Showing papers on "Hartley transform published in 2012"
TL;DR: The system parameters of Arnold transform and gyrator transform in each channel serve as additional keys in color image encryption and decryption, and hence enhances the security of the system.
97 citations
Book•
08 Mar 2012TL;DR: In this paper, a systematic treatment of orthogonal transform methods for signal processing, data analysis and communications is presented, with a discussion of how each transform method can be applied to real-world experimental problems.
Abstract: A systematic, unified treatment of orthogonal transform methods for signal processing, data analysis and communications, this book guides the reader from mathematical theory to problem solving in practice. It examines each transform method in depth, emphasizing the common mathematical principles and essential properties of each method in terms of signal decorrelation and energy compaction. The different forms of Fourier transform, as well as the Laplace, Z-, Walsh–Hadamard, Slant, Haar, Karhunen–Loeve and wavelet transforms, are all covered, with discussion of how each transform method can be applied to real-world experimental problems. Numerous practical examples and end-of-chapter problems, supported by online Matlab and C code and an instructor-only solutions manual, make this an ideal resource for students and practitioners alike.
94 citations
TL;DR: This paper investigates the Wigner-Ville distribution in the linear canonical transform domain and proposes the applications of the newly defined Wigne-Villes transform in thelinear-frequency-modulated signal detection.
Abstract: The linear canonical transform is shown to be one of the most powerful tools for nonstationary signal processing. Based on the properties of the linear canonical transform and the classical Wigner-Ville transform, this paper investigates the Wigner-Ville distribution in the linear canonical transform domain. Firstly, unlike the classical Wigner-Ville transform, a new definition of Wigner-Ville distribution associated with the linear canonical transform is given. Then, the main properties of the newly defined Wigner-Ville transform are investigated in detail. Finally, the applications of the newly defined Wigner-Ville transform in the linear-frequency-modulated signal detection are proposed, and the simulation results are also given to verify the derived
theory.
75 citations
TL;DR: The windowed Fourier transform is generalized to the windowed linear canonical transform by substituting the Fouriertransform kernel with the linear canonicaltransform kernel in the windowing Fourier Transform definition.
73 citations
TL;DR: The convolution theorem in FT domain is shown to be a special case of the achieved results, and the correlation theorem is derived, which is also a one dimensional integral expression.
Abstract: As a generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) plays an important role in many fields of optics and signal processing. Many properties for this transform are already known, but the correlation theorem, similar to the version of the Fourier transform (FT), is still to be determined. In this paper, firstly, we introduce a new convolution structure for the LCT, which is expressed by a one dimensional integral and easy to implement in filter design. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, based on the new convolution structure, the correlation theorem is derived, which is also a one dimensional integral expression. Last, as an application, utilizing the new convolution theorem, we investigate the sampling theorem for the band limited signal in the LCT domain. In particular, the formulas of uniform sampling and low pass reconstruction are obtained.
72 citations
Journal Article•
TL;DR: In this paper, the Natural transform is derived from the Fourier Integral and it converges to Laplace and Sumudu transform, it is shown it to the theoretical dual of Laplace, and it is proved Natural-multiple shift theorems, Bromwich contour integral and Heviside's Expansion formula for Inverse Natural transform.
Abstract: Nature of the Natural transform is, it converges to Laplace and Sumudu transform. The theme of this paper is to give detailed study of Natural transform. The Natural transform is derived from the Fourier Integral. We showed it to the theoretical dual of Laplace and Sumudu transforms. This work includes Natural-multiple shift theorems, Bromwich contour integral and Heviside's Expansion formula for Inverse Natural transform.
66 citations
TL;DR: A multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball is introduced, which achieves floating-point precision.
