Showing papers on "Hartley transform published in 2010"
TL;DR: A general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier and S- transforms, is presented, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm.
Abstract: Examining the frequency content of signals is critical in many applications, from neuroscience to astronomy. Many techniques have been proposed to accomplish this. One of these, the S-transform, provides simultaneous time and frequency information similar to the wavelet transform, but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements. It has shown promise in many medical imaging applications but has high computational requirements. This paper presents a general transform that describes Fourier-family transforms, including the Fourier, short-time Fourier, and S- transforms. A discrete, nonredundant formulation of this transform, as well as algorithms for calculating the forward and inverse transforms are also developed. These utilize efficient sampling of the time-frequency plane and have the same computational complexity as the fast Fourier transform. When configured appropriately, this new algorithm samples the continuous S-transform spectrum efficiently and nonredundantly, allowing signals to be transformed in milliseconds rather than days, as compared to the original S-transform algorithm. The new and efficient algorithms make practical many existing signal and image processing techniques, both in biomedical and other applications.
126 citations
TL;DR: In this article, the authors proposed an optical orthogonal frequency division multiplexing (O-OFDM) scheme, suitable for intensity-modulated direct-detection systems, where the modulation/demodulation processing takes advantage of the fast Hartley transform algorithm.
Abstract: We present a novel optical orthogonal frequency division multiplexing (O-OFDM) scheme, suitable for intensity-modulated direct-detection systems, where the modulation/demodulation processing takes advantage of the fast Hartley transform algorithm. Due to the properties of the discrete Hartley transform (DHT), the conventional transmission scheme can be streamlined. We demonstrate that asymmetrically clipping (AC) technique can also be applied to DHT-based OFDM; the signal can be transmitted without the need of a DC bias, resulting in a power-efficient system, not affected by clipping noise. Hermitian symmetry is not required for the input signal. Therefore, this technique supports the double of input symbols compared to both AC and DC-biased O-OFDM, based on standard Fourier processing. The analysis in an additive white Gaussian noise channel shows that the same performance can be achieved by replacing 4, 16, and 64 QAM (quadrature-amplitude modulation) AC optical-OFDM with a simpler system based on DHT, using binary phase-shift keying (BPSK), 4 and 8 PAM (pulse-amplitude modulation), respectively.
115 citations
TL;DR: Two random angle shifts are introduced to rotate the color vectors composed by the three color components in discrete Hartley transform domains in image encryption process, and the corresponding rotation shifts of the two angles can serve as the key of the scheme.
72 citations
TL;DR: In this paper, the Sumudu transform was applied to solve a one-dimensional wave equation having a singularity at the initial conditions, and the authors also applied this transform to the distribution of the wave equation.
Abstract: In this paper, we generalize the concepts of a new integral transform, namely the Sumudu transform, to distributions and study some of their properties. Further, we also apply this transform to solve one-dimensional wave equation having a singularity at the initial conditions.
68 citations
TL;DR: Some numerical simulations have validated the feasibility of the proposed image encryption scheme and the parameters in the affine transform and the gyrator transform are regarded as the key for the encryption algorithm.
Abstract: We propose a kind of double-image-encryption algorithm by using the affine transform in the gyrator transform domain. Two original images are converted into the real part and the imaginary part of a complex function by employing the affine transform. And then the complex function is encoded and transformed into the gyrator domain. The affine transform, the encoding and the gyrator transform are performed twice in this encryption method. The parameters in the affine transform and the gyrator transform are regarded as the key for the encryption algorithm. Some numerical simulations have validated the feasibility of the proposed image encryption scheme.
65 citations
TL;DR: The novel discrete transform has several advantages over existing transforms, such as lower redundancy ratio, hierarchical data structure and ease of implementation.
Abstract: An implementation of the discrete curvelet transform is proposed in this work. The transform is based on and has the same order of complexity as the Fast Fourier Transform (FFT). The discrete curvelet functions are defined by a parameterized family of smooth windowed functions that satisfies two conditions: i) 2π periodic; ii) their squares form a partition of unity. The transform is named the uniform discrete curvelet transform (UDCT) because the centers of the curvelet functions at each resolution are positioned on a uniform lattice. The forward and inverse transform form a tight and self-dual frame, in the sense that they are the exact transpose of each other. Generalization to M dimensional version of the UDCT is also presented. The novel discrete transform has several advantages over existing transforms, such as lower redundancy ratio, hierarchical data structure and ease of implementation.
