# Evaluation of a Framework for Robust Image Reversible Watermarking
## 2 Evaluation of the Robust Framework for Robust Reversible Image Watermarking
### Perceptual Distortion Evaluation  
The peak-signal-to-noise ratio (PSNR) measures the similarity of two signals, a reference signal, and a processed version of it, and this ratio is given in decibels [28; 29] as some signals hold a very wide dynamic range. The PSNR is a metric complementary to distortion. The higher the PSNR, the lower the distortion is.  
In image processing, a typically PSNR is measured in an 8 bit grayscale version. For an image \(f\) and its processed version \(g\), both of size \(M\times N\), the PSNR between \(f\) and \(g\) is computed by:  
\[\text{PSNR(f,g)}=10\cdot\log_{10}\biggl{(}\frac{255^{2}}{MSE(f,g)}\biggr{)} \tag{14}\]  
\[MSE(f,g)=\frac{1}{(M\cdot N)}\sum_{i=1}^{M}\sum_{j=1}^{N}(f_{ij}-g_{ij})^{2} \tag{15}\]  
As for the mean-squared error (MSE), the difference between the pixels \(f_{ij}\) and \(g_{ij}\) is considered an error that generates image quality loss. The lower the MSE, the higher the PSNR; therefore, the higher the PSNR(f, g) values, the higher the image quality is. For digital images, sometimes, a PSNR \(>35\) dB is considered as good quality [30].  
The Watson metric, which quantifies the distortion of an image based on just noticeable differences [31], is another tool to evaluate the distortion of a processed image. This metric measures the errors for each DCT coefficient in each block by its corresponding sensitivity threshold. That threshold considers contrast sensitivity, luminance masking, and contrast masking.  
In the Watson model, for each \(ij\) DCT coefficient, the relation between the luminance and frequency is considered, as follows:  
\[t_{ijk}=t_{ij}\left(\frac{c_{00k}}{\bar{c}_{00}}\right)^{a_{t}} \tag{16}\]  
where \(t_{ij}\) is the threshold for the smallest frequency coefficient that yields a visible signal, \(c_{00k}\) is the DC coefficient of block \(k\), \(\bar{c}_{00}\) is the DC coefficient corresponding to the mean luminance of the display (1024 for an 8 bit image), and \(a_{t}\) determines the degree of masking (set to 0.65) [32].  
When the visibility of a pattern is reduced by the presence of another pattern in the image, this phenomenon is known as texture masking. [PERSON] extends the results of luminance and frequency masking to include texture masking as follows:  
\[m_{ijk}=\max[t_{ijk,}\mid c_{ijk}\mid^{w_{ij}}t_{ijk}^{1-w_{ij}}] \tag{17}\]  
where \(m_{ijk}\) is the masked threshold and \(w_{ij}\) determines the degree of texture masking. Typically, \(w_{00}=0\) and \(w_{ij}=0.7\) for all other coefficients. The perceptual error in each frequency of each block is given by  
\[d_{ijk}=\frac{e_{ijk}}{m_{ijk}} \tag{18}\]  
where \(e_{ijk}\) is the quantization error.  
To associate the errors in the model, the [PERSON] metric is used as follows:  
\[p_{ij}=\left(\sum_{k}\mid d_{ijk}\mid^{\bar{\beta}_{s}}\right)^{\frac{1}{\bar{ \beta}_{s}}} \tag{19}\]  
where different values of the exponent \(\beta_{s}\) implement different types or degrees of pooling. In practice, \(\beta_{s}=100\) is commonly used.  
A typical threshold for the Watson metric is 0.4, since measures below this point guarantee visual imperceptibility [33].  
The following section presents the experimental results obtained after implementing the framework using the fragile reversible watermarking schemes, and self-recovery schemes were selected.