Showing papers on "Hartley transform published in 1996"

Localization of the complex spectrum: the S transform

Abstract: The S transform, which is introduced in the present correspondence, is an extension of the ideas of the continuous wavelet transform (CWT) and is based on a moving and scalable localizing Gaussian window. It is shown to have some desirable characteristics that are absent in the continuous wavelet transform. The S transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates.

Localisation of the complex spectrum : The S transform

Abstract: The S transform, an extension to the ideas of the Gabor transform and the Wavelet transform, is based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms. The S transform is fully convertible both forward and inverse from the time domain to the 2-D frequency translation (time) domain and to the familiar Fourier frequency domain. Parallel to the translation (time) axis, the S transform collapses as the Fourier transform. The amplitude frequency-time spectrum and the phase frequency-time spectrum are both useful in defining local spectral characteristics. The superior properties of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis while the localising scalable Gaussian window dilates and translates. As a result, the phase spectrum is absolute in the sense that it is always referred to the origin of the time axis, the fixed reference point. The real and imaginary spectrum can be localised independently with a resolution in time corresponding to the period of the basis functions in question. Changes in the absolute phase ofa constituent frequency can be followed along the time axis and useful information can be extracted. An analysis of a sum of two oppositely progressing chirp signals provides a spectacular example of the power of the S transform. Other examples of the applications of the Stransform to synthetic as well as real data are provided.

Digital computation of the fractional Fourier transform

Abstract: An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.

Fourier transforms of colour images using quaternion or hypercomplex, numbers

Abstract: The 2D quaternion, or hypercomplex, Fourier transform is introduced. This transform makes possible the handling of colour images in the frequency domain in a holistic manner, without separate handling of the colour components, and it thus makes possible very wide generalisation of monochrome frequency domain techniques to colour images.

The radon transform and local tomography

Abstract: Introduction Brief Description of New Results and the Aims of the Book Review of Some Applications of the Radon Transform Properties of the Radon Transform and Inversion Formulas Definitions and Properties of the Radon Transform and Related Transforms Inversion Formulas for R Singular Value Decomposition of the Radon Transform Estimates in Sobolev Spaces Inversion Formulas for the Backprojection Operator Inversion Formulas for X-Ray Transform Uniqueness Theorems for the Radon and X-Ray Transforms Attenuated and Exponential Radon Transforms Convergence Properties of the Inversion Formulas on Various Classes of Functions Range Theorems and Reconstruction Algorithms Range Functions for R on Smooth Functions Range Functions for R on Sobolev Spaces Range Theorems for R* Range Theorem for X-Ray Transform Numerical Solution of the Equation Rf = g with Noisy Data Filtered Backprojection Algorithm Other Reconstruction Algorithms Singularities of the Radon Transform Introduction Singular Support of the Radon Transform The Relation Between S and S (WE NEED A "HAT" OVER THE LAST S. See hard copy of toc for details) The Envelopes and the Duality Law Asymptotics of Rf Near S Singularities of the Radon Transform: An Alternative Approach Asymptotics of the Fourier Transform Wave Front Sets Singularities of X-Ray Transform Stable Calculation of the Legendre Transform Local Tomography Introduction A Family of Local Tomography Functions Optimization of Noise Stability Algorithm for Finding Values of Jumps of a Function Using Local Tomography Numerical Implementation Local Tomography for the Exponential Radon Transform Local Tomography for the Generalized Radon Transform Local Tomography for the Limited-Angle Data Asymptotics of Pseudodifferential Operators, Acting on a Piecewise-Smooth Function, f, Near the Singular Support of f Pseudolocal Tomography Introduction Definition of a Pseudolocal Tomography Function Investigation of the Convergence frc(x) Ae f(x) as r Ae 0 More Results on Functions frc, fr, and on convergence frc Ae f A Family of Pseudolocal Tomography Functions Numerical Implementation of Pseudolocal Tomography Pseudolocal Tomography for the Exponential Radon Transform Geometric Tomography Basic Idea Description of the Algorithm and Numerical Experiments Inversion of Incomplete Tomographic Data Inversion of Incomplete Fourier Transform Data Filtered Backprojection Method for Inversion of the Limited-Angle Tomographic Data The Extrapolation Problem The Davison-Grunbaum Algorithm Inversion of Cone-Beam Data Inversion of the Complete Cone-Beam Data Inversion of Incomplete Cone-Beam Data An Exact Algorithm for the Cone-Beam Circle Geometry g-Ray Tomography Radon Transform of Distributions Main Definitions Properties of the Test Function Spaces Examples Range Theorem for the Radon Transform on e' A Definition Based on Spherical Harmonics Expansion When Does the Radon Transform on Distributions Coincide with the Classical Radon Transform? The Dual Radon Transform on Distributions Abel-Type Integral Equation The Classical Abel Equation Abel-Type Equations Reduction of the Equation to a More Stable One Finding Locations and Values of Jumps of the Solution to the Abel Equation Multidimensional Algorithm for Finding Discontinuities of Signals from Noisy Discrete Data Introduction Edge Detection Algorithm Thin Line Detection Algorithm Generalization of the Algorithms Justification of the Edge Detection Algorithm Justification of the Algorithm for Thin Line Detection Justification of the General Scheme Numerical Experiments Proof of Auxiliary Results Test of Randomness and Its Applications Introduction Consistency of Rank Test Against Change Points (Change Surfaces) Alternative Consistency of Rank Test Against Trend in Location Auxiliary Results Abstract and Functional Spaces Distribution Theory Pseudodifferential and Fourier Integral Operators Special Functions Asymptotic Expansions Linear Equations in Banach Spaces Ill-Posed Problems Examples of Regularization of Ill-Posed Problems Radon Transform and PDE Statistics Research Problems Bibliographical Notes References Index List of Notations