Abstract: We develop an exact wavelet transform on the three-dimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial half-line using damped Laguerre polynomials and develop a corresponding quadrature rule. Combined with the spherical harmonic transform, this approach leads to a sampling theorem on the ball and a novel three-dimensional decomposition which we call the Fourier-Laguerre transform. We relate this new transform to the well-known Fourier-Bessel decomposition and show that band-limitedness in the Fourier-Laguerre basis is a sufficient condition to compute the Fourier-Bessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the Fourier-Laguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels are well localised in real and Fourier-Laguerre spaces and their angular aperture is invariant under radial translation. We introduce a multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball. Our implementation of these new tools achieves floating-point precision and is made publicly available. We perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example.
65 citations
TL;DR: A new approach for image encryption based on the real-valuedness of the reality-preserving multiple-parameter fractional Fourier transform and the decorrelation property of chaotic maps is proposed in order to meet the requirements of the secure image transmission.
62 citations
TL;DR: A double image encryption method is proposed by utilizing discrete multiple-parameter fractional Fourier transform and chaotic maps to encrypt two original images scrambled by one chaotic map with one original image as its phase.
61 citations
TL;DR: In this article, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2).
Abstract: In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these transforms is given, with emphasis on aspects such as eigenfunctions and spectrum of the transform, characterization of the integral kernel and connection with various special functions.
06 Nov 2012
TL;DR: The literature review of the Natural transform and existing definitions and connections to the Laplace and Sumudu transforms are discussed in this article, where the relation of Bessel's function to Natural transform is defined.
Abstract: The literature review of the Natural transform and the existing definitions and connections to the Laplace and Sumudu transforms are discussed in this communication. Along with the complex inverse Natural transform and Heaviside's expansion formula, the relation of Bessel's function to Natural transform (and hence Laplace and Sumudu transforms) are defined.
TL;DR: In this paper, the authors present techniques for implementing a fast algorithm for implementing odd-type DCTs and DSTs, which include a mapping between the real-valued data sequence to an intermediate sequence to be used as an input to a DFT.
Abstract: This disclosure presents techniques for implementing a fast algorithm for implementing odd-type DCTs and DSTs. The techniques include the computation of an odd-type transform on any real-valued sequence of data (e.g., residual values in a video coding process or a block of pixel values of an image coding process) by mapping the odd-type transform to a discrete Fourier transform (DFT). The techniques include a mapping between the real-valued data sequence to an intermediate sequence to be used as an input to a DFT. Using this intermediate sequence, an odd-type transform may be achieved by calculating a DFT of odd size. Fast algorithms for a DFT may be then be used, and as such, the odd-type transform may be calculated in a fast manner
01 Jan 2012
TL;DR: In this correspondence paper, biometrics is chosen as the primary application; and hence, a new technique is proposed for securing fingerprints during communication and transmission over insecure channel, i.e., fractional random wavelet transform (FrRnWT).
Abstract: In this correspondence paper, the wavelet transform, which is an important tool in signal and image processing, has been generalized by coalescing wavelet transform and fractional random transform. The new transform, i.e., fractional random wavelet transform (FrRnWT) inherits the excellent mathematical properties of wavelet transform and fractional random transform. Possible applications of the proposed transform are in biometrics, image compression, image transmission, transient signal processing, etc. In this correspondence paper, biometrics is chosen as the primary application; and hence, a new technique is proposed for securing fingerprints during communication and transmission over insecure channel.
TL;DR: In this article, a new representation of the Schrodinger operator of a free particle by using the short-time Fourier transform was proposed, and its applications were described. But this representation is restricted to the case of a single particle.
Abstract: We propose a new representation of the Schrodinger operator of a free particle by using the short-time Fourier transform and give its applications.
TL;DR: An efficient algorithm for finding the number of conjugate points to every point of the fundamental domain of F^{e}_M, together with its continuous interpolation, is described in full generality.
Abstract: Ten types of discrete Fourier transforms of Weyl orbit functions are developed. Generalizing one-dimensional cosine, sine and exponential, each type of the Weyl orbit function represents an exponential symmetrized with respect to a subgroup of the Weyl group. Fundamental domains of even affine and dual even affine Weyl groups, governing the argument and label symmetries of the even orbit functions, are determined. The discrete orthogonality relations are formulated on finite sets of points from the refinements of the dual weight lattices. Explicit counting formulas for the number of points of the discrete transforms are deduced. Real-valued Hartley orbit functions are introduced and all ten types of the corresponding discrete Hartley transforms are detailed.
