Showing papers on "Hartley transform published in 2022"

Discrete quaternion linear canonical transform

Abstract: Fourier transform (FT), and its generalizations, the fractional Fourier transform (FrFT) and linear canonical transform (LCT) are integral transforms that are useful in optics, signal processing, and in many other fields. In the applications, the performance of LCT is superior because of its three extra degrees of freedom as compared to no degree of freedom for FT and one degree of freedom for FrFT. Recently, quaternion linear canonical transform (QLCT), an extension of the LCT in quaternion algebra, has been derived and since received noticeable attention, thanks to its elegance and expressive power in the study of multi-dimensional signals/images. To the best of our knowledge computation of the QLCT by using digital techniques is not possible now, because a discrete version of the QLCT is undefined. It initiated us to introduce the two-dimensional (2D) discrete quaternion linear canonical transform (DQLCT) that is analogous to the 2D discrete quaternion Fourier transform (DQFT). The main properties of the 2D DQLCT, including the basic properties, reconstruction formula and Rayleigh-Plancherel theorem, are obtained. Importantly, the convolution theorem and fast computation algorithm of 2D DQLCT, which are key to engineering usage, are considered. Finally, we demonstrate applications, illustrate simulations, and discuss some future prospects of the DQLCT.

Windowed linear canonical transform: its relation to windowed Fourier transform and uncertainty principles

Abstract: Abstract The windowed linear canonical transform is a natural extension of the classical windowed Fourier transform using the linear canonical transform. In the current work, we first remind the reader about the relation between the windowed linear canonical transform and windowed Fourier transform. It is shown that useful relation enables us to provide different proofs of some properties of the windowed linear canonical transform, such as the orthogonality relation, inversion theorem, and complex conjugation. Lastly, we demonstrate some new results concerning several generalizations of the uncertainty principles associated with this transformation.

Bargmann Transform on the Space of Hyperplanes

Abstract: We introduce an integral transform on the space of hyperplanes applying the Plancherel formula of the Radon transform to the definition of the semiclassical Bargmann transform on the Euclidean space. This is similar to the semiclassical Bargmann transform and some basic facts on microlocal analysis are also discussed.

A graph signal processing approach to Fourier-like number-theoretic transforms

Abstract: In this paper, we employ a graph signal processing approach to redefine Fourier-like number-theoretic transforms, which includes the Fourier number transform itself, the Hartley number transform and specific types of the cosine number transform and the sine number transform. Our strategy basically consists in identifying graphs whose Laplacian or adjacency matrix has an eigenbasis coinciding with the basis employed to define each of the aforementioned transforms. We then demonstrate how to extend this idea to multidimensional cases and provide a general definition, which corresponds to the graph Fourier number transform (GFNT). We develop illustrative examples and perform a preliminary investigation regarding the use of the GFNT in an image encryption scenario.

On the Zak Transform-based Interpretation of OTFS Modulation

Abstract: In this paper, we present a Zak transform-based development of the recently proposed orthogonal time frequency space (OTFS) modulation scheme. Unlike previous works, we focus on the interpretation of the spreading function as the Zak transform of the “impulse train response” for general underspread linear time-varying (LTV) channels. For an underspread channel, the Zak transform of the output signal is given by the twisted convolution of the spreading function with the Zak transform of the input signal. This twisted convolution relationship provides a Zak domain input-output relationship for general underspread LTV channels. We extend these results to the discrete case, by presenting a development of the discrete Zak transform (DZT) similar to the one provided by Mohammed for the continuous Zak transform. We argue that the discrete Zak domain twisted convolution relationship for LTV channels provides a simple and concise input-output relationship for OTFS modulation, analogous to the frequency domain multiplication relationship for orthogonal frequency division multiplexing (OFDM) over linear time invariant (LTI) channels. Lastly, we discuss the impact of adding a cyclic prefix and zero-padding in delay and Doppler in the discrete Zak domain.