62 citations
01 Jan 2010
TL;DR: In this article, a new integral transform, namely Sumudu transform, was applied to solve linear ordinary differential equation with and without constant coefficients having convolution terms, in particular, to solve Spring-Mass systems, population growth and financial problem.
Abstract: In this work a new integral transform, namely Sumudu transform was applied to solve linear ordinary differential equation with and without constant coefficients having convolution terms. In particular we apply Sumudu transform technique to solve Spring-Mass systems, Population Growth and financial problem. Mathematics Subject Classification: Primary 35G15, 44A85; Secondary 44A35
62 citations
16 Aug 2010
TL;DR: An orthogonal multiplication-free transform of order that is an integral power of two by an appropriate extension of the well-known fourthorder integer discrete cosine transform is proposed and an efficient algorithm for its fast computation is developed.
Abstract: In this paper, we propose an orthogonal multiplication-free transform of order that is an integral power of two by an appropriate extension of the well-known fourthorder integer discrete cosine transform. Moreover, we develop an efficient algorithm for its fast computation. It is shown that the computational and structural complexities of the algorithm are similar to that of the Hadamard transform. By applying the proposed transform to image compression, we show that it outperforms the existing transforms having complexities similar to that of the proposed one.
57 citations
TL;DR: A joint space-wavenumber localized quaternion S transform is presented in this study for a simultaneous determination of the local color image spectra using a two-dimensional Gaussian localizing window that scales with wavenumbers.
57 citations
TL;DR: This paper structure certain types of non-bandlimited signals based on two ladder-shape filters designed in the LCT domain based on the phase function of the nonlinear Fourier atom which is the boundary value of the Mobius transform.
52 citations
TL;DR: An advanced Radon transform is developed using a multilayer fractional Fourier transform, a Cartesian-to-polar mapping, and 1-D inverse Fourier transforms, followed by peak detection in the sinogram.
Abstract: The Hough transform (HT) is a commonly used technique for the identification of straight lines in an image. The Hough transform can be equivalently computed using the Radon transform (RT), by performing line detection in the frequency domain through use of central-slice theorem. In this research, an advanced Radon transform is developed using a multilayer fractional Fourier transform, a Cartesian-to-polar mapping, and 1-D inverse Fourier transforms, followed by peak detection in the sinogram. The multilayer fractional Fourier transform achieves a more accurate sampling in the frequency domain, and requires no zero padding at the stage of Cartesian-to-polar coordinate mapping. Our experiments were conducted on mix-shape images, noisy images, mixed-thickness lines and a large data set consisting of 751 000 handwritten Chinese characters. The experimental results have shown that our proposed method outperforms all known representative line detection methods based on the standard Hough transform or the Fourier transform.
Book•
20 Oct 2010
TL;DR: Low level algorithms as mentioned in this paper use bit wizardry and permutations and their operations to find paths in directed graphs and search paths for directed graphs in directed graph graphs, using the GP language.
Abstract: Low level algorithms.- Bit wizardry.- Permutations and their operations.- Sorting and searching.- Data structures.- Combinatorial generation.- Conventions and considerations.- Combinations.- Compositions.- Subsets.- Mixed radix numbers.- Permutations.- Multisets.- Gray codes for string with restrictions.- Parenthesis strings.- Integer partitions.- Set partitions.- Necklaces and Lyndon words.- Hadamard and conference matrices.- Searching paths in directed graphs.- Fast transforms.- The Fourier transform.- Convolution, correlation, and more FFT algorithms.- The Walsh transform and its relatives.- The Haar transform.- The Hartley transform.- Number theoretic transforms (NTTs).- Fast wavelet transforms.- Fast arithmetic.- Fast multiplication and exponentiation.- Root extraction.- Iterations for the inversion of a function.- The AGM, elliptic integrals, and algorithms for computing.- Logarithm and exponential function.- Computing the elementary functions with limited resources.- Numerical evaluation of power series.- Cyclotomic polynomials, product forms, and continued fractions.- Synthetic Iterations.-. Algorithms for finite fields.- Modular arithmetic and some number theory.- Binary polynomials.- Shift registers.- Binary finite fields.- The electronic version of the book.- Machine used for benchmarking.- The GP language.- Bibliography.- Index.
