Discrete Hartley Transforms Research Articles

An image encryption algorithm using hybrid sea lion optimization and chaos theory in the hartley domain

ABSTRACT This paper implements a hybrid encryption algorithm that combines multiple chaotic systems with the Sea Lion Optimization (SLO) algorithm and Discrete Hartley Transforms. Before converting the input image to the frequency domain, it undergoes confusion using the Lorenz chaotic system and is divided into four equal parts. Once the four sections are converted into the frequency domain, the confusion operations are carried out on each section using the chaotic sequences generated by the Uruk 4D chaotic system. The SLO algorithm incorporates a third phase of confusion to enhance randomness and disrupt the statistical correlation among the image components. The ultimate image is encrypted utilizing a bit stream derived from a chaotic Nahrain 3D system. The results of security tests and analyses conducted with the new hybrid algorithm demonstrate that it exhibited a superior level of protection against predictable statistical attacks compared to conventional encryption techniques.

Reducing peak to average power ratio (PAPR) for visible light communications (VLC) system using Walsh–Hadamard precoded discrete Hartley transforms

Abstract In visible light communication (VLC), intensity modulation with direct detection (IM/DD) based on DC-biased optical orthogonal frequency division multiplexing (DCO-OFDM) is used to transfer data at a high rate. Because OFDM signals are modulated using IM/DD, the peak of the transmitted signals is scaled up, resulting in a high peak-to-average power ratio (PAPR). In this article, the Walsh–Hadamard transform (WHT) is used as a precoder for discrete Hartley transform (DHT) to decrease the PAPR and the DC bias and achieve a good bit error rate (BER) performance in such a system. PAPR reduction results from the decrease of superposition of the transmitted signals owing to the use of WHT as a precoder. The proposed scheme and conventional DCO-OFDM indoor VLC system based on discrete Hartley transform (DHT) are compared to assure the new system’s benefits and efficiency.

A spiral based image watermarking scheme using Karhunen–Loeve and discrete hartley transforms

The main objective of developing the image watermarking technique is satisfying the robustness and the imperceptibility of multimedia protection. To achieve this objective, a novel hybrid image watermarking scheme based on Karhunen Loeve transform (KLT) and discrete hartley transform (DHT) and is proposed. In the proposed scheme (KL---DHT based watermarking scheme), the original image is divided into blocks using spiral scan, then the DHT is calculated for each block, after that, the KLT is applied for each DHT block. The watermark embedding process is carried out during the KLT transformation process. In the watermark extracting process, the watermarked image is distorted using several image attacks such as JPEG image compression, image rotation, image resizing, adding noise (Gaussian, impulsive or speckle) or image contrast adjustment using adaptive histogram equalization. Furthermore, the proposed scheme is carried out using block by block image scan and the achieved results are compared with those obtained from other watermarking schemes such as discrete cosine transform/singular value decomposition (DCT---SVD) based watermarking scheme, discrete wavelet transform/SVD (DWT---SVD) based watermarking scheme and SVD watermarking in the homomorphic domain watermarking scheme. The achieved results proved that, the superiority of the KL---DHT based watermarking scheme with spiral scan compared to other watermarking schemes in the presence of different image attacks.

Two‐band fast Hartley transform

Efficient algorithms have been developed over the past 30 years for computing the forward and inverse discrete Hartley transforms (DHTs). These are similar to the fast Fourier transform (FFT) algorithms for computing the discrete Fourier transform (DFT). Most of these methods seek to minimise the complexity of computations and/or the number of operations. A new approach for the computation of the radix-2 fast Hartley transform (FHT) is presented. The proposed algorithm, based on a two-band decomposition of the input data, possesses a very regular structure, avoids the input or out data shuffling, requires slightly less multiplications than the existing approaches, but increases the number of additions.

Arbitrary-length Fast Hartley Transform without Multiplications

Discrete Hartley transform (DHT) is an important tool in digital signal processing. In this paper, a multilierless algorithm to compute discrete Hartley transforms is proposed, which can deal with arbitrary length input signals. The proposed algorithm can be implemented by integer additions of fixed points in binary system. Besides, an efficient and regular systolic array is designed to implement the proposed method, followed by the complexity analysis. Being different to other fast Hartley transforms, our algorithm can deal with arbitrary length signals and get high precision. The proposed method is easily implemented by hardware and very suited to a real-time processing.

