Proposed quaternion fractional dual-Hahn moments for color image reconstruction and encryption

Moments are essential descriptors for capturing fundamental characteristics of a signal, such as its shape and texture, thereby enabling a compact and easily analyzable representation. This article introduces a new family of discrete fractional moments, the quaternion Cartesian fractional dual-Hahn moments (QCFrDHOMs). These moments are derived from the fractional dual-Hahn moments (FrDHOMs), which are constructed from the matrix of fractional dual-Hahn orthogonal polynomials (FrDHOPs), obtained through the spectral decomposition of the classical dual-Hahn orthogonal polynomials (DHOPs). To ensure the stability of the computations, particularly for high-degree polynomials, a recursive method is proposed to calculate the initial terms of the DHOPs, thereby reducing the risk of numerical instability. The FrDHOMs are then generalized into QCFrDHOMs for efficient analysis of color images using quaternion algebra. Experimental results demonstrate that the QCFrDHOMs outperform classical DHOMs in terms of robustness and reconstruction capability. Additionally, an encryption and decryption scheme using QCFrDHOMs and chaotic systems is presented. Tests show that this scheme provides significant resistance to various attacks while maintaining nearly intact quality in the decrypted images. This not only highlights the effectiveness of the encryption scheme but also the enhanced security and robustness of the approach. Compared to other existing methods, our scheme stands out for its exceptional reliability and robustness, making a significant contribution to the secure protection of color images.