# Asymmetric Encryption of Invisible Structured Light 3D Imaging
## 3 Simulation Results and Performance Analyses
### Noise Attack Analysis  
In the process of image transmission, the ciphertext image may be injected with noise by hackers. Therefore, salt and pepper noise with 0.1 density is introduced to detect the anti-noise ability of the system. Salt and pepper noise, also known as impulse noise, is the noise often seen in images. It is a kind of random white or black dot. Figure 12(a)-h shows the ciphertext after introducing noise. Then, the original 8 fringe images are median \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Correlation Coefficients**} & \multicolumn{8}{c}{**Original Images**} \\ \cline{2-9}  & **O1** & **O2** & **O3** & **O4** & **O5** & **O6** & **O7** & **O8** \\ \hline Horizontal & 0.9921 & 0.9921 & 0.9921 & 0.9921 & 0.9917 & 0.9916 & 0.9917 & 0.9916 \\ \hline Vertical & 0.9921 & 0.9921 & 0.9921 & 0.9921 & 0.9917 & 0.9916 & 0.9917 & 0.9916 \\ \hline Diagonal & 0.9688 & 0.9688 & 0.9688 & 0.9688 & 0.9681 & 0.9680 & 0.9681 & 0.9680 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlation coefficients of adjacent pixels of original images. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Correlation Coefficients**} & \multicolumn{8}{c}{**Ciphertext Images**} \\ \cline{2-9}  & **C1** & **C2** & **C3** & **C4** & **C5** & **C6** & **C7** & **C8** \\ \hline Horizontal & 0.0000 & 0.0002 & 0.0017 & \(-0.0005\) & 0.0013 & 0.0016 & 0.0006 & 0.0008 \\ \hline Vertical & \(-0.0056\) & \(-0.0107\) & 0.0036 & \(-0.0055\) & \(-0.0045\) & \(-0.0035\) & \(-0.0001\) & \(-0.0088\) \\ \hline Diagonal & 0.0129 & 0.0079 & 0.0117 & 0.0156 & 0.0142 & 0.0064 & 0.0137 & 0.0171 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation coefficients of adjacent pixels of ciphertext images. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Original Images**} & **Ciphertext Images** & **Theoretical Value** \\ \hline
4.6799 & 7.9901 & 8 \\ \hline
4.6838 & 7.9894 & 8 \\ \hline
4.6831 & 7.9891 & 8 \\ \hline
4.6838 & 7.9896 & 8 \\ \hline
5.6854 & 7.9918 & 8 \\ \hline
5.6981 & 7.9912 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The information entropy of the four original images and four ciphertext images. filtered, as shown in Figure 14a-h. The images after filtering are shown in Figure 14i-p. The measured object is reconstructed, as shown in Figure 15. Obviously, the original 3D object can still be reconstructed even when noise exists. The results show that the imaging system has high anti-noise ability. Next, we use structural similarity (_SSIM_) to detect the anti-noise ability of the eight images under different salt and pepper noise densities. Among them, _SSIM_ is an index used to measure the similarity of two images. The specific expression of _SSIM_ is as follows:  
\[SSIM=\left(\frac{2\mu_{x}\mu_{y}+Q_{1}}{\mu_{x}^{2}+\mu_{y}^{2}+Q_{1}}\right) \left(\frac{2\sigma_{x}\sigma_{y}+Q_{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}+Q_{2}} \right)\left(\frac{\sigma_{xy}+Q_{3}}{\sigma_{x}\sigma_{y}+Q_{3}}\right) \tag{8}\]  
where \(\mu_{x}\) and \(\mu_{x}\) are the average values of the original pattern and the decrypted pattern, respectively; \(\sigma_{x}\) and \(\sigma_{y}\) are the variance of the original pattern and the decrypted pattern, respectively; \(\sigma_{xy}\) is the covariance between the original pattern and the decrypted pattern; and \(Q_{1}\), \(Q_{2}\), and \(Q_{3}\) are constants. The value of _SSIM_ is a value from 0 to 1. The closer the value is to 1, the smaller the gap between the original pattern and the decrypted pattern, thus the image reconstruction quality is enhanced. _SSIM_ of the 8 images under different salt and pepper noise densities is shown in Table 4. It can be concluded from the data in the table that the eight images have a strong anti-noise ability when different salt and pepper noises are introduced.