# Robust Image Embedded Watermarking Using DCT and Listless SPIHT
## 4 Simulation Results and Analysis  
The experiments have been conducted on test images of size 512 \(\times\) 512 pixels. MATLAB 14.1 tool is used for experiments on an Intel core i5 processor, 2.6 GHz CPU and 6 GB RAM. The watermark used in the experiment is a binary image of 62 \(\times\) 64 pixels. In order to validate the objective judgement of the extracting fidelity, the performance comparison of the proposed method is compared with other  
Figure 3: Watermark extraction process.  
schemes in terms of PSNR, Tamper assessment function (TAF) and Normalized Correlation (NC). For a watermark of size \(m\times n\), the TAF in percentage is defined as:  
\[\text{TAF}(\%)=\frac{1}{mn}\left[\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{n}w(i, j)\oplus\overline{w}(i,j)\right]\times 100, \tag{4}\]  
and, Normalized Correlation (NC) is defined as:  
\[\text{NC}=\frac{\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{n}w(i,j)\times \overline{w}(i,j)}{\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{n}\left[w(i,j) \right]^{2}}, \tag{5}\]  
where \(w(i,j)\) and \(\overline{w}(i,j)\) represent the original and extracted watermarks at position \((i,j)\), respectively; and, \(\oplus\) is exclusive or operation. TAF represents the number of bits of the extracted watermark that are different from the original watermark. The acceptance level of TAF is usually 20% as the extracted watermark is not recognizable beyond this value. Similarly, the NC value is 1 for highest-quality watermark and 0 for lowest-quality watermark. Watermark having NC below 0.3-0.4 is not recognizable.  
PSNR is a quantitative measure of image quality between original host image and watermarked image. The value of PSNR usually ranges between 20 dB (low quality) and 40 dB (high quality). For 8-bit grayscale images PSNR is given as:  
\[\text{PSNR}(\text{dB})=10\ \log_{10}\left[\frac{255^{2}}{\frac{1}{MN}\sum \limits_{i=1}^{M}\sum\limits_{j=1}^{N}\left[\text{X}(i,j)-\overline{X}(i,j) \right]^{2}}\right], \tag{6}\]  
where \(X\) represents the host image, and \(\overline{X}\) represents the watermarked image. \(M\times N\) represents the size of the two images.  
Structural similarity (SSIM) index is used for measuring similarity between two images. SSIM is designed to improve on traditional methods such as PSNR, which has been proven to be inconsistent with human visual perception. The measure between two images \(X\) and \(\overline{X}\) of same size \(M\times N\) is given by:  
\[\text{SSIM}=\frac{(2\mu_{x}\mu_{\overline{x}}+c_{1})(2\sigma_{x\overline{x}}+ c_{2})}{(\mu_{x}^{2}+\mu_{\overline{x}}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{x}^{2}+c_{2 })}, \tag{7}\]  
where \(\mu_{x}\) and \(\sigma_{x}^{2}\) are the average and variance of image \(X\) respectively, \(\mu_{\overline{x}}\) and \(\sigma_{\overline{x}}^{2}\) are the average and variance of watermarked image \(\overline{X}\) respectively. \(\sigma_{x\overline{x}}\) is the covariance. \(c_{1}\) and \(c_{2}\) are two variables to stabilize the denominator. Its range is from 0 to 1. If two images are perfectly matched, SSIM = 1.  
The performance comparison is carried out considering the following performance criteria:  
* Quality of the watermarked images.
* Robustness of the scheme against common attacks such as compression, crop, histogram manipulation, auto contrast, noise, inversion and filtering.