Difference operators and generalized discrete fractional transforms in signal and image processing

The fractional Fourier transform (FrFT) is a major tool in signal and image processing. Since its computation for analog signals includes the evaluation of improper integrals involving e−x2,x∈R, several methods have been proposed to approximate the FrFT for various signals. These methods include spectral decomposition techniques, which are based on the theory of second-order self-adjoint operators. This approach led to a tremendous stream of research on various spectral decomposition methods, including multiparameter and randomized transforms. In this paper, we introduce generalized discrete transforms that extend the known discrete-type transforms and introduce new types as well. The derivations are carried out in both unitary and non-unitary settings. The strengths of the proposed transforms are demonstrated through numerical simulations and applications in image encryption and watermarking.