) Boat; (**d**) Airplane; (**e**) Peppers; (**f**) Baboon; (**g**) Clock; (**h**) Original watermark. correlation between the original watermark and the extracted watermark. For a watermark image \(\mathbf{W}\) with a size of \(N\,\times\,N\), the definition of NCC can be formulated as:  
\[\text{NCC}=\frac{\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\left[\mathbf{W}(i,j)\, \times\,\mathbf{W}^{*}(i,j)\right]}{\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} \left[\mathbf{W}(i,j)\right]^{2}}, \tag{17}\]  
where \(\mathbf{W}^{*}\) is the extracted watermark. Both the values of SSIM and NCC range from 0 to 1. In addition, a greater value of SSIM indicates a better watermark invisibility, while a greater value of NCC implies a better robustness. Table 1 lists the SSIM and NCC values of the proposed scheme under different quantization steps. From Table 1, we can notice that the SSIM is decreased with the increase of quantization step. When the quantization step \(\Delta\) lies between 4 and 16, the quality of the watermarked image does not degrade too much, and the NCC values are equal to 1. Therefore, the quantization step \(\Delta\) should be selected between 4 and 16.