TL;DR: In this paper, biometric is chosen as the primary application and hence a new technique is proposed for securing biometrics during communication and transmission over insecure channel, i.e. the fractional dual tree complex wavelet transform (FrDT-CWT).
Journal Article•
TL;DR: In this paper, the authors proposed a one-shot sparse Fourier transform (SFT) algorithm, which estimates the k largest coefficients in a single shot, in a manner akin to sketching/streaming algorithms.
Abstract: We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory.We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and Dolph-Chebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating "large" coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in "one shot", in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice.
TL;DR: In this article, the authors discuss Hartley-Hilbert transform and Fourier transform on generalized functions and show that they are well-defined, one-to-one, linear and continuous mappings with respect to δ and Δ convergence.
Abstract: In this paper, we discuss Hartley-Hilbert and Fourier-Hilbert transforms on a certain class of generalized functions. The extended transforms considered in this article are shown to be well-defined, one-to-one, linear and continuous mappings with respect to δ and Δ convergence. Certain theorems are also established.
25 Apr 2012
TL;DR: H Hilbert transform can be interpreted as a way to represent a narrow-band signal in terms of amplitude and frequency modulation and is useful for diverse purposes such as latency analysis in neuro-physiological signals.
Abstract: When x(t) is narrow-banded, |z(t)| can be regarded as a slow-varying envelope of x(t) while the phase derivative ∂t[tan −1(y/x)] is an instantaneous frequency. Thus, Hilbert transform can be interpreted as a way to represent a narrow-band signal in terms of amplitude and frequency modulation. The transform is therefore useful for diverse purposes such as latency analysis in neuro-physiological signals (Recio-Spinoso et al., 2011; van Drongelen, 2007), design of bizarre stimuli for psychoacoustic experiments (Smith et al., 2002), speech data compression for communication (Potamianos & Maragos, 1994), regularization of convergence problems in multi-channel acoustic echo cancellation (Liu & Smith, 2002), and signal processing for auditory prostheses (Nie et al., 2006).
TL;DR: This paper considers the inverse source problem for time-harmonic acoustic or electromagnetic wave propagation in the two-dimensional free space and sets up a filtered backprojection algorithm to recover information on the unknown source term from a single far field measurement.
Abstract: We consider the inverse source problem for time-harmonic acoustic or electromagnetic wave propagation in the two-dimensional free space Given the radiated far field pattern of the solution to the Helmholtz equation for a certain source term, we find that the windowed Fourier transform of this far field is related to an exponential Radon transform with purely imaginary exponent of a smoothed approximation of the source Based on this observation we set up a filtered backprojection algorithm to recover information on the unknown source term from a single far field measurement We analyze this algorithm and provide extensive numerical results that illustrate our theoretical findings As one outcome the method is shown to work better the larger the wave number Possible extensions of the reconstruction method to limited aperture data and to inverse obstacle scattering problems are briefly sketched
TL;DR: In this article, the concept of fractional S transform (FRST) of a signal, based on the idea of the fractional Fourier transform and S transform, was defined, and the S transform was extended to the time-fractional frequency domain from the timefrequency domain to obtain the inverse transform.
Abstract: The S transform, which is a time-frequency representation known for its local spectral phase properties in signal processing, uniquely combines elements of wavelet transforms and the short-time Fourier transform (STFT). The fractional Fourier transform is a tool for non-stationary signal analysis. In this paper, we define the concept of the fractional S transform (FRST) of a signal, based on the idea of the fractional Fourier transform (FRFT) and S transform (ST), extend the S transform to the time-fractional frequency domain from the time-frequency domain to obtain the inverse transform, and study the FRST mathematical properties. The FRST, which has the advantages of FRFT and ST, can enhance the ST flexibility to process signals. Compared to the S transform, the FRST can effectively improve the signal time-frequency resolution capacity. Simulation results show that the proposed method is effective.
