## 4 Experiments and Analysis of Results
### Data Embedding Performance  
The embedding rate (ER) serves as a critical metric for gauging the effectiveness of an information hiding algorithm, quantifying the average amount of information embeddable per pixel in an image. The embedding rate can be calculated as follows:  
\[\text{ER}=\frac{\text{Total number of embedded bits}}{\text{Total number of pixels of the encrypted image}} \tag{20}\]  
We can theoretically calculate the ER of the algorithm proposed in this paper:  
\[\text{ER}=\frac{8(r-1)(MN-3)+8(1-r+n)\left\lfloor Load/8\right\rfloor}{nMN} \tag{21}\]  
where \(Load\) represents the total load capacity of the intermediate image, higher pixel prediction accuracy corresponds to a greater \(Load\) value. It is evident that with other parameters unchanged, as the image size \(MN\) increases, ER also increases. In practice, as the image size grows, the \(Load\) value also increases, thereby increasing ER further. Additionally, the threshold values \((r,n)\) will cause nonlinear effects on ER depending on the actual sizes of other parameters. Upon receiving the encrypted image, the information hider embeds the secret data by replacing bits in the image's noisy regions. The quantity of embedded information correlates with the load capacity value of the intermediate image. Table 3 presents the load capacities for various images when \(B=64\). The load capacity for an image is not merely the sum of the pixel blocks' load capacities, as additional space is required to store edge information, ensuring the algorithm's ability to fully recover the original image. As seen in Table 3, the load capacity varies across images due to their distinct texture features. By  
Figure 7: Comparison chart of PSNR variations at different ERs. [5; 7; 14; 18; 21]. employing the (3, 4)-threshold algorithm proposed in this paper for image embedding, it is clear that the algorithm demonstrates varying ERs across different images. To assess the universality of the embedding capabilities of the algorithms proposed in this paper, we selected 10,000 grayscale images of size \(512\times 512\) from each of the BOSSBase and BOWS-2 databases, along with 1338 grayscale images of sizes \(512\times 384\) or \(384\times 512\) from the UCID database. These images were used to embed randomly generated information. According to the experimental findings presented in Table 4, the minimum embedding rate of the algorithm for the (3, 4) threshold-exceeds 4 bpp, and the average embedding rate is over 5.5 bpp across all databases. For the (6, 6)-threshold, the minimum embedding rate exceeds 6.5 bpp, with an average rate over 7 bpp. Furthermore, the PSNRs for the algorithm are infinite, ensuring reversibility. The performance of the algorithms, in terms of embedding rates across the three image databases, is illustrated in Figure 8.  
Next, we delve into the impact of the threshold values in secret sharing on the image embedding rate. Table 5 illustrates the comparative embedding rates of our algorithm in embedding experiments on Man image under different parameters. The experimental findings reveal that with a constant threshold value, smaller block sizes result in higher embedding rates. This phenomenon arises because the AMED prediction relies on texture features from different blocks; thus, the smaller the block size, the higher the prediction accuracy. Moreover, when the block size remains constant, holding an equal \(r\) value, lower \(n\) values correspond to higher embedding rates, and when \(n\) values remain the same, higher \(r\) values lead to higher embedding rates. The maximum embedding rate achieved by our algorithm in this experiment reaches up to 7.5330 bpp. In practical applications, it is essential to align the \(r\) and \(n\) values of the algorithm as closely as possible to ensure both resilience and a high embedding rate.