Showing papers on "Hartley transform published in 2002"
Book•
06 Nov 2002
TL;DR: The Breadth and Depth of DSP Statistics, Probability and Noise ADC and DAC DSP Software Linear Systems Convolution Properties of Convolution The Discrete Fourier Transform Applications of the DFT Fourier transform Properties Fouriertransform Pairs The Fast Fouriers Transform Continuous Signal Processing Introduction to Digital Filters
Abstract: The Breadth and Depth of DSP Statistics, Probability and Noise ADC and DAC DSP Software Linear Systems Convolution Properties of Convolution The Discrete Fourier Transform Applications of the DFT Fourier Transform Properties Fourier Transform Pairs The Fast Fourier Transform Continuous Signal Processing Introduction to Digital Filters Moving Average Filters Windowed-Sinc Filters Custom Filters FFT Convolution Recursive Filters Chebyshev Filters Filter Comparison Audio Processing Image Formation and Display Linear Image Processing Special Imaging Techniques Neural Networks (and more!) Data Compression Digital Signal Processors Getting Started with DSPs Complex Numbers The Complex Fourier Transform The Laplace Transform The z-Transform Index
594 citations
TL;DR: A new architecture is proposed that encodes a primary image to white noise based on iterative fractional Fourier transform that can provide additional keys for encryption to make the code more difficult to break.
174 citations
TL;DR: This paper proposes a fast approximate algorithm for the associated Legendre transform by means of polynomial interpolation accelerated by the Fast Multipole Method (FMM), and shows that the algorithm is stable and is faster than the direct computation for N ≥ 511.
Abstract: The spectral method with discrete spherical harmonics transform plays an important role in many applications. In spite of its advantages, the spherical harmonics transform has a drawback of high computational complexity, which is determined by that of the associated Legendre transform, and the direct computation requires time of O(N3) for cut-off frequency N. In this paper, we propose a fast approximate algorithm for the associated Legendre transform. Our algorithm evaluates the transform by means of polynomial interpolation accelerated by the Fast Multipole Method (FMM). The divide-and-conquer approach with split Legendre functions gives computational complexity O(N2 log N). Experimental results show that our algorithm is stable and is faster than the direct computation for N ≥ 511.
101 citations
TL;DR: The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects.
Abstract: The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed out.
84 citations
TL;DR: This paper derives several new transforms that are the generalization of the cosine, sine, or Hartley transform and shows that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided signal processing.
Abstract: In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. We introduce several new transforms. They are all the generalization of the cosine, sine, or Hartley transform. We first derive the fractional cosine, sine, and Hartley transforms (FRCT/FRST/FRHT). They are analogous to the FRFT. Then, we derive the canonical cosine and sine transforms (CCT/CST). They are analogous to the LCT. We also derive the simplified fractional cosine, sine, and Hartley transforms (SFRCT/SFRST/SFRHT). They are analogous to the SFRFT and have the advantage of real-input-real-output. We also discuss the properties, digital implementation, and applications (e.g., the applications for filter design and space-variant pattern recognition) of these transforms. The transforms introduced in this paper are very efficient for digital implementation. We can just use one half or one fourth of the real multiplications required for the FRFT and LCT to implement them. When we want to process even, odd, or pure real/imaginary functions, we can use these transforms instead of the FRFT and LCT. Besides, we also show that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided (t /spl isin/ [0, /spl infin/]) signal processing.
73 citations
17 Jun 2002
55 citations
TL;DR: A flexible system for time-frequency signal analysis based on the S-method, which has a significant advantage in implementation since it can involve, as a key intermediate step, the Short-time Fourier transform or the Hartley transform, each widely studied and commonly used in practice.
Abstract: A flexible system for time-frequency signal analysis is presented. It is based on the S-method, which has a significant advantage in implementation since it can involve, as a key intermediate step, the Short-time Fourier transform or the Hartley transform, each widely studied and commonly used in practice. Signal invariant and signal dependent system forms are presented. Hardware design, for a fixed-point arithmetic, is well-structured and suitable for vlsi implementation. The same hardware, without additional time requirements, may be shared for the simultaneous realization of the fourth order L-Wigner distribution, as well as for the realization of the cross-terms free fourth order polynomial Wigner-Ville distribution. This possibility makes the designed hardware suitable for wide range of the applications. The proposed hardware is applied to the realization of time-varying filtering, as well. Finally, it has been implemented with fpga chips (Field Programmable Gate Array) in order to verify the results on real devices.