Asymmetric double-image encryption using twin decomposition in fractional Hartley domain

Abstract: Twin decomposition, consisting of equal and random modulus decompositions, not only makes a cryptosystem asymmetric but also resists special attack. A new double-image asymmetric cryptosystem using twin decomposition in fractional Hartley domain is proposed. An input grayscale image, bonded with another grayscale image as its phase mask, is transformed via fractional Hartley transform. Equal modulus decomposition is applied on the resulting image, giving us two intermediate images. One of them is subjected to another fractional Hartley transform followed by random modulus decomposition, whereas the other serves as the first private key. The application of random modulus decomposition also results in two images: encrypted image and the second private key. During the process of decryption, firstly the encrypted image is combined with second private key and thereafter it is subjected to inverse fractional Hartley transform. The resulting image is then combined with the first private key, and followed by another inverse fractional Hartley transform, thus recovering the two original images. The proposed cryptosystem is validated for pairs of grayscale images.

Discrete Cosine Transform: A Revolutionary Transform That Transformed Human Lives [CAS 101]

Abstract: The widely known Discrete Fourier Transform (DFT), the frequency-domain representation of a finite-length time-domain sequence is an orthogonal transform and has been known for a very long time and has found many application. The popularity of the DFT increased tremendously after the publication of the Fast Fourier Transform (FFT) algorithm by Cooley and Tukey in 1965 [1] . Orthogonal transforms offer many advantages, namely fast computational speeds, less storage space, less rounding off errors, etc. The benefits of the orthogonal transforms stem from the fact that they can be factored based on matrix computations. Data compression, digital filter design, speech processing, image and video processing are applications where the orthogonal transforms have made a significant impact.

Characterizations of the gyrator transform via the fractional Fourier transform

Abstract: ABSTRACT In this note, we will explain the relationship between the fractional Fourier transform and the gyrator transform. In particular, we will show the properties of the gyrator transform, which is getting the eigenfunction and eigenvalue of the gyrator transform, recursion formula, the relation between the Wigner distribution and the gyrator transform, the differential equation satisfied with the gyrator transform of some functions, and the representation of the gyrator transform as the self-adjoint operator. Moreover, we will consider the generalized gyrator transform of tempered distributions.

Integral Transforms Characterized by Convolution

Abstract: Abstract Inspired by Jaming’s characterization of the Fourier transform on specific groups via the convolution property, we provide a novel approach which characterizes the Fourier transform on any locally compact abelian group. In particular, our characterization encompasses Jaming’s results. Furthermore, we demonstrate that the cosine transform as well as the Laplace transform can also be characterized via a suitable convolution property.

A new complex SEL integral transform and its applications on ordinary differential equations

Abstract: This paper introduces a new complex integral transformation obtained by inserting a complex parameter into the well-known Rangaig integral transform kernel function. The new integral transform is denoted by the acronym SEL and is called the Complex (Serifenur-Emad-Luay) integral transform. The proposed SEL integral transform features are explained and shown to correspond to some fundamental functions. The application of the SEL transform to finding the solution of some differential equations, including those arising in some real-world practical applications, is discussed as an illustration of the actual fields that could benefit from this novel transform.

One result on boundedness of the Hilbert transform in Marcinkiewics spaces

Abstract: In mathematics and in signal theory, the Hilbert transform is an important linear operator that takes a real-valued function and produces another real-valued function. The Hilbert transform is a linear operator which arises from the study of boundary values of the real and imaginary parts of analytic functions. Also, it is a widely used tool in signal processing. The Cauchy integral is a figurative way to motivate the Hilbert transform. The complex view helps us to relate the Hilbert transform to something more concrete and understandable. Moreover, the Hilbert transform is closely connected with many operators in harmonic analysis such as Laplace and Fourier transforms which have numerous application in partial and ordinary differential equations. In this paper, we study boundedness properties of the classical (singular) Hilbert transform acting on Marcinkiewicz spaces. More precisely, we obtain if and only if condition for boundedness of the Hilbert transform in Marcinkiewicz function spaces.

