Showing papers on "Fractional Fourier transform published in 1999"
23 Mar 1999
TL;DR: This paper proposes to use Haar Wavelet Transform for time series indexing and shows that Haar transform can outperform DFT through experiments, and proposes a two-phase method for efficient n-nearest neighbor query in time series databases.
Abstract: Time series stored as feature vectors can be indexed by multidimensional index trees like R-Trees for fast retrieval. Due to the dimensionality curse problem, transformations are applied to time series to reduce the number of dimensions of the feature vectors. Different transformations like Discrete Fourier Transform (DFT) Discrete Wavelet Transform (DWT), Karhunen-Loeve (KL) transform or Singular Value Decomposition (SVD) can be applied. While the use of DFT and K-L transform or SVD have been studied on the literature, to our knowledge, there is no in-depth study on the application of DWT. In this paper we propose to use Haar Wavelet Transform for time series indexing. The major contributions are: (1) we show that Euclidean distance is preserved in the Haar transformed domain and no false dismissal will occur, (2) we show that Haar transform can outperform DFT through experiments, (3) a new similarity model is suggested to accommodate vertical shift of time series, and (4) a two-phase method is proposed for efficient n-nearest neighbor query in time series databases.
1,160 citations
TL;DR: The proposed DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT and will provide similar transform and rotational properties as those of continuous fractional Fourier transforms.
Abstract: The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.
291 citations
TL;DR: A fast algorithm based on the Fractional Fourier transform allow accurate evaluation of the Fresnel integral from object to Fraunhofer domain in a single step.
251 citations
235 citations
TL;DR: It is observed that the chirplet decomposition and the related TFD provide more compact and precise representation of signal inner structures compared with the commonly used time-frequency representations.
Abstract: A new four-parameter atomic decomposition of chirplets is developed for compact and precise representation of signals with chirp components. The four-parameter chirplet atom is obtained from the unit Gaussian function by successive applications of scaling, fractional Fourier transform (FRFT), and time-shift and frequency-shift operators. The application of the FRFT operator results in a rotation of the Wigner distribution of the Gaussian in the time-frequency plane by a specified angle. The decomposition is realized by using the matching pursuit algorithm. For this purpose, the four-parameter space is discretized to obtain a small but complete subset in the Hilbert space. A time-frequency distribution (TFD) is developed for clear and readable visualization of the signal components. It is observed that the chirplet decomposition and the related TFD provide more compact and precise representation of signal inner structures compared with the commonly used time-frequency representations.
222 citations
15 Mar 1999
TL;DR: This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions and supports confidence that it will be accepted as the definitive definition of this transform.
Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.
210 citations
TL;DR: The generalised S transform is described, a variant of the wavelet transform which allows calculation of the instantaneous phase of a signal, and its application to the decomposition of vibration signals from mechanical systems such as gearboxes for the early detection of failure.
177 citations
TL;DR: This work applies the language of the unified FT to develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity.
Abstract: The fractional Fourier transform (FRT) is an extension of the ordinary Fourier transform (FT). Applying the language of the unified FT, we develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity. The FRT sampling theorem is derived as an extension of its ordinary counterpart.
152 citations
TL;DR: The fractional Fourier transform (FFT) as discussed by the authors is a generalization of the ordinary FFT with an order parameter a, and it is used to interpolate between a function f(u) and its FFT F(μ).
Abstract: Publisher Summary This chapter is an introduction to the fractional Fourier transform and its applications. The fractional Fourier transform is a generalization of the ordinary Fourier transform with an order parameter a . Mathematically, the a th order fractional Fourier transform is the a th power of the Fourier transform operator. The a = 1st order fractional transform is the ordinary Fourier transform. In essence, the a th order fractional Fourier transform interpolates between a function f(u) and its Fourier transform F(μ) . The 0th order transform is simply the function itself, whereas the 1st order transform is its Fourier transform. The 0.5th transform is something in between, such that the same operation that takes us from the original function to its 0.5 th transform will take us from its 0.5th transform to its ordinary Fourier transform. More generally, index additivity is satisfied: The a 2 th transform of the a 1 th transform is equal to the ( a 2 + a 1 )th transform. The –1th transform is the inverse Fourier transform, and the – a th transform is the inverse of the a th transform.