47 citations
Patent•
18 Oct 2002TL;DR: In this article, a method for coding in frequency, module and phase a digital representation, in the space field, of a ring-shaped element, including the steps of: applying to any point of the element a polar conversion at constant angle, whereby the element is unfolded in rectangular form; transferring, to the frequency field, any points of the converted rectangular shape by means of a Fourier transform; filtering the discrete data resulting from the transfer by at least one real, bidimensional, band-pass filter, oriented along the phase axis; applying a Hilbert transform to the filtering results
Abstract: A method for coding in frequency, module and phase a digital representation, in the space field, of a ring-shaped element, including the steps of: applying to any point of the element a polar conversion at constant angle, whereby the element is unfolded in rectangular form; transferring, to the frequency field, any point of the converted rectangular shape by means of a Fourier transform; filtering the discrete data resulting from the transfer by means of at least one real, bidimensional, band-pass filter, oriented along the phase axis; applying a Hilbert transform to the filtering results; applying an inverse Fourier transform to the results of the Hilbert transform; and extracting phase and module information in the space field.
39 citations
26 citations
TL;DR: An integral transform involving a Fox's H-function is introduced in this article, which is closely related to a multi-index analogue of the classical Mittag-Leffler function.
Abstract: An integral transform involving a Fox's H-function is introduced. This integral transform is closely related to a multi-index analogue of the classical Mittag-Leffler function. Along with the basic operational and mapping properties of this transform, the new results presented here include complex and real inversion formulas and a convolution theorem.
25 citations
TL;DR: In this paper, a Fourier transform based algorithm for the reconstruction of functions from their nonstandard sampled Radon transform is proposed, which incorporates recently developed fast Fourier transforms for nonequispaced data.
Abstract: In this paper, we suggest a new Fourier transform based algorithm forthe reconstruction of functions from their nonstandard sampled Radon transform. The algorithm incorporates recently developed fast Fourier transforms for nonequispaced data. We estimate the-corresponding aliasing error in dependence on the sampling geometry of the Radon transform and confirm our theoretical results by numerical examples.
TL;DR: The proposed FHTn method is an efficient approach for numerical evaluation of an arbitrary integer order of the Hankel transform (HT) and is applied to the analysis of cylindrical EM field propagation through a diffractive microlens.
Abstract: We present a fast Hankel transform (FHTn) method for direct numerical evaluation of electromagnetic (EM) field propagation through an axially symmetric system. Comparing with the vector-based plane-wave spectrum (VPWS) method, we present an alternative approach to implement the fast Hankel transform which does not require an additional coordinate transformation for Fourier transform. The proposed FHTn method is an efficient approach for numerical evaluation of an arbitrary integer order of the Hankel transform (HT). As an example to demonstrate the effectiveness of the proposed method, we apply the FHTn technique to the analysis of cylindrical EM field propagation through a diffractive microlens.
TL;DR: The Hartley transform as mentioned in this paper is an integral transform similar to the Fourier transform and has most of the characteristics of the FFT, but it has better properties and a faster algorithm than FFT.
Abstract: The measurement principle of particle image velocimetry (PIV) is based on cross-correlation analysis of flow images. By using the cross-correlation property and fast algorithm of the Fourier transform, the analysis in PIV can be implemented easily and quickly. The Hartley transform is an integral transform similar to the Fourier transform and has most of the characteristics of the Fourier transform. The Hartley transform also has some better properties and a faster algorithm than the Fourier transform. The cross-correlation property of the Hartley transform based on separable kernels is presented in detail and the application to PIV analysis is introduced. The advantage of the Hartley transform can be shown from the comparison of numbers of operations in theory and computation time in practice.