TL;DR: A new method for color image encryption by wavelength multiplexing on the basis of two-dimensional (2-D) generalization of 1-D fractional Hartley transform that has been redefined recently in search of its inverse transform is proposed.
TL;DR: After analyzing the properties of WFRFT, a typical scheme for modulation/demodulation is proposed, which could make the statistics properties of the real and image part on both of the time and frequency domain and the phase properties have a significant variation.
Abstract: The paper reveals the relationship between the weighting coefficients and weighted functions via the research of coefficients matrix and based on the original definition of 4-weighted fractional Fourier transform (4-WFRFT). The multi-parameters expression of weighting coefficients are given. Moreover, the 4-WFRFT of discrete sequences is defined by introducing DFT into it, which makes it suitable for digital communication systems. After analyzing the properties of WFRFT, a typical scheme for modulation/demodulation is proposed, which could make the statistics properties of the real and image part on both of the time and frequency domain and the phase properties have a significant variation. Such a variation could be controlled by the adjustment of transform parameters. If the WFRFT of multi-parameters is implemented, it will be more difficult to intercept and capture the modulated signals than normal.
TL;DR: In this article, an extended fractional complex transform is proposed to convert some kinds of fractional differential equations with the modified Riemann-Liouville derivatives into ordinary differential equations.
Abstract: An extended fractional complex transform is proposed to convert some kinds of fractional differential equations with the modified Riemann-Liouville derivatives into ordinary differential equations.
01 Jan 2010
TL;DR: This chapter introduces the DHT and discusses those aspects of its solution, as obtained via the FHT, which make it an attractive choice for applying to the real-data DFT problem.
Abstract: This chapter introduces the DHT and discusses those aspects of its solution, as obtained via the FHT, which make it an attractive choice for applying to the real-data DFT problem. This involves first showing how the DFT may be obtained from the DHT, and vice versa, followed by a discussion of those fundamental theorems, common to both the DFT and DHT algorithms, which enable the input data sets to be similarly related to their respective transforms and thus enable the DHT to be used for solving those DSP-based problems commonly addressed via the DFT, and vice versa. The limitations of existing FHT algorithms are then discussed bearing in mind the ultimate objective of mapping any subsequent solution onto silicon-based parallel computing equipment. A discussion is finally provided relating to the results obtained in the chapter.
TL;DR: In this paper, a fractional Fourier transform with modified fractional integrals and derivatives is studied for solution of a partial diffusion-type differential equation of fractional order in a special space of functions.
Abstract: The paper is devoted to the study of a fractional Fourier transform in a special space of functions by Lizorkin. Connections of such a transform with differentiation operators are established and an inverse operator for such a transform if constructed. Compositions of the fractional Fourier transform with modified fractional integrals and derivatives are proved. Application to solution of a partial diffusion-type differential equation of fractional order is given.
TL;DR: A new method for image encryption using improper Hartley transform and chaos theory and two chaotic random intensity masks is proposed and the mean square error has been calculated.
TL;DR: An intimate mathematical relation between Hartley and Hilbert transforms is given here in contrast with the well known Fourier and Hilbert transform relations as discussed by the authors, and the feasibility of Hartley-Hilbert transform for a straight forward interpretation, total magnetic anomaly due to a thin plate from Tejpur, India and self potential data of the Sulley monkey anomaly in the Ergani Copper district, Turkey are illustrated in contrast to the Fourier-HILbert transform.
TL;DR: In this paper, an elementary proof is presented to show that a connection exists between the Esscher-Girsanov transform and the Wang transform, which is called Wang transform.
Abstract: An elementary proof is presented to show that a connection exists between the Esscher–Girsanov transform and the Wang transform.
TL;DR: It is proved that the new detector can be generalized to the integration of the n th-power modulus of the fractional Fourier transform via mathematical derivation and computer simulation results have confirmed the effectiveness of the proposed detector in LFM-signal detection.
