This paper proposes a new set of discrete orthogonal separable moments of fractional order, named Fractional Charlier–Meixner Moments (FrCMMs). The latter are constructed from fractional Charlier polynomials (FrCPs) and fractional Meixner polynomials (FrMPs) proposed in this paper. The proposed FrMPs are constructed algebraically using the spectral decomposition of classical Meixner polynomials and singular value decomposition (SVD). The proposed FrCMMs generalize the separable moments of Charlier–Meixner of integer order (CMMs). In addition, FrCMMs are characterized by the polynomial parameters and by the fractional orders of the two fractional kernel functions of Charlier and Meixner, which allows them to be used efficiently for different applications such as local and global image reconstruction and image watermarking. Based on the proposed FrCMMs, a new watermarking scheme for copyright protection of digital images in the transform domain is proposed where the watermark is embedded in the FrCMM coefficients leading to an efficient watermarking scheme in terms of imperceptibility, robustness and security. The performances of the proposed moments are evaluated and compared with discrete fractional moments existing in the literature and with classical separable moments of integer order.

本文提出了一组新的分数阶离散正交可分离矩，称为分数Charlier–Meixner矩（FrCMM）。后者由分数Charlier多项式（FrCP）和分数Meixner多项式（FrMP）构成。使用经典Meixner多项式的频谱分解和奇异值分解（SVD），以代数方式构建提出的FrMP。提出的FrCMM概括了Charlier–Meixner整数阶（CMM）的可分离矩。此外，FrCMM的特征在于多项式参数以及Charlier和Meixner的两个分数核函数的分数阶，这使它们可以有效地用于不同的应用程序，例如局部和全局图像重建和图像水印。基于提出的FrCMM，提出了一种在变换域中用于数字图像版权保护的新水印方案，其中将水印嵌入到FrCMM系数中，从而在不易察觉性，鲁棒性和安全性方面提供了一种有效的水印方案。评估了提出的矩的性能，并将其与文献中存在的离散分数矩以及经典的整数阶可分离矩进行了比较。