Fractional Hilbert transform.

Abstract: We have generalized the Hilbert transform by defining the fractional Hilbert transform (FHT) operation. In the first stage, two different approaches for defining the FHT are suggested. One is based on modifying only the spatial filter, and the other proposes using the fractional Fourier plane for filtering. In the second stage, the two definitions are combined into a fractional Hilbert transform, which is characterized by two parameters. Computer simulations are presented.

Handbook of Function and Generalized Function Transformations

Abstract: Preliminaries Special Functions Generalized Functions Function Transformations The Laplace Transform The Two-Sided Laplace Transform The Borel Transform The Stieltjes Transform The Lambert Transform The Mellin Transform The Mellin-Type Transform The Fourier Transform The Hartley Transform The Hilbert Transform The Boas Transform The Mittag-Leffler Transform The Convolution Transform The Weierstrass Transform The Abel Transform The Riemann-Liouville and Weyl Fractional Integrals The Hankel and Hankel-Type Transforms (The I, Y and H-Transforms) The Hardy Transform The K(Meijer)-Transform The Kontorovich-Lebedev Transform The Mehler-Fock Transform The Sturm-Liouville-Type Transforms Miscellaneous Transforms: Kummer, Erdelyi, Hypergeometric, E, and G Transforms The Bargmann Transform The Zak Transform The Gabor Transform The Ambiguity Transformation and Wigner Distribution The Wavelet Transform The Radon Transform Sequence Transformations: The Discrete Fourier, Fast Fourier, Z, and Walsh Transforms

The discrete rotational Fourier transform

Abstract: We define a discrete version of the angular Fourier transform and present the properties of the transform that show it to be a rotation in time-frequency space, this new transform is a generalization of the DFT. Efficient algorithms for its computation can then be based on the FFT and the eigenstructure of the DFT.

Comparisons of spectral techniques for geoid computations over large regions

Abstract: The Stokes formula is efficiently evaluated by the one-and two- dimensional (1D, 2D) fast Fourier transform (FFT) technique in the plane and on the sphere in order to obtain precise geoid determinatiover a large area such as Europe. Using a high-pass filtered spherical harmonic reference model (OSU91A truncated to different degrees), gridded gravity anomalies and geoid heights were produced and the anomalies were used as input in the FFT software. Various tests were performed with respect to the different kernel functions used, to the spherical computations in bands, as well as to windowing, edge effects and extent of the area. It is thus demonstrated that, in geoid computations over large regions, the 1D spherical FFT and the 2D multiband spherical FFT in combination with discrete spectra for the kernel functions and 100% zero-padding give better results than those obtained by the other transform techniques. Additionally, numerical tests were carried out at the same test area using the planar fast Hartley transform (FHT) instead of the FFT and the results obtained by the two attractive alternatives were compared regarding the requirements in both computer time and computer memory needed in geoid height computations.

Walsh-like functions and their relations

Abstract: A new discrete transform, the 'Haar-Walsh transform', has been introduced. Similar to well known Walsh and non-normalised Haar transforms, the new transform assumes only +1 and -1 values, hence it is a Walsh-like function and can be used in different applications of digital signal and image processing. In particular, it is extremely well suited to the processing of two-valued binary logic signals. Besides being a discrete transform on its own, the proposed transform can also convert Haar and Walsh spectra uniquely between themselves. Besides the fast algorithm that can be implemented in the form of in-place flexible architecture, the new transform may be conveniently calculated using recursive definitions of a new type of matrix, a 'generator matrix'. The latter matrix can also be used to calculate some chosen Haar-Walsh spectral coefficients which is a useful feature in applications of the new transform in logic synthesis.