TL;DR: A signal-dependent wavelet transform based on the lifting scheme is proposed and results indicate that the proposed method is superior to the S+P method.
Abstract: A signal-dependent wavelet transform based on the lifting scheme is proposed. The transform can be made reversible (i.e. an integer-to-integer transform). The reversible transform, followed by arithmetic coding, is applied to lossless image compression. Simulation results indicate that the proposed method is superior to the S+P method.
TL;DR: The ath-order fractional Fourier transform as mentioned in this paper is a generalization of the ordinary Fourier Transform (OFT) and it can be seen as a form of commontransformer.
Abstract: The ath-order fractional Fourier transform is a generalization ofthe ordinary Fourier transform such that the zeroth-order fractionalFourier transform operation is equal to the identity operation and thefirst-order fractional Fourier transform is equal to the ordinaryFourier transform. This paper discusses the relationship of thefractional Fourier transform to harmonic oscillation; both correspondto rotation in phase space. Various important properties of thetransform are discussed along with examples of commontransforms. Some of the applications of the transform are brieflyreviewed.
TL;DR: New fast algorithms for multidimensional discrete Hartley transform (MD-DHT) are presented, based on the index mapping and multiddimensional polynomial transform (PT), which achieves considerable savings on the number of operations.
Patent•
01 Nov 2002TL;DR: In this article, a set of probabilities is estimated for values of a hidden variable and a Fourier transform is determined for the set of probability values and then the inverse Fourier transformation is used to form an estimated prototype pattern.
Abstract: The present invention provides a method of constructing recognition models. Under the method, a set of probabilities is estimated for values of a hidden variable. A Fourier transform is determined for the set of probabilities and is used to determine a Fourier transform of an estimated prototype pattern. The inverse Fourier transform is then determined for the Fourier transform of the estimated prototype pattern to form an estimated prototype pattern.
TL;DR: A three- dimensional algorithm for fast computation of the three-dimensional discrete Hartley transform is developed and it is found that this algorithm offers substantial savings in both the number of multiplications and additions.
TL;DR: The purpose of this letter is to compare the recognition e5ectiveness of the proposed feature representation approach and of the Hartley transform approach for Gaussian signal recognition.
TL;DR: A new and more efficient algorithm is presented that acts directly on a real signal via the real Zak transform (RZT) relation between a signal and its Hartley transform, leading, in effect, to approximately a four-fold reduction in the computational complexity of the complex Zak space approach.
Abstract: A new formulation of the Gerchberg-Papoulis (1974, 1975) algorithm for extrapolation of bandlimited signals was introduced. The new formulation was obtained by translating the fundamental operations of the GP procedure, the truncation, and the Fourier transform into the language of the finite Zak (1967) transform. However, the Zak transform formulation of the GP algorithm assumes complex-valued signals, whereas the GP procedure is usually applied to real signals. We present a new and more efficient algorithm that acts directly on a real signal via the real Zak transform (RZT) relation between a signal and its Hartley transform, leading, in effect, to approximately a four-fold reduction in the computational complexity of the complex Zak space approach.
04 Aug 2002
TL;DR: This paper reviews the available linear transforms and derives three new transforms, namely the centered step invariant transform, the local cubic spline invariant Transform, and the scaled impulse invariants transform.
Abstract: In the modeling of continuous time systems, there is a need to convert the analog system to a discrete time system The transform frequently encountered in the literature is the impulse invariant transform This transform has proven to be inadequate for biological system modeling In this paper, we review the available linear transforms and derive three new transforms, namely the centered step invariant transform, the local cubic spline invariant transform, and the scaled impulse invariant transform
TL;DR: The subspaces of FMmlet transform are investigated and it is shown that some of the existing transforms like the Fourier transform, short-time Fouriertransform, Gabor transform, wavelet transform, chirplet transform, the mean of signal, and the FM−1let transform and the butterfly subspace are all special cases of FM mlet transform.
Abstract: The subspaces of FMmlet transform are investigated. It is shown that some of the existing transforms like the Fourier transform, short-time Fourier transform, Gabor transform, wavelet transform, chirplet transform, the mean of signal, and the FM−1let transform, and the butterfly subspace are all special cases of FMmlet transform. Therefore the FMmlet transform is more flexible for delineating both the linear and nonlinear time-varying structures of a signal.
