Phase image encryption in the fractional Hartley domain using Arnold transform and singular value decomposition

An innovative fractional order LMS based on variable initial value and gradient order

Real transforms require less complexity for computations and less memory for storages than complex transforms. However, discrete fractional Fourier and Hartley transforms are complex transforms. In this paper, we propose reality-preserving fractional versions of the discrete Fourier, Hartley, generalized Fourier, and generalized Hartley transforms. All of the proposed real discrete fractional transforms have as many as $O(N^{2})$ parameters and thus are very flexible. The proposed real discrete fractional transforms have random eigenvectors and they have only two distinct eigenvalues 1 and $-$ 1. Properties and relationships of the proposed real discrete fractional transforms are investigated. Besides, for the real conventional discrete Hartley and generalized discrete Hartley transforms, we propose their alternative reality-preserving fractionalizations based on diagonal-like matrices to further increase their flexibility. The proposed real transforms have all of the required good properties to be discrete fractional transforms. Finally, since the proposed new transforms have random outputs and many parameters, they are all suitable for data security applications such as image encryption and watermarking.

Real Discrete Fractional Fourier, Hartley, Generalized Fourier and Generalized Hartley Transforms With Many Parameters

In this paper, we construct discrete fractional Fourier transforms (DFrFT) using recently introduced closed-form Hermite–Gaussian-like (HGL) eigenvectors. With respect to such eigenvectors, we discuss the convergence of their components to samples of the corresponding continuous Hermite–Gaussian functions and propose solutions to deal with some restrictions related to their construction. This allows us to give new procedures for obtaining orthonormal bases of HGL eigenvectors, which are used to fractionalize the discrete Fourier transform. We illustrate the application of the resulting DFrFT in the scenario of filtering in the fractional domain and compare the results with existing DFrFT approaches.

Discrete Fractional Fourier Transforms Based on Closed-Form Hermite–Gaussian-Like DFT Eigenvectors

Optimized sparse fractional Fourier transform: Principle and performance analysis

In this paper, two new real fractional transforms with many parameters are constructed. They are the real discrete fractional Fourier transform (RDFRFT) and the real discrete fractional Hartley transform (RDFRHT). The eigenvectors of these two new transforms are all random, and they both have only two distinct eigenvalues: 1 or -1. Real eigenvectors of both two transforms are constructed from random DFT-commuting matrices. We also propose an alternative definition of RDFRHT based on a diagonal-like matrix. All of the proposed new transforms have required good properties to be fractional transforms. Finally, since outputs of proposed new transforms are random, they can be applied in image encryptions.

Wei-Ching Chang

Papers

Real Discrete Fractional Fourier, Hartley, Generalized Fourier and Generalized Hartley Transforms With Many Parameters

Multiple-parameter real discrete fractional Fourier and Hartley transforms