151 citations
TL;DR: An algorithm to reconstruct a high- resolution image from multiple aliased low-resolution images, which is based on the generalized deconvolution technique, and it is shown that the artifact caused by inaccurate motion information is reduced by regular- ization.
Abstract: While high-resolution images are required for various applica- tions, aliased low-resolution images are only available due to the physi- cal limitations of sensors. We propose an algorithm to reconstruct a high- resolution image from multiple aliased low-resolution images, which is based on the generalized deconvolution technique. The conventional approaches are based on the discrete Fourier transform (DFT) since the aliasing effect is easily analyzed in the frequency domain. However, the useful solution may not be available in many cases, i.e., the underdeter- mined cases or the insufficient subpixel information cases. To compen- sate for such ill-posedness, the generalized regularization is adopted in the spatial domain. Furthermore, the usage of the discrete cosine trans- form (DCT) instead of the DFT leads to a computationally efficient recon- struction algorithm. The validity of the proposed algorithm is both theo- retically and experimentally demonstrated. It is also shown that the artifact caused by inaccurate motion information is reduced by regular- ization. © 1999 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(99)00508-5)
142 citations
TL;DR: In this article, it was shown that the stationary-phase method gives a very accurate expression for the Fourier transform of the gravitational-wave signal produced by an inspiraling compact binary.
Abstract: We prove that the oft-used stationary-phase method gives a very accurate expression for the Fourier transform of the gravitational-wave signal produced by an inspiraling compact binary. We give three arguments. First, we analytically calculate the next-order correction to the stationary-phase approximation, and show that it is small. This calculation is essentially an application of the steepest-descent method to evaluate integrals. Second, we numerically compare the stationary-phase expression to the results obtained by fast Fourier transform. We show that the differences can be fully attributed to the windowing of the time series, and that they have nothing to do with an intrinsic failure of the stationary-phase method. And third, we show that these differences are negligible for the practical application of matched filtering. @S0556-2821~99!00414-2#
TL;DR: The goal of this article is to develop two absent schemes of fractional Fourier analysis methods that are the generalizations of Fourier series and discrete-time Fourier transform (DTFT), respectively.
Abstract: Conventional Fourier analysis has many schemes for different types of signals. They are Fourier transform (FT), Fourier series (FS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). The goal of this article is to develop two absent schemes of fractional Fourier analysis methods. The proposed methods are fractional Fourier series (FRFS) and discrete-time fractional Fourier transform (DTFRFT), and they are the generalizations of Fourier series (FS) and discrete-time Fourier transform (DTFT), respectively.
TL;DR: In this paper, the Fourier transform is used for averaging lemmas in the L2 framework with the help of the Fouriers transform in variables x and v, but not t.
Abstract: We prove classical averaging lemmas in the L2 framework with the help of the Fourier transform in variables x and v, but not t. This method is then used to study discretized problems arising out of the numerical analysis of kinetic equations.
TL;DR: In this article, the authors compare the finite Fourier (-exponential) and Fourier-Kravchuk transform, which is a canonical transform whose fractionalization is well defined.
TL;DR: Two new sampling formulae for reconstructing signals that are band limited or time limited in the fractional Fourier transform sense are obtained, each taken at half the Nyquist rate.
TL;DR: The Data Encryption Standard (DES) can be regarded as a nonlinear feedback shift register (NLFSR) with input and the properties of the S-boxes of DES under the Fourier transform, Hadamard transform, extended Hadamards transform, and the Avalanche transform are investigated.
