## 4 Experimental Results
Figure 5: Comparative PSNR of SI\({}_{1}\) with different ERs for Lena image in different schemes [14,22,23]. Figure 6: Comparative PSNR of SI\({}_{2}\) with different ERs for Lena image in different schemes [14,22,23]. Figure 7 expands on the traditional visual analysis by presenting detailed histograms comparing the original image of [PERSON] with the two displayed images \(\Delta I_{1}\) and \(\Delta I_{2}\). This deliberate arrangement facilitates a direct visual comparison and provides a clear benchmark for evaluating the effectiveness of our data hiding methods. In particular, Figure 7a highlights the subtle differences between the histograms of the original image and that displayed in \(\Delta I_{1}\). Similarly, Figure 7b provides a comprehensive comparison between the original image and that displayed in \(\Delta I_{2}\), with both marked Lena images containing 50,000 hidden bits. As depicted in Figure 7a,b, the histogram of the displayed image closely matches the histogram of the original image with only minor deviations. This observation validates the non-intrusive nature of our data hiding technique and clearly differentiates our approach from conventional data hiding methods. Our method, which efficiently conceals data while preserving the integrity of the original image's histogram, signifies a substantial advancement in the field of secure data embedding techniques. We underscore the adaptability of our method across a broad spectrum of applications in digital media security, prepared to lead innovation and enhance the safeguarding of digital communications. As we delve deeper into the realms of data hiding and digital image processing, we anticipate developing even more secure and efficient solutions in the future. We elaborate on the methodology employed to augment the resilience of digital shadows against regular singular (RS) analysis attacks. Within the scope of RS analysis, a shadow is partitioned into the following three distinct subsets: the Regular subset (denoted as \(R_{M}\) or \(R_{-M}\)), the Singular subset (\(S_{M}\) or \(S_{-M}\)), and the Unaltered subset \(U\). The terms \(R_{M}\) and \(S_{M}\) (or \(R_{-M}\) and \(S_{-M}\)) quantitatively denote the percentage of Regular and Singular subsets, which are masked by \(M\) or its complement \(-M\), respectively. The experimental framework is designed to group pixels into quartets and apply a mask \(M\), which is defined by the matrix \([0,1,1,0]\). This mask plays a crucial role in distinguishing between the Regular and Singular subsets. For a configuration to be considered as valid within the RS analysis paradigm, the interrelation between the Regular and Singular subsets must adhere to the constraints as outlined by Equations (9) and (10). \[R_{M}+S_{M}\leq 1\text{ and }R_{-M}+S_{-M}\leq 1 \tag{9}\]  
\[R_{M}\cong R_{-M}\text{ and }S_{M}\cong S_{-M} \tag{10}\]  
Figure 7: Comparison of histograms of the original Lena image and the two marked images. Figure 8 provides a visual accompaniment to the RS analysis for each shadow derived from the benchmark images 'Lena'. The RS analysis ensures that the following conditions are met: Firstly, \(R_{M}+S_{M}\leq 1\): The sum of the regular and singular pixels under the influence of mask \(M\) should be less than or equal to one. This condition suggests that after the mask's application, the overall alteration to the image is minimized to reduce the likelihood of detection. Secondly, \(R_{M}\approx R_{-M}\) and \(S_{M}\approx S_{-M}\): The regular and singular pixels affected by the mask \(M\) should closely correspond to those affected by the inverse mask \(-M\). This balance ensures the message's uniform distribution throughout the image, thereby preventing concentrations in any specific area that could facilitate detection. By adhering to these conditions, RS analysis enables the careful encoding of a message in an image while preserving its visual integrity. The approach effectively balances concealing information with the risk of image quality degradation due to alterations.