## 3 Experimental Results and Comparison  
In this section, the experimental results and comparisons are provided to verify the effectiveness of the proposed method. The experiments were performed on eight standard test images, as shown in Figure 5[49]. The typical and state-of-the-art RDHEI methods in [15; 17; 18; 21; 41; 23] were used as the competitors. Figure 7 shows the encrypted image, the marked encrypted image, the marked decrypted image, and the recovered image of _Lena_ in the proposed method. Figure 7b is the encrypted image after the image has been preprocessed and encrypted. Figure 7c is the marked encrypted image with the embedding rate of 1 bpp after embedding the data. Figure 7d is the marked decrypted image generated from Figure 7c. The PSNR value of Figure 7d is 51.10 dB. Figure 7e is the recovered image, which is the same as Figure 7a. Table 1 shows the embedding rates of the proposed method and the methods in [15; 17; 18; 21; 41; 41] on the eight test images in Figure 5. The embedding rates are measured by bit per pixel (bpp), i.e., the ratio of the number of secret bits to the number of image pixels. In the proposed method, 128 bits were used to represent the linear regression-based predictor (32 bits for each coefficient), and 9 bits and 5 bits were used to represent the prediction error value and the length of the Huffman codeword, respectively, in the Huffman codebook. In [PERSON]'s method [34], the block size was set to \(4\times 4\). In [PERSON] and [PERSON]'s method [41], the pixel group consisted of seven pixels, and the 6 th LSB plane was used for embedding data. Because extracted-bit errors may occur in [PERSON]'s method [34] and in [PERSON] and [PERSON]'s method [41], the pure embedding rates in these methods are calculated by multiplying the original embedding rate by \((1-H(p))\times ER_{\text{O}}\), where \(ER_{\text{O}}\) is the original embedding rate, and \(H(p)\) is the binary entropy for the extract-bit error rate \(p\)[15]. As shown in the table, the proposed method for the eight images can achieve higher embedding rate than the other methods. For the images that contain large smooth regions, such as _Airplane_, _Crowd_, and _Lena_, the proposed method can achieve embedding rates higher than 3.4 bpp. For the images containing many complex regions, such as _Baboon_, the proposed method still works efficiently to achieve an embedding rate higher than 1.7 bpp, which is much higher than the other methods. Since the proposed method directly respectively trains the specific linear regression-based predictor for each original image, the predictor can be more accurate than the conventional predictor for all imagesm such as median edge detector. In smooth images such as _Airplane_, the predictor can be very precise. The prediction errors  
Figure 7: Experimental images of _Lena_ in the proposed method. can be highly concentrated in a small range so that they can be compressed efficiently. In complex images such as _Baboon_, it is hard to predict pixel values precisely, and the prediction errors cannot be as concentrated as those in smooth images. However, the predictor still reduces the errors as much as possible for higher capacity. Therefore, the proposed method can efficiently use the redundancy of the original image to vacate large room for data embedding. Table 2 shows an average time of the proposed method in training linear regression-based predictor, vacating room, image encryption, and data hiding (1 bpp). The experimental environment is a computer with Intel i9-10920X 3.5 GHz CPU, 32 GB RAM, and MATLAB R2017a. As shown in the table, training a linear regression-based predictor on an original image for vacating room is quick and practical. Figures 8 and 9 show the comparison of the marked decrypted image quality measured by peak signal to noise ratio (PSNR) in the proposed method and the methods in [15; 17; 18; 21; 34; 41]. PSNR is a standard for image distortion evaluation.