Optical illustration of a varied fractional Fourier-transform order and the Radon–Wigner display

Abstract: Based on an all-optical system, a display of a fractional Fourier transform with many fractional orders is proposed. Because digital image-processing terminology is used, this display is known as the Radon–Wigner transform. It enables new aspects for signal analysis that are related to time- and spatial-frequency analyses. The given approach for producing this display starts with a one-dimensional input signal although the output signal contains two dimensions. The optical setup for obtaining the fractional Fourier transform was adapted to include only fixed free-space propagation distances and variable lenses. With a set of two multifacet composite holograms, the Radon–Wigner display has been demonstrated experimentally.

The errors in FFT estimation of the Fourier transform

Abstract: The problem of determining the error in approximating the Fourier transform by the discrete Fourier transform is studied. Exact formulas for the relative error are established for classes of functions, called canonical-k (k/spl ges/0), and asymptotic error formulas are established for a much wider class of functions, called order-k. The formulas are dependent only on the class and not on the function in the class whose Fourier transform is being approximated, and this facilitates the application of the results.

A CORDIC-based unified systolic architecture for sliding window applications of discrete transforms

Abstract: A CORDIC-based, unified systolic architecture for sliding window applications of the discrete Fourier transform (DFT), the discrete Hartley transform (DHT), the discrete cosine transform (DCT), and the discrete sine transform (DST) is proposed. Compared to earlier works, the proposed scheme offers significant reduction in hardware, particularly for DHT. For an N-point DHT, it requires only [N/2]+1 processing elements, each consisting of one CORDIC processor and two adders.

Enhanced fast fourier transform technique on vector processor with operand routing and slot-selectable operation

Abstract: An apparatus and a method perform an N-point Fast Fourier Transform (FFT) on first and second arrays having real and imaginary input values using a processor with a multimedia extension unit (MEU), wherein N is a power of two. The invention repetitively sub-divides the N-point Fourier Transform into N/2-point Fourier Transforms until only a 2-point Fourier Transform remains. Next, it vector processes the 2-point Fourier Transform using the MEU and cumulates the results of the 2-point Fourier Transforms from each of the sub-divided N/2 Fourier Transforms to generate the result of the N-point Fourier Transform.

Deconvolution of overlapping chromatographic peaks by means of fast Fourier and hartley transforms

Abstract: Chromatographic peaks usually exhibit an exponentially Gaussian modified peak profile that is characterized by a tailing behaviour at the end of the peaks. As a result of this modification of the ideal peak shape, the peaks lose their symmetry and, most importantly, become broader. An unfortunate consequence of this broadening effect is that the resolution between adjacent peaks decreases as compared to the ideal case in which purely Gaussian symmetrical peaks would be considered. In addition to various chemometrical techniques developed to increase the separation of overlapping peaks, methods based on deconvolution in the frequency domain can be exploited. The principle of this latter family of deconvolution methods rests on the division of the frequency spectrum of the signal to be deconvoluted by the frequency spectrum of a judiciously chosen deconvoluting signal. In this paper, the analytical characteristics and utility of deconvolution is assessed with emphasis on aspects of resolution enhancement, data distortion, linearity and signal-to-noise ratios.

Inverse transform narrow band/broad band sound synthesis

Abstract: An additive sound synthesis process for generating complex, realistic sounds is realized in a computationally efficient manner. In accordance with one aspect of the invention, polyphony is efficiently achieved by dosing the energy of a given partial between separate transform sums corresponding to different channels. In accordance with another aspect of the invention, noise (87) is injected by randomly perturbing the phase of the sound, either on a per-partial basis or on a transform-sum basis. In the latter instance, the phase is perturbed in different regions of the spectrum to a degree determined by the amount of energy present in the respective regions of the spectrum. In accordance with yet another aspect of the invention, a transform sum (83) representing a sound is processed in the transform domain to achieve with great economy effects achievable only at much greater expense outside the transform domain. Other transforms besides the Fourier transform may be used to advantage. For example, use of the Hartley transform produces comparable results but allows transforms to be computed at approximately twice the speed as the Fourier transform.

Fractional Radon transform: definition.

Abstract: In this paper a novel fractional transformation for which we coin the term the fractional Radon transform is defined. This transform generalizes the Radon transform and combines it with the fractional Fourier transform. Both transformations are useful tools for invariant pattern recognition, tomography, and signal processing. Some of the properties of the new transformation, as well as further directions for investigation, are presented.

Fractional wave packet transform

Abstract: The short-time Fourier transform (STFT), or windowed Fourier transform, is the most widely used method in signal processing. We introduce the concept of the fractional wave packet transform (FRWPT), based on the idea of the fractional Fourier transform (FRFT) and wave packet transform (WPT). We show a version of the resolution of the identity and some properties of FRWPT connected with those of the FRFT and WPT.

