4.3.1. Correlation Coefficient Analysis

In plain images, the correlation between adjacent pixels is fairly strong, and the correlation between adjacent pixels can be used by the attacker to obtain some useful information. Therefore, after image encryption, the correlation between adjacent pixels of the encrypted image is closer to 0, indicating that the pixel distribution is random. We selected 4000 pairs of adjacent pixels in plain images and encrypted images, and then calculated the correlation coefficient of two horizontal, vertical and diagonal adjacent pixels using Equation (16):

$$\mathcal{C}\_{xy} = \frac{E\{ [\mathbf{x} - E(\mathbf{x})][y - E(y)] \}}{\sqrt{D(\mathbf{x})}\sqrt{D(y)}}\tag{16}$$

where *E*(*x*) and *D*(*x*) represent the expectation and variance of variable *x*, respectively. Table 1 shows the experimental results of the tested images by performing the encryption in two rounds. The correlation coefficient of three directions is close to 0 after the encryption.

Figure 8 shows the correlation of the Lena image and its cipher image. The adjacent pixel pairs of the plain image in all directions are densely on the line of *y* = *x*, and the adjacent pixel pairs of the cipher image in all directions are evenly distributed in the rectangular area.

**Figure 8.** *Cont.*

**Figure 8.** Adjacent pixels correlation analysis: the correlation between two horizontal, vertical, and diagonal pixels in (**a**–**c**) a plain image and (**d**–**f**) an encrypted image.


**Table 1.** The correlation coefficient in three directions for tested images.