TL;DR: In this paper, a vector-valued version of the classical continuous wavelet transform is investigated, where the analyzing vector consists of the first elements of the basis of admissible functions, namely the functions whose Fourier transform is a Laguerre function.
Abstract: We investigate a vector-valued version of the classical continuous wavelet transform. Special attention is given to the case when the analyzing vector consists of the first elements of the basis of admissible functions, namely the functions whose Fourier transform is a Laguerre function. In this case, the resulting spaces are, up to a multiplier isomorphism, poly-Bergman spaces. To demonstrate this fact, we introduce a new map and call it the polyanalytic Bergman transform. Our method of proof uses Vasilevski’s restriction principle for Bergman-type spaces. The construction is based on the idea of multiplexing of signals.
TL;DR: In this article, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters, and a series expansion for the kernel of the resulting integral transform is derived.
Abstract: In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension, also an explicit expression for the kernel in terms of Bessel functions is obtained. Finally, the analytic properties of this new integral transform are studied in detail.
16 Jun 2012
TL;DR: This work exploits the linearity of the Schrödinger equation to design fast discrete convolution methods using the FFT to compute the distance transform, derive the histogram of oriented gradients (HOG) via the squared magnitude of the Fourier transform of the wave function.
Abstract: Despite the ubiquitous use of distance transforms in the shape analysis literature and the popularity of fast marching and fast sweeping methods — essentially Hamilton-Jacobi solvers, there is very little recent work leveraging the Hamilton-Jacobi to Schrodinger connection for representational and computational purposes. In this work, we exploit the linearity of the Schrodinger equation to (i) design fast discrete convolution methods using the FFT to compute the distance transform, (ii) derive the histogram of oriented gradients (HOG) via the squared magnitude of the Fourier transform of the wave function, (iii) extend the Schrodinger formalism to cover the case of curves parametrized as line segments as opposed to point-sets, (iv) demonstrate that the Schrodinger formalism permits the addition of wave functions — an operation that is not allowed for distance transforms, and finally (v) construct a fundamentally new Schrodinger equation and show that it can represent both the distance transform and its gradient density — not possible in earlier efforts.
02 Apr 2012
TL;DR: In this article, the Yang-Fourier transform (YFT) in fractal signals was investigated in detail, and both properties and theorems of the discrete YFT were investigated.
Abstract: The Yang-Fourier transform (YFT) in fractal space is a generation of Fourier transform based on the local fractional calculus. The discrete Yang-Fourier transform (DYFT) is a specific kind of the approximation of discrete transform, used in Yang- Fourier transform in fractal space. This paper points out new standard forms of discrete Yang-Fourier transforms (DYFT) of fractal signals, and both properties and theorems are investigated in detail. Keywords -Fractal, Signal, Discrete, Yang-Fourier transforms
TL;DR: The proposed algorithm can minimize the numerical error arising from interpolation operations and requires fewer chirp multiplications and the required computation time is also less.
Abstract: The two-dimensional nonseparable linear canonical transform (2D NSLCT), which is a generalization of the fractional Fourier transform and the linear canonical transform, is useful for analyzing optical systems. However, since the 2D NSLCT has 16 parameters and is very complicated, it is a great challenge to implement it in an efficient way. In this paper, we improved the previous work and propose an efficient way to implement the 2D NSLCT. The proposed algorithm can minimize the numerical error arising from interpolation operations and requires fewer chirp multiplications. The simulation results show that, compared with the existing algorithm, the proposed algorithms can implement the 2D NSLCT more accurately and the required computation time is also less.
TL;DR: In this paper, the authors define the fractional Fourier transform over finite fields (GFrFT) and present a modified version of the finite field Fourier Transform (FFT) for trigonometry in finite fields.