Multichannel transforms for signal/image processing

Abstract: This paper presents a novel approach to the Fourier analysis of multichannel time series. Orthogonal matrix functions are introduced and are used in the definition of multichannel Fourier series of continuous-time periodic multichannel functions. Orthogonal transforms are proposed for discrete-time multichannel signals as well. It is proven that the orthogonal matrix functions are related to unitary transforms (e.g., discrete Hartley transform (DHT), Walsh-Hadamard transform), which are used for single-channel signal transformations. The discrete-time one-dimensional multichannel transforms proposed in this paper are related to two-dimensional single-channel transforms, notably to the discrete Fourier transform (DFT) and to the DHT. Therefore, fast algorithms for their computation can be easily constructed. Simulations on the use of discrete multichannel transforms on color image compression have also been performed.

Time-frequency transform vs. Fourier transform for radar imaging

Abstract: Conventional radar imaging systems use the Fourier transform for reconstruction of radar images. To use the Fourier transform adequately, some restrictions must be applied. Due to target's motion and manoeuvring, the reconstructed image by using Fourier transform becomes smeared. Therefore, some sophisticated motion compensation procedures must be applied to produce a clear image. However, the restrictions of Fourier transform can be lifted if a high resolution time-frequency transform can be used to retrieve the Doppler information. By replacing the conventional Fourier transform with the joint time-frequency transform, a 2-D range-Doppler Fourier frame becomes a 3-D time-range-Doppler cube. With sampling in time, a sequence of clear 2-D range-Doppler images can be simply reconstructed without using sophisticated motion compensation.

Deconvolution of analytical peaks by means of the fast Hartley transform

Abstract: A peak is one of the commonest and more desirable forms of analytical response. In multicomponent systems, where each individual component might give rise to one or more peaks, overlap between adjacent peaks often takes place. Depending on the degree of overlap, the extraction of qualitative and quantitative data can be difficult, and in extreme cases, impossible. In this paper, a method for deconvoluting overlapping peaks, based on the implementation of the fast Hartley transform (FHT), is described. The FHT is superior to the traditionally used fast Fourier transform in terms of speed and memory requirements. The program for deconvolution was developed in an object-oriented, icon-based software development tool (LabVIEW for Windows). The suggested method was applied to electroanalytical data involving overlapping peaks.

Transform domain adaptive linear phase filter

Abstract: Describes a new adaptive linear-phase filter whose weights are updated by the normalized least-mean-square (LMS) algorithm in the transform domain. This algorithm provides a faster convergence rate compared with the time domain linear phase LMS algorithm. Various real-valued orthogonal transforms are investigated such as the discrete cosine transform (DCT), discrete Hartley transform (DHT), and power of two (PO2) transform, etc. By using the symmetry property of the transform matrix, an efficient implementation structure is proposed. A system identification example is presented to demonstrate its performance.

Minimization of adders in fast FIR digital filters and its application to filter design

Abstract: This paper proposes fast FIR digital filter structures using the minimal number of adders. Filter coefficients are expressed with canonic signed digit (CSD) code and Hartley's technique is used to minimize the number of adders and subtractors. The proposed filters implemented as wired logic are fast because the structure having the shortest critical path is selected. An algorithm is given to obtain such fast structures. In many examples the critical path length of the filter structures obtained using the proposed method is equal to that of the conventional CSD structures. This paper also presents a new design method of FIR filters using MILP. Utilization of common expressions in Hartley's technique widen the CSD coefficient space. Thus the mixed integer linear programming (MILP) may lead to better frequency responses. Superior frequency responses are actually obtained in many simulations.

Efficient calculation of finite Gabor transforms

Abstract: The Gabor transform may be viewed as a collection of localized Fourier transforms and as such is useful for analysis of nonstationary signals and images We present a new approach to analyzing the Gabor transform and use it to study the various critically sampled discretizations that form the infinite-discrete, periodic finite-discrete, and nonperiodic finite-discrete versions of the transform In particular, we distinguish between the analysis and synthesis forms of the transform, and introduce an intermediate operation that decomposes both forms into collections of independent Toeplitz operators In the continuous, the infinite-discrete, and the periodic finite-discrete cases, this decomposition allows us to show that, for appropriate windows, the analysis and synthesis transforms are inverses of each other In the nonperiodic finite-discrete case this relation no longer holds, but we are still able to use the decomposition and results on Toeplitz matrices to show that both the transform and the inverse transform of P discrete samples are computable in O(P log P) operations (after a setup cost of O(Plog/sup 2/P)) Furthermore, we use the decomposition to study in detail the differences between the periodic and nonperiodic versions of the transform and to compare their conditioning