Integral transforms characterized by convolution

Abstract: Inspired by Jaming's characterization of the Fourier transform on specific groups via the convolution property, we provide a novel approach which characterizes the Fourier transform on any locally compact abelian group. In particular, our characterization encompasses Jaming's results. Furthermore, we demonstrate that the cosine transform as well as the Laplace transform can also be characterized via a suitable convolution property.

Some properties of fractional hartley transform

Abstract: This paper is motivated by the ideas of fractional Fourier transform and Hartley transform. Looking towards the practicality and demanding attention of fractional Hartley transform we take keen interest into it. In this paper, we deal with inverse theorem of FRHT and some important properties of fractional Hartley transform like exponential rule, multiplication rule, transform of derivative and derivative of transform, which play a very crucial role in the development of fractional Hartley transform.

The uncertainty principle for the octonion Fourier transform

Abstract: The octonion Fourier transform (OFT) is a hypercomplex Fourier transform that generalizes the quaternion Fourier transform. However, in octonion algebra, there are two major obstacles that are presented in the loss of associativity and commutativity. Researchers have been trying to extend the results of the Euclidean Fourier transform to quaternion‐valued signals using special techniques to overcome these two problems. In this context, we intend to generalize the Heisenberg uncertainty principles associated with covariance and Hardy's uncertainty principle for octonion multivector valued signals over ℝ3$$ {\mathbb{R}}^3 $$ using the polar form of an octonion, the quaternion decomposition, and the relationship between the OFT and the three‐dimensional (3D)‐Clifford‐Fourier transform.

Encryption and Decryption of Image for Secured Data Transmission

Abstract: This work combines Arnold transform method with the spatial domain to encrypt a covered image and also recovers the covered image by the application of inverse Arnold transform. First, transform a cover image into subparts which consists of eight binary images by decimal value to eight-digit binary operation. Then, transform eight binary images into sub-blocks of eight binary scrambled images by the Arnold transform, respectively. Further, recombine the sequence of the eight binary scrambled matrices into a scrambled matrix with 256 Gy levels according to the specific club. Discrete wavelet transform (DWT) is used to perform image compression on the input image and secretly hidden image which is done using alpha blending. Finally, derive an encrypted image from the scrambled image by the Hartley transform. Second, decode the encrypted image using inverse Arnold Transform. Inverse DWT is performed to regain the compressed images. Simulations indicate that the hybrid new technique has more security, more robustness counter to occlusion and attacks about with high degree of image scrambling.

A Complex Integral Transform “Complex EE Transform” and Its Applications

Abstract: In this paper, a new complex integral transform under the name EE (Emad-Elaf) transform had been suggested and applied to solve linear ordinary differential equations with constant coefficient, the new EE transform had been tested by applying it on some practical problems, and it proved to be an efficient and powerful transform.

Multiple Transform Selection for Chroma Channel

Abstract: For chroma signal, the current VVC standard can either perform transform using DCT-II kernel or skip transform, which is much simpler than for the luma. Since video of 444 color format can have much diverse chroma characteristics, the VVC scheme employing the very simple transform cannot represent the rich chroma information effectively. In this paper, we propose a new Multiple Transform Selection (MTS) scheme for chroma to improve its transform efficiency.

Integral Transforms of the Hilfer-Type Fractional Derivatives

Abstract: In this paper, some important properties concerning the Hilfer-type fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform, Mellin transform and Fourier transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. An application is get with the Jafari transform and nite Hankel transform to obtain the analytical solution to fractional radial diffusion equation in terms of the \(\kappa\)-Hilfer fractional derivative.

Windowed linear canonical transform: its relation to windowed Fourier transform and uncertainty principles

Abstract: Abstract The windowed linear canonical transform is a natural extension of the classical windowed Fourier transform using the linear canonical transform. In the current work, we first remind the reader about the relation between the windowed linear canonical transform and windowed Fourier transform. It is shown that useful relation enables us to provide different proofs of some properties of the windowed linear canonical transform, such as the orthogonality relation, inversion theorem, and complex conjugation. Lastly, we demonstrate some new results concerning several generalizations of the uncertainty principles associated with this transformation.