Image watermarking using separable fractional moments of Charlier–Meixner

Abstract

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.