## 3 Experimental Results
### Image Pixel Correlation  
The pixel correlation of an image refers to the similarity or correlation between different pixels in the image. By analyzing the brightness differences or common variations between each pixel and its surrounding pixels, features such as texture, structure, and patterns can be revealed in the image. This kind of correlation analysis is crucial for many tasks in the field of image processing, such as image recognition, segmentation, enhancement, and compression, aiding in understanding and extracting information from the image, thus enabling various applications. In the security evaluation of image encryption schemes, the pixel correlation coefficient is a critically important indicator. It reflects the correlation between neighboring pixels in the image. Specifically, this refers to the relationship between the values of neighboring pixels in the image. We can evaluate the effectiveness of the encryption scheme by comparing the correlation between neighboring pixels in the original image and the encrypted image by using Equations (13)-(15). \[E(x)=\frac{1}{W\times H}\sum_{i=1}^{W\times H}x_{i_{i}}D(x)=\frac{1}{W\times H }\sum_{i=1}^{W\times H}(x_{i}-E(x_{i}))^{2}. \tag{13}\]  
\[cov(x,y)=\frac{1}{W\times H}\sum_{i=1}^{W\times H}(x_{i}-E(x_{i}))(y_{i}-E(y_{ i})). \tag{14}\]  
\[p_{xy}=\frac{cov(x,y)}{\sqrt{D(X)D(Y)}}\enspace. \tag{15}\]  
Here, \(x\) and \(y\) are the pixel values of neighboring pixels in an image, and \(W\) and \(H\) is the width and height of the original image or encrypted image. It is noted that Equation (13) defines the mean \(E(x)\) and variance \(D(x)\) of pixel values in an image, where \(E(x)\) is the average value of the image's pixel values, where \(W\times H\) is the total number of pixels, and \(x_{i}\) represents the value of the i-th pixel. This mean value represents the overall brightness or grayscale level of the image. \(D(x)\) describes the distribution of pixel values, i.e., the degree to which pixel values deviate from the mean. A larger variance indicates that the pixel values are more dispersed, which corresponds to higher contrast in the image. Equation (14) defines the covariance \(cov(x,y)\) between two variables \(x\) and \(y\), where \(x\) and \(y\) represent the values of adjacent pixels \(x\) and \(y\). The greater the correlation between \(x\) and \(y\), the larger the value of the covariance. In image pixel correlation analysis, covariance is used to measure the correlation between adjacent pixels. A larger covariance indicates a higher correlation between adjacent pixel values, while a covariance close to 0 indicates a lower correlation. Equation (15) defines the correlation coefficient \(p_{xy}\) between pixels, which represents the correlation between adjacent pixels by dividing the covariance by the product of the standard deviations. The range of the correlation coefficient \(p_{xy}\) is between \(-1\) and \(1\). If \(p_{xy}\) approaches 1, it indicates a high positive correlation between the two pixels, meaning that adjacent pixel values in the original image tend to change in a similar manner. If \(p_{xy}\) approaches 0, it indicates that the two pixels are almost uncorrelated, which usually occurs in encrypted images because encryption disrupts the correlation between adjacent pixels, making the image appear more random. If \(p_{xy}\) approaches \(-1\), it indicates a highly negative correlation, though this situation is rare in images. By comparing the correlation of adjacent pixels in the image before and after encryption, the effectiveness of the encryption scheme can be analyzed. In the original image, adjacent pixels exhibit significant linear correlation, while after encryption, the randomization of pixel values reduces the correlation between adjacent pixels. Therefore, the encrypted image should show a lower correlation coefficient \(p_{xy}\), which helps validate the effectiveness of the encryption scheme.