Abstract: The Data Encryption Standard (DES) can be regarded as a nonlinear feedback shift register (NLFSR) with input. From this point of view, the tools for pseudo-random sequence analysis are applied to the S-boxes in DES. The properties of the S-boxes of DES under the Fourier transform, Hadamard transform, extended Hadamard transform, and the Avalanche transform are investigated. Two important results about the S-boxes of DES are found. The first result is that nearly two-thirds of the total 32 functions from GF (2/sup 6/) to GF(2) which are associated with the eight S-boxes of DES have the maximal linear span G3, and the other one-third have linear span greater than or equal to 57. The second result is that for all S-boxes, the distances of the S-boxes approximated by monomial functions has the same distribution as for the S-boxes approximated by linear functions. Some new criteria for the design of permutation functions for use in block cipher algorithms are discussed.
TL;DR: A new convolution structure for the FRFT is introduced that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.
TL;DR: In this paper, a fitting method is described which enables one to model real line profiles intermediate between Lorentzian and Gaussian by an analytical function which has an analytical counterpart in the time domain.
TL;DR: The introduction of this new virtual instrument for time-frequency analysis may be of help to the scientists and practitioners in signal analysis.
Abstract: A virtual instrument for time-frequency analysis is presented. Its realization is based on an order recursive approach to the time-frequency signal analysis. Starting from the short time Fourier transform and using the S-method, a distribution having the auto-terms concentrated as high as in the Wigner distribution, without cross-terms, may be obtained. The same relation is used in a recursive manner to produce higher order time-frequency representations without cross-terms. Thus, the introduction of this new virtual instrument for time-frequency analysis may be of help to the scientists and practitioners in signal analysis. Application of the instrument is demonstrated on several simulated and real data examples.
KEMA1
TL;DR: In this paper, the authors considered a matrix representation of the Radon transform and formulated the corresponding discrete consistency conditions in the form of the orthogonal projection of the data vector onto the orthOGonal complement of the column space of the matrix.
Abstract: The attenuated Radon transform serves as a mathematical tool for single-photon emission computerized tomography (SPECT). The identification problem for the attenuated Radon transform is to find the attenuation coefficient, which is a parameter of the transform, from the values of the transform alone. Previous attempts to solve this problem used range theorems for the continuous attenuated/exponential Radon transform. We consider a matrix representation of the transform and formulate the corresponding discrete consistency conditions in the form of the orthogonal projection of the data vector onto the orthogonal complement of the column space of the matrix. The singular value decomposition is applied to compute the orthogonal projector and its Frechet derivative. The numerical algorithm suggested is based on the Newton method with the Tikhonov regularization. Results of numerical experiments and inversion of the measured SPECT data are considered.
TL;DR: The chirp‐z transform can reconstruct NMR images directly onto the ultimate grid instead of reconstructing onto the original grid and then applying interpolation to get the final real‐space image in the conventional way.
Abstract: A quick and accurate way to rotate and shift nuclear magnetic resonance (NMR) images using the two-dimensional chirp-z transform is presented. When the desired image grid is rotated and shifted from the original grid due to patient motion, the chirp-z transform can reconstruct NMR images directly onto the ultimate grid instead of reconstructing onto the original grid and then applying interpolation to get the final real-space image in the conventional way. The rotation angle and shift distances are embedded in the parameters of the chirp-z transform. The chirp-z transform implements discrete sinc interpolation to get values at grid points that are not exactly on the original grid when applying the inverse Fourier transform. Therefore, the chirp-z transform is more accurate than methods such as linear or bicubic interpolation and is more efficient than direct implementation of sinc interpolation because the sinc interpolation is implemented at the same time as reconstruction from k-space.
TL;DR: In this paper, the authors established an inversion formula for the Fourier transform for smooth functions on a semisimple symmetric space and showed that this transform is injective on the space C 1 c (X) of smooth functions.
Abstract: Let X be a semisimple symmetric space. In previous papers, [8] and [9], we have dened an explicit Fourier transform for X and shown that this transform is injective on the space C 1 c (X) ofcompactly supported smooth functions on X. In the present paper, which is a continuation of these papers, we establish an inversion formula for this transform.
