## 4 Simulation Results and Analysis
### Correlation  
Correlation analysis is a statistical method used to evaluate the strength and direction of the relationship between two variables. The correlation coefficient, a measure of association, quantifies the degree to which two variables are linearly related. The Pearson correlation coefficient is the most commonly used measure of association and can range from \(-1\) to \(1\), with \(-1\) indicating a perfect negative correlation, \(1\) indicating a perfect positive correlation, and \(0\) indicating no correlation. When the correlation coefficient value is significantly different from \(0\) on the encrypted image, it becomes difficult or impossible to interpret the data without using the appropriate decryption key or method. An effective image encryption scheme should result in a low correlation between adjacent pixels, which are randomly distributed in order to ensure secure protection of the image, as seen in Figure 13d-f. In the case of a plain image, the correlation among variables is concentrated, making it easier to predict the original image, as seen in Figure 13a-c. Equation (17) shows how to calculate the correlation coefficient of two variables in an image. Table 6 presents the correlation coefficient for plain and encrypted images of Lena, House, and Cameraman in various sizes and the average value of the horizontal, vertical, and diagonal correlation and compares the results to those of previous studies on the encrypted Lena image. This demonstrates the effectiveness of our proposed encryption techniques and their potential for use in image encryption.  
\[r_{xy}=\frac{\text{cov}(x,y)}{\sqrt{(x)}\sqrt{(y)}} \tag{17}\]  
where \(\text{var}(x)=\frac{1}{N}\sum\limits_{i=1}^{N}\left(x_{i}-E(x)^{2}\right)\), \(\text{cov}(x,y)=\text{E}([x-E(X)][y-E(Y)])\), \(E(x)=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}\), \(E(y)=\frac{1}{N}\sum\limits_{i=1}^{N}y_{i}\).  
In our proposed scheme, the variables \(x_{i}\) and \(y_{i}\) represent the grayscale values of two adjacent pixels, with \(E(x)\) denoting the average of \(x_{i}\) and \(E(y)\) denoting the average of \(y_{i}\). By applying Equation (16) to the proposed system, we observed a significant difference between the original and encrypted images. This indicates that the proposed system is resistant to statistical attacks.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Algorithm** & **[PERSON] (128\(\times\)128)** & **[PERSON] (64\(\times\)64)** & **[PERSON] (64\(\times\)64)** & **[PERSON] (64\(\times\)64)** & **[PERSON] (32\(\times\)32)** & **[PERSON] (32\(\times\)32)** & **[PERSON] (256\(\times\)256)** \\ \hline Plain Image & 6.4971 & 6.4512 & 6.7457 & 7.3801 & 7.5264 & 6.5335 & 7.5468 \\ \hline Encrypted image & 7.9981 & 7.9978 & 7.9991 & 7.9954 & 7.9991 & 7.8546 & 7.9989 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Information entropy values of Ciphertext images.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Proposed (Average)** & **Ref [19]** & **Ref [29]** & **Ref [33]** & **Ref [35]** & **Ref [37]** & **Ref [38]** \\ \hline Entropy & 7.9992 & 7.999284 & 7.9022 & 7.996513 & 7.9977 & 7.9970 & 7.9974 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Information entropy comparison.