TL;DR: The Fourier transform and Radon transform are both simple and effective tools for solving this problem, which extract the information in spectral domain and spatial domain, respectively, respectively as discussed by the authors.
Abstract: Orientation and spacing are important features of fringe patterns, especially in speckle photography when the displacement information of the object is determined by pointwisely filtering the double-exposure specklegram The Fourier transform and Radon transform are both simple and effective tools for solving this problem, which extract the information in spectral domain and spatial domain, respectively A hybrid method combining Radon transform and Fourier transform is also possible The results can be further improved by enhanced Fourier transform and enhanced Radon transform The theories, as well as real applications, are given in this paper
13 May 2002
TL;DR: A unified theory for arithmetic transform of a variety of discrete trigonometric transforms is proposed and it is shown that the interpolation method determines the transform to be computed.
Abstract: In this paper, we propose a unified theory for arithmetic transform of a variety of discrete trigonometric transforms. The main contribution of this work is the elucidation of the interpolation process required in arithmetic transforms. We show that the interpolation method determines the transform to be computed. Several kernels were examined and asymptotic interpolation formulae were derived. Using the arithmetic transform theory, we also introduce a new algorithm for computing the discrete Hartley transform.
TL;DR: Fast algorithms for a wide class of nonseparable n-dimensional (n-D) discrete unitary /spl Kscr/ transforms (DKTs) are introduced and lead to decrease of multiplicative complexity by the factor of n, compared with the classical row/column separable approach.
Abstract: Fast algorithms for a wide class of nonseparable n-dimensional (n-D) discrete unitary /spl Kscr/ transforms (DKTs) are introduced. They need fewer 1-D DKTs than in the case of the classical radix-2 FFT-type approach. The method utilizes a decomposition of the n-D K transform into the product of a new n-D discrete Radon transform and of a set of parallel/independ 1-D K transforms. If the n-D K transform has a separable kernel (e.g., the case of the discrete Fourier transform), our approach leads to decrease of multiplicative complexity by the factor of n, compared with the classical row/column separable approach.
07 Aug 2002
TL;DR: The theory of the unified discrete Fourier-Hartley transform is developed and the GDFHT is generalized to the case of non-constant coefficients and can be used to compute the modified discrete cosine transform for both non-window and window modes.
Abstract: In this paper, the theory of the unified discrete Fourier-Hartley transform (UDFHT) is developed. The UDFHT includes the discrete Fourier and Hartley transforms of types I-IV as special cases, and with little modification, it also includes the discrete cosine and sine transforms of types II-IV. A unified efficient structure using fast Fourier transform is proposed. The UDFHT is then generalized (GDFHT) to the case of non-constant coefficients. Sufficient conditions for orthogonality of the transform are presented. The GDFHT can be used to compute the modified discrete cosine transform for both non-window and window modes as illustrated in the paper.
Posted Content•
TL;DR: In this article, the standard proof of the Springer correspondence in positive characteristic (via Deligne-Fourier transform) works verbatim in characteristic zero, up to replacing Deligne Fourier transform by another etale Fourier Transform introduced by the author in a previous work.
Abstract: We show that the standard proof of the Springer correspondence in positive characteristic (via Deligne-Fourier transform) works verbatim in characteristic zero, up to replacing Deligne-Fourier transform by another etale Fourier transform introduced by the author in a previous work. The construction of this Fourier transform uses methods from p-adic analytic geometry.
TL;DR: In this article, the linearity, time shifting, time scaling, and time inversion properties of FMmlet transform are proved, and the frequency shifting property of one of the subspaces of the chirplet transform is presented.
Abstract: The linearity, time shifting, time scaling, and time inversion properties of FMmlet transform are proved, and the frequency shifting property of one of the subspaces of FMmlet transform, namely the chirplet transform is presented. Moreover, it is proved that in the process of FMm let based atomic signal decomposition, the residual signals decay exponentially.