## 1 Introduction  
The widespread adoption of the Internet has transformed nearly every aspect of modern life. With billions of users connected globally, the Internet has become an essential part of daily life, enabling instant communication and access to a vast array of resources. However, with this increase in connectivity comes the growing concern for privacy and secret communication. Data hiding that involves concealing specific information to protect it from unauthorized access or manipulation provides a satisfactory solution for secret communication. Hiding secret data in the cover image has been proposed for a variety of applications, including image copyright protection, annotation, and image authentication. Inevitably, hiding data in the cover image changes the image content, though the hiding distortion is very low and imperceptible to human eyes. Reversible data hiding (RDH) [1; 2; 3; 4; 5; 6; 7; 8; 9; 10] allows data to be embedded into an image in such a way that both the hidden data and the original image can be fully recovered, which is needed for situations where the integrity of the original image must be preserved after extracting the embedded data. It is especially crucial in fields like medical images, legal documents, and scientific data, where any alteration or distortion of the original data could lead to incorrect interpretations or harmful consequences. [PERSON] et al. [1] introduced a lossless data embedding scheme using a lossless compression algorithm, which compresses bits that are impacted by embedding to reserve space for hiding data. The secret data, along with the compressed bits, are then embedded within the reserved space in the cover image. In 2003, [PERSON] [2] developed a high-capacity reversible data hiding method utilizing a difference expansion (DE) technique, embedding secret data in the pixel differences on high-pass bands of the 1-D Haar wavelet transform. This DE method enables a high bit rate of 0.15 to 1.97 bpp, marking a significant enhancement in hiding capacity. [PERSON] and [PERSON] [3] later introduced an improved version of the DEscheme, called prediction-error expansion, which leverages local pixel correlations more effectively than DE. Their method demonstrated a doubled maximum hiding capacity compared to DE in the experimental results. In 2006, [PERSON] et al. [4] introduced a histogram-based reversible data hiding method that uses pairs of peak and zero points to embed data into histogram bins, achieving low embedding distortion. To further enhance hiding capacity, [PERSON] et al. [5] proposed a multilevel reversible data hiding scheme using difference image histogram modification, combining peak-point detection with a multilevel hiding strategy to ensure reversibility while increasing capacity. [PERSON] et al. [6] extended histogram modification by using pixel differences and introduced a binary tree structure to eliminate the need for communicating overhead information to the recipient. Reversible data-hiding techniques have also been explored in the compression domain, including JPEG [7], SMVQ [8], and wavelet domains [9]. Reversible data hiding based on image interpolation [11; 12; 13; 14; 15; 16; 17] has been proposed to ensure both integrity and reversibility of the image. Interpolated images, generated by scaling or resolution enhancement, provide space for data embedding. [PERSON] et al. [12] used various interpolation techniques to create a hiding room, embedding the decimal-converted secret data into cover pixels by adding or subtracting values. [PERSON] and [PERSON] [13] employed enhanced neighbor mean interpolation (ENMI) and modified neighbor mean interpolation (MNMI) to scale up images before embedding, improving hiding capacity. A novel two-layer secure scheme [14] combined interpolation-based data hiding with difference expansion, utilizing the maximum pixel difference to provide high-capacity data hiding without compromising image quality. [PERSON] et al.