## 4 Experimental Results  
The hardware device used by our proposed recurrent robust reversible data hiding (triple-RDH) method is i7 with a solid-state disk with 16 GB of memory. The operating system is Win10, and the tool developed is MATLAB. Ten 512 \(\times\) 512 gray-scale images (\"Lena\", \"Baboon\", \"Barbara\", \"Boat\", \"Elaine\", \"F16\", \"House\", \"Pepper\", \"Sailboat\", and \"Tiffany\") are used as the test images, as shown in Figure 8. The secret data are binary string generated by a random function. The following two evaluation metrics are used in our experiment: embedding capacity (EC) and peak signal-to-noise ratio (PSNR). EC represents the amount of secret data that can be embedded in the image, usually calculated in bits. The maximum embedding capacity refers to the maximum amount of data that can be embedded in an image. The PSNR computes the peak signal-to-noise ratio, in decibels, between two images. The higher the PSNR, the better, indicating that the stego image is highly undetectable. The calculation method of PSNR is \(PSNR=10\times\log_{10}\left(\frac{MAX^{2}}{MSE}\right)\), where \(MAX\) is the maximum possible pixel value of the image. The method in this article is mainly based on the gray-scale image, so \(MAX\) is 255, and \(MSE\) is the mean square error, calculated by \(MSE=\frac{1}{H\times W}\sum\limits_{i=0}^{H-1}\sum\limits_{j=0}^{W-1}\left[I( i,j)-I^{\prime}(i,j)\right]^{2}\), where \(H\) is the length of the picture, \(W\) is the width of the picture, \(i,\ j\) are the pixel value positions, \(I\) is the cover image, and \(I^{\prime}\) is the stego image. Figure 9 presents the experimental results of our proposed recurrent robust reversible data hiding (triple-RDH) method. Among the predictor methods used, predictor method 2 shows better experimental results than other predictor methods, so the proposed triple-RDH method chooses the predictor method 2 in all experiments. Additionally, we choose to compare with SBDE (significant-bit-difference expansion) [26], TLE (two-layer embedding) [27], and IPVO (improved pixel-value ordering) [22], because the TLE method, SBDE method, IPVO method, and proposed triple-RDH method are all based on prediction error expansion (PEE) technique. Figure 8: Ten test images which are (**a**) Lena (**b**) Baboon (**c**) Barbara (**d**) Boat (**e**) Elaine (**f**) F16 (**g**) House (**h**) Pepper (**i**) Sailboat (**j**) Tiffany respectively. Figure 9: Cont. In Figure 9, compared with the other three methods, it can be clearly seen that the EC of our proposed method is significantly larger than the other methods. In terms of PSNR, in addition to the IPVO method, the triple-RDH method we proposed is also slightly higher than that of TLE method and SBDE method. Take the \"Lena\" image in Figure 9a and Table 5 as an example; you can see that the maximum EC of the TLE method has 214,116 bits, the maximum EC of the SBDE method has 179,793 bits, the maximum EC of IPVO method is 38,755 bits, and the maximum EC of our proposed triple-RDH method can be as high as 310,732 bits, which is 1.45 times and 1.73 times more than that of the TLE method and the SBDE method, respectively; the maximum EC of our proposed method is eight times that of the IPVO method. Figure 9: Comparison of PSNRs and ECs between the proposed triple-RDH, TLE, SBDE, and IPVO methods using ten test images (**a**) Lena (**b**) Baboon (**c**) Barbara (**d**) Boat (**e**) Elaine (**f**) F16 (**g**) House (**h**) Pepper (**i**) Sailboat (**j**) Tiffany. When the \"Lena\" image in Figure 9a has the EC = 20,000 bits, the PSNR values of the TLE method, the SBDE method, the IPVO method, and the proposed triple-RDH methods are 43.72, 42.41, 54.91, and 43.94 dB, respectively. When the embedded information is added up to 40,000 bits, the PSNR values of the TLE method, the SBDE method, and our proposed triple-RDH method are 40.52, 39.19, and 40.81 dB, respectively. At this time, the IPVO method is not capable of hiding 40,000 bits into the \"Lena\" image.