# Robust Image Watermarking in Spatial Domain Utilizing Features Equivalent to SVD Transform
## 4 Experimental Results Discussion  
In this section, a comparison of the performance of the proposed watermarking scheme with the performance of other relevant watermarking techniques that have been developed by [PERSON] et al. [48], [PERSON] et al. [49], [PERSON] et al. [50], [PERSON] et al. [47], and [PERSON] et al. [26] is carried out. All these schemes have, in some way or another, utilized an idea that is similar to the proposed scheme. Ten grayscale standard test images (_Image_1 to _Image_10) with a size of 512 \(\times\) 512 and four binary watermarks (_W_1, \(W\)2, \(W\)3, and \(W\)4) with a size of 64 \(\times\) 64 were considered for evaluating the performance of the proposed scheme and are given in Figures 4 and 5, respectively. These test images were compiled from a variety of publicly accessible open-source image libraries. To verify the robustness of the proposed approach, the quality of the watermarked image was degraded using a variety of standard image distortion attacks, as shown in Table 1. The proposed scheme was tested on a personal computer (_PC_) configured with MATLAB, an Intel core i7 processor, NVIDIA GeForce MX450 graphic card, Windows 11, and 16 GB of RAM. The results are given in Table 6 and Figures 12 and 13 for the comparison of algorithms. The best result corresponding to each row is highlighted in bold. Figure 4: Test images: Ten test images labeled from _Image_1 to _Image_10. The peak signal-to-noise ratio (PSNR) is a widely used metric for evaluating the quality of a watermarked image among the different quality metrics reported in the literature [49]. A higher _PSNR_ value points toward the higher imperceptibility of the watermarked image. The _PSNR_ of 8-bit image \(A\) with a size of \(m\times m\) and its watermarked image \(Aw\) is given by the following equation. \[PSNR(A,Aw)=10\log_{10}\left(\frac{\left(255\right)^{2}}{\frac{1}{m\times m} \sum_{i=1}^{m}\sum_{j=1}^{m}\left(A_{i,j}-Aw_{i,j}\right)^{2}}\right)(dB) \tag{22}\]  
Another metric for the imperceptibility measurement is the structural similarity (_SSIM_) index [49], which measures the similarity between original image \(A\) and watermarked image \(Aw\). It models any image distortion as a combination of three factors, which are a loss of correlation \(s(A,A_{w})\), contrast distortion \(c(A,A_{w})\), and luminance distortion \(l(A,A_{w})\). The human visual system (HVS) quality perception and structural similarity are correlated. It is defined as the follows:  
\[SSIM(A,A_{w})=l(A,A_{w})c(A,A_{w})s(A,A_{w}) \tag{23}\]  
\[l(A,A_{w})=\frac{2\mu_{A}\mu_{Aw}+C_{1}}{\mu_{A}^{2}+\mu_{Aw}^{2}+C_{1}},\ c(A,A_{w})=\frac{2 \sigma_{A}\sigma_{Aw}+C_{2}}{\sigma_{A}^{2}+\sigma_{Aw}^{2}+C_{2}},\ s(A,A_{w} )=\frac{\sigma_{AAu}+C_{3}}{\sigma_{A}\sigma_{Aw}+C_{3}} \tag{24}\]  
where \(\mu_{A},\ H_{Aw},\ \sigma_{A},\ \sigma_{Aw},\ \sigma_{Aw}\) and \(\sigma_{AAu}\) are the mean, the variance of image \(A\) and its watermarked image \(Aw\), and the covariance between these images, while constants \(C_{1}\), \(C_{2}\), and \(C_{3}\) are stability constants used to avoid division by zero. The extracted watermark (\(W\)) may not be identical to the embedded watermark (\(W\)) with a size of \(n\times n\) due to the implementation of the image distortion attacks on the watermarked image. Consequently, a standard is necessary to evaluate the robustness of the watermarking scheme, and often, the normalized correlation coefficient (NCC) and bit