## 4 Experimental Results  
In this study, we have meticulously conducted a comprehensive comparative analysis to highlight the performance distinctions between our innovative RDH approach and various established methodologies. At the heart of our evaluation framework is a set of nine carefully selected grayscale images, each with a resolution of \(512\times 512\) pixels, as illustrated in Figure 4.  
Sourced from the esteemed USC-SIPI image database [26], these images form the empirical foundation for our experimental analysis. To anchor our evaluation in rigorous quantitative metrics, we utilize the following two critical performance indicators: embedding capacity (EC) and peak signal-to-noise ratio (PSNR) [16]. Furthermore, PSNR acts as a well-known measure of image quality after data embedding. As defined in Equation (7), PSNR provides a mathematical framework for assessing how well the quality is preserved during the steganographic process, enabling an in-depth investigation into any potential degradation of image quality caused by the data hiding techniques under evaluation. \[PSNR=10 log_{10}\frac{255^{2}}{MSE} \tag{7}\]  
These metrics are vital for a detailed comparison of the efficacy of our proposed solution against established techniques in the domain. EC measures the number of secret bits that can be seamlessly integrated into a cover image without compromising its visual perception. This analysis aims to shed light on the relative advantages and possible limitations of our method, laying a strong foundation for ongoing innovation and enhancement in the realms of digital image processing and secure data embedding. More precisely, PSNR is defined by an equation that quantifies the relationship between the maximum possible power of a signal and the impact of noise that compromises its integrity upon reproduction. To measure PSNR, the mean square error (MSE) (Equation (8)) is utilized, indicating the average squared deviation in intensity levels between the compressed and the original (or reference) image. \[MSE(x,y)=\frac{1}{N\times N}\sum_{i=1}^{N^{2}}(x_{i}-y_{i})^{2} \tag{8}\]  
High image quality is indicated when the MSE of the compressed image is low, which suggests only a minor deviation from the original. MSE is determined by comparing thefollowing two images: the original, denoted as \(x\), and the distorted or compressed version, represented as \(y\). Table 2 presents a comparison of the data hiding performance of our proposed methods with that of existing methods, including [PERSON] & [PERSON] (2013) ([PERSON] & [PERSON], 2015), [PERSON] et al. (2015) ([PERSON] & [PERSON], 2019), [PERSON] & [PERSON] (2019) ([PERSON] & [PERSON], 2019), [PERSON] et al. (2019) ([PERSON] & [PERSON], 2020), and [PERSON] & [PERSON] (2022) ([PERSON] & [PERSON], 2020). Among these comparisons, [PERSON] & [PERSON] (2013) and [PERSON] et al. (2019) report a performance of 1.07 bits per pixel (bpp), showing comparable outcomes. However, our analysis finds no significant overall difference in performance among the existing methods. Noteworthy is that [PERSON] et al. (2019) ([PERSON] & [PERSON], 2020) employ an EMD-based method, whereas [PERSON] & [PERSON] (2022) ([PERSON] & [PERSON], 2020) utilize a LSB matching revisited approach. Our proposed method exhibits superior performance, achieving 1.10 bpp, thus surpassing the bpp values reported by the previously mentioned methods. Figure 4: Test grayscale images (512 \(\times\) 512): (**a**) Baboon, (**b**) Barbara, (**c**) Boat, (**d**) Goldhill, (**e**) Airplane, (**f**) Lena, (**g**) Peppers, (**h**) Tiffany, and (**i**) Zelda. To assess the performance of data quality and embedding capacity (EC), we conducted simulations on both the existing methods and our proposed method using randomly generated data on selected cover images. The results are showcased in Table 3, where the superiority of our proposed method is highlighted.