New Parametric Discrete Fourier and Hartley Transforms, and Algorithms for Fast Computation

In this paper, we propose a new reciprocal-orthogonal parametric discrete Fourier transform (DFT) by appropriately replacing some specific twiddle factors in the kernel of the classical DFT by independent parameters that can be chosen arbitrarily from the complex plane. A new class of parametric unitary transforms can be obtained from the proposed transform by choosing all its independent parameters from the unit circle. One of the special cases of this class is then exploited for developing a new one-parameter involutory discrete Hartley transform (DHT). The proposed parametric DFT and DHT can be computed using the existing fast algorithms of the DFT and DHT, respectively, with computational complexities similar to those of the latter. Indeed, for some special cases, the proposed transforms require less number of operations. In view of the fact that the transforms of small sizes are used in some image and video compression techniques and employed as building blocks for larger size transform algorithms, we develop new algorithms for the proposed small-size transforms. The proposed parametric DFT and DHT, in view of the introduction of the independent parameters, offer more flexibility in achieving better performance compared to the classical DFT and DHT. As examples of possible applications of the proposed transforms, image compression, Wiener filtering, and spectral analysis are considered.

Fast Algorithms for the Computation of Sliding Discrete Hartley Transforms

Fast algorithms for computing generalized discrete Hartley transforms in a sliding window are proposed. The algorithms are based on second-order recursive relations between subsequent local transform spectra. New efficient inverse algorithms for signal processing in a sliding window are also presented. The computational complexity of the algorithms is compared with that of known fast discrete Hartley transforms and running recursive algorithms

Fast and numerically stable algorithms for discrete Hartley transforms and applications to preconditioning

The discrete Hartley transforms (DHT) of types I -- IV and the related matrix algebras are discussed. We prove that any of these DHTs of length $N=2^t$ can be factorized by means of a divide--and--conquer strategy into a product of sparse, orthogonal matrices where in this context sparse means at most two nonzero entries per row and column. The sparsity joint with orthogonality of the matrix factors is the key for proving that these new algorithms have low arithmetic costs equal to $\frac52 N\log_2 (N)+O(N)$ arithmetic operations and an excellent normwise numerical stability. Further, we consider the optimal Frobenius approximation of a given symmetric Toeplitz matrix generated by an integrable symbol in a Hartley matrix algebra. We give explicit formulas for computing these optimal approximations and discuss the related preconditioned conjugate gradient (PCG) iterations. By using the matrix approximation theory, we prove the strong clustering at unity of the preconditioned matrix sequences under the sole assumption of continuity and positivity of the generating function. The multilevel case is also briefly treated. Some numerical experiments concerning DHT preconditioning are included.

A Novel Algorithm for Computing the 1-D Discrete Hartley Transform

An efficient split algorithm for calculating the one-dimensional discrete Hartley transforms, by using a special partitioning in the frequency domain, is introduced. The partition determines a fast paired transform that splits the 2/sup r/-point unitary Hartley transform into a set of 2/sup r-n/-point odd-frequency Hartley transforms, n=1:r. A proposed method of calculation of the 2/sup r/-point Hartley transform requires 2/sup r-1/(r-3)+2 multiplications and 2/sup r-1/(r+9)-r/sup 2/-3r-6 additions.

Fast DHT algorithms for length N=q*2/sup m/

This article presents an improved split-radix algorithm that can flexibly compute the discrete Hartley transforms (DHT) of length-q*2/sup m/ where q is an odd integer. Comparisons with previously reported algorithms show that savings on the number of arithmetic operations can be made. Furthermore, a wider range of choices on different DHT lengths is naturally provided.

Fast radix-3/9 discrete Hartley transform

An efficient algorithm for computing radix-3/9 discrete Hartley transforms (DHTs) is presented. It is shown that the radix-3/9 fast Hartley transform (FHT) algorithm reduces the number of multiplications required by a radix-3 FHT algorithm for nearly 50%. For the computation of real-valued discrete Fourier transforms (DFTs) with sequence lengths that are powers of 3, it is shown that the radix-3/9 FHT algorithm reduces the number of multiplications by 16.2% over the fastest real-valued radix-3/9 fast Fourier transform (FFT) algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>