## 4 Experiments and Analysis of Results
### Reversibility  
In this study, we implemented the proposed algorithm, based on the (3, 4)-threshold, to conduct embedding simulation experiments using the Jetplane image. Figure 6 illustrates images generated at different stages of the experiments. The sequence begins with the original image shown in Figure 6a, followed by the transitional image post-pixel prediction and encoding in Figure 6b. Figure 6c-f display the marked encrypted images produced after embedding random data, from which human eyes cannot discern any identifiable information about the secret data or the original image. Figure 6g visualizes the recovered image.  
To ascertain whether the recovered image exhibits significant distortion, we utilize the peak signal-to-noise ratio (PSNR) as a comparative metric between the recovered image and the original image, measuring the reversibility of the algorithm. Typically, when the PSNR approaches infinity, the original image is considered fully recovered. The PSNR is calculated using the following equation:  
\[\text{PSNR}=10\times\log_{10}\left[\frac{\left(2^{k}-1\right)^{2}}{MSE}\right] \tag{18}\]  
where 4 is the image pixel depth and MSE is the mean square error, which can be derived from the following equation:  
\[MSE=\frac{1}{M\times N}\sum_{i=0}^{M-1}\sum_{j=0}^{N-1}\left[T_{i,j}-T_{i,j}^{ \prime}\right]^{2} \tag{19}\]  
where \(M\) and \(N\) denote the image size, \(T_{ij}\) and \(T_{ij}^{\prime}\) are the pixels of the original image and the pixels of the recovered image, respectively.  
Figure 7 shows a comparison of the rate distortion curves of the algorithms on the Lena image. The algorithm developed by [PERSON] et al. [5] sustains a stable PSNR of approximately  
Figure 6: Images generated during the experiment. (**a**) Original image. (**b**) Encrypted image. (**c**) Marked encrypted image I. (**d**) Marked encrypted image II. (**e**) Marked encrypted image III. (**f**) Marked encrypted image IV. (**g**) Recovered image.  
40 dB during image recovery. The algorithm by [PERSON] et al. [7] employs matrix encoding for embedding, resulting in distortion during image recovery, with an average PSNR of 72.52 dB. Additionally, the algorithm by [PERSON] et al. [14] achieves a PSNR of 58.7 dB at an embedding rate of 1 bpp. The algorithm by [PERSON] et al. [18] discards certain data regarding the original image during encoding and embedding, which leads to image distortion. Similarly, the approach by [PERSON] et al. [21] experiences difficulty in fully recovering the original pixel due to bit flipping, causing image distortion. In contrast, the algorithm presented in this paper is tested in Matlab simulation experiments, and the PSNRs obtained are all \(\infty\) with full reversibility. In the theoretical derivation, Equations (11) and (14) are inverse operations to each other, which ensures that the extracted pixel \(\textbf{IR}(i,j)\) has the same value as before encryption, and the process of encoding the original image \(\textbf{I}\) into the transitional image \(\textbf{IR}\) is reversible, so that the recovered image is the same as the original image. The algorithm in this study achieves lossless image recovery and outperforms existing methodologies.