Abstract: A new LFM-signal detector formulated by the integration of the 4th-power modulus of the fractional Fourier transform is proposed. It has similar performance to the modulus square detector of Radon-ambiguity transform because of the equivalence relationship between them. But the new detector has much lower computational complexity in the case that the number of the searching angles is far less than the length of the signal. Moreover, it is proved that the new detector can be generalized to the integration of the nth-power (2 ≶ n) modulus of the fractional Fourier transform via mathematical derivation. Computer simulation results have confirmed the effectiveness of the proposed detector in LFM-signal detection.
18 Jul 2010
TL;DR: New hardware architecture for implementing a Discrete Fractional Fourier Transform (DFrFT) which requires hardware complexity of O(4N), where N is transform order is proposed.
Abstract: Since decades, fractional Fourier transform has taken a considerable attention for various applications in signal and image processing domain. On the evolution of fractional Fourier transform and its discrete form, the real time computation of discrete fractional Fourier transform is essential in those applications. On this context, we have proposed new hardware architecture for implementing a Discrete Fractional Fourier Transform (DFrFT) which requires hardware complexity of O(4N), where N is transform order. This proposed architecture has been simulated and synthesized using verilogHDL, targeting a FPGA device (XLV5LX110T). The simulation results are very close to the results obtained by using MATLAB. The result shows that, this architecture can be operated on a maximum frequency of 217MHz.
TL;DR: In this paper, the relationship between the instantaneous frequencies of a signal in polar form and the phase of the corresponding Stockwell transform is analyzed using the stockwell transform and the corresponding results using a reciprocal Morlet wavelet transform are given for comparisons.
Abstract: The phase of a signal is analyzed using the Stockwell transform. In particular, the relationships between the instantaneous frequencies of a signal in polar form and the phase of the corresponding Stockwell transform are given. The corresponding results using a reciprocal Morlet wavelet transform are given for comparisons.
TL;DR: Several types of atomic orbital functions are transformed with the proposed method to illustrate its accuracy and efficiency, demonstrating its applicability for transforms of general order with high accuracy.
TL;DR: The proposed image encryption scheme based on double random amplitude coding technique by using random Hartley transform, which is defined according to the random Fourier transform has enhanced security and the correct information of original image can be well protected under bare decryption, blind decryption and brute force attacks.
TL;DR: In this article, the Fourier-Bessel transform was used to express the Clifford-Hermite transform in terms of Fourier Bessel transform, leading to a closed form of the Clifford Fourier integral kernel.
TL;DR: This paper proposes a newer version of Fourier transform, namely, Distributed Multiresolution Discrete Fourier Transform (D-MR-DFT) and its application in digital watermarking and shows better visual imperceptibility and resiliency of the proposed scheme against intentional or unintentional variety of attacks.
Abstract: The Fourier transform is undoubtedly one of the most valuable and frequently used tools in signal processing and analysis but it has some limitations. In this paper, we rectify these limitations by proposing a newer version of Fourier transform, namely, Distributed Multiresolution Discrete Fourier Transform (D-MR-DFT) and its application in digital watermarking. The core idea of the proposed watermarking scheme is to decompose an image into four frequency sub-bands using D-MR-DFT and then singular values of every sub-band are modified with the singular values of the watermark. The experimental results show better visual imperceptibility and resiliency of the proposed scheme against intentional or unintentional variety of attacks.
TL;DR: It is shown that NCHT is a natural-ordered version of complex Hadamard transform whereas SCHT shows the sequency ordering, which is parallel to their real-valued counterparts, the WHT and thesequency-ordered Walsh transform (SOWT).
TL;DR: This transform was published earlier by Artyom Grigoryan in 1984 and 1986 in the USSR and was known as the tensor, or vector transform and was introduced as an orthogonal 2-D discrete transform which was completely defined by the direction binary functions.
Abstract: In the above paper, the new discrete periodic Radon transform was proposed. We would like to prove that this transform was published earlier by Artyom Grigoryan in 1984 and 1986 in the USSR and was known as the tensor, or vector transform. The applications of the tensor and more advanced, paired transforms for image reconstruction of discrete images were also described by Artyom and Merughan Grigoryan in many publications from 1986 through 1990 and later. The paired transform was introduced as an orthogonal 2-D discrete transform which was completely defined by the direction binary functions. The paired transform showed a way of collecting the discrete linear integrals (or projection data) to compose the unitary 2-D transform and reconstruct the image.