TL;DR: A Cormack-type inversion of the exponential Radon transform is derived by employing the circular harmonic transform directly in the projection space and the image space instead of the Fourier space, which greatly mitigates the difficulty of image reconstruction due to the complicated collimator geometry.
Abstract: A variety of inversions of exponential Radon transform has been derived based on the circular harmonic transform in Fourier space by several research groups. However, these inversions cannot be directly applied to deal with the reconstruction for fan-beam or variable-focal-length fan-beam collimator geometries in single photon emission computed tomography (SPECT). In this paper, the authors derived a Cormack-type inversion of the exponential Radon transform by employing the circular harmonic transform directly in the projection space and the image space instead of the Fourier space. Thus, a unified reconstruction framework is established for parallel-, fan-, and variable-focal-length fan-beam collimator geometries. Compared to many existing algorithms, the presented one greatly mitigates the difficulty of image reconstruction due to the complicated collimator geometry and significantly reduces the computational burden of the special functions, such as Chebyshev or Bessel functions. By the well-established fast-Fourier transform (FFT), the authors' algorithm is very efficient, as demonstrated by several numerical simulations.
TL;DR: In this paper, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier Transform and by performing a novel error-removal procedure.
Abstract: This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT.
TL;DR: An efficient direct method for the computation of a length-N discrete cosine transform (DCT) given two adjacent length-(N/2) DCT coefficients, which is lower than the traditional approach for lengths N>8.
Abstract: An efficient direct method for the computation of a length-N discrete cosine transform (DCT) given two adjacent length-(N/2) DCT coefficients, is presented. The computational complexity of the proposed method is lower than the traditional approach for lengths N>8. Savings of N memory locations and 2N data transfers are also achieved.
TL;DR: In this paper, an advanced boundary element/fast Fourier transform (BE/FFT) methodology for axisymmetric acoustic wave scattering and radiation problems with non-axismmetric boundary conditions is reported.
Abstract: An advanced boundary element/fast Fourier transform (BE/FFT) methodology for solving axisymmetric acoustic wave scattering and radiation problems with non-axisymmetric boundary conditions is reported. The boundary quantities of the problem are expanded in complex Fourier series with respect to the circumferencial direction. Each of the expanding coefficients satisfies a surface integral equation which, due to axisymmetry, is reduced to a line integral along the surface generator of the body and an integral over the angle of revolution. The first integral is evaluated through Gauss quadrature by employing a two-dimensional boundary element methodology. The integration over the circumferencial direction is performed simultaneously for all the Fourier coefficients through the FFT. The singular and hyper-singular integrals are computed directly by employing highly accurate three-dimensional integration techniques. The accuracy of the proposed boundary element methodology is demonstrated by means of representative numerical examples.
TL;DR: An algorithm for decreasing the effect of film-grain noise in image compression by using the orthogonal wavelet transform and the simplicity in calculating the filter coefficients is proposed.
Abstract: An algorithm for decreasing the effect of film-grain noise in image compression is proposed. The algorithm operates in the transform domain in conjunction with quantisation. Although any orthogonal transform is suitable for this application, the orthogonal wavelet transform is preferred due to the simplicity in calculating the filter coefficients.
TL;DR: An analytical form that provides a computationally efficient algorithm for numerical evaluation of the Hankel transform of nth order by fast-Fourier-transform techniques is presented and tested with some well-known functions.
Abstract: An analytical form that provides a computationally efficient algorithm for numerical evaluation of the Hankel transform of nth order by fast-Fourier-transform techniques is presented and tested with some well-known functions.
TL;DR: In this paper, a Fourier transform is defined for the quantum double D(G) of a finite group G. The characters form a ring over the integers under both algebra multiplication and its dual, with the latter encoding the fusion rules.
Abstract: We define a Fourier transform S for the quantum double D(G) of a finite group G. Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2,). The characters form a ring over the integers under both the algebra multiplication and its dual, with the latter encoding the fusion rules of D(G). The Fourier transform relates the two ring structures. We use this to give a particularly short proof of the Verlinde formula for the fusion coefficients.