Showing papers on "Hartley transform published in 2004"
TL;DR: This note makes a critical comparison of some matlab programs for the digital computation of the fractional Fourier transform that are freely available and describes the own implementation that lters the best out of the existing ones.
134 citations
Book•
31 Jan 2004
TL;DR: In this paper, two-dimensional Fourier Transform of the Shah Function was used for line detection in two dimensions, one dimension and the other dimension of the Hartley Transform of a Shah Function.
Abstract: 1 Introduction.- Summary of the Chapters.- Notation.- Teaching a Course from This Book.- The Problems.- Aspects of Imaging.- Computer Code.- Literature References.- Recommendation.- 2 The Image Plane.- Modes of Representation.- Some Properties of a Function of Two Variables.- Projection of Solid Objects.- Image Distortion.- Operations in the Image Plane.- Binary Images.- Operations on Digital Images.- Reflectance Distribution.- Data Compression.- Summary.- Appendix: A Contour Plot Programt.- Literature Cited.- Further Reading.- Problems.- 3 Two-Dimensional Impulse Functions.- The Two-Dimensional Point Impulse.- Rules for Interpreting Delta Notation.- Generalized Functions.- The Shah Functions iii and 2III.- Line Impulses.- Regular Impulse Patterns.- Interpretation of Rectangle Function of f(x).- Interpretation of Rectangle Function of f(x,y).- General Rule for Line Deltas.- The Ring Impulse.- Impulse Function of f(x,y).- Sifting Property.- Derivatives of Impulses.- Summary.- Literature Cited.- Problems.- 4 The Two-Dimensional Fourier Transform.- One Dimension.- The Fourier Component in Two Dimensions.- Three or More Dimensions.- Vector Form of Transform.- The Corrugation Viewpoint.- Examples of Transform Pairs.- Theorems for Two-Dimensional Fourier Transforms.- The Two-Dimensional Hartley Transform.- Theorems for the Hartley Transform.- Discrete Transforms.- Summary.- Literature Cited.- Further Reading.- Problems.- 5 Two-Dimensional Convolution.- Convolution Defined.- Cross-Correlation Defined.- Feature Detection by Matched Filtering.- Autocorrelation Defined.- Understanding Autocorrelation.- Cross-Correlation Islands and Dilation.- Lazy Pyramid and Chinese Hat Function.- Central Value and Volume of Autocorrelation.- The Convolution Sum.- Computing the Convolution.- Digital Smoothing.- Matrix Product Notation.- Summary.- Literature Cited.- Problems.- 6 The Two-Dimensional Convolution Theorem.- Convolution Theorem.- An Instrumental Caution.- Point Response and Transfer Function.- Autocorrelation Theorem.- Cross-Correlation Theorem.- Factorization and Separation.- Convolution with the Hartley Transform.- Summary.- Problems.- 7 Sampling and Interpolation in Two Dimensions.- What is a Sample?.- Sampling at a Point.- Sampling on a Point Pattern, and the Associated Transfer Function.- Sampling Along a Line.- Curvilinear Sampling.- The Shah Function.- Fourier Transform of the Shah Function.- Other Patterns of Sampling.- Factoring.- The Two-Dimensional Sampling Theorem.- Undersampling.- Aliasing.- Circular Cutoff.- Double-Rectangle Pass Band.- Discrete Aspect of Sampling.- Interpolating Between Samples.- Interlaced Sampling.- Appendix: The Two-Dimensional Fourier Transform of the Shah Function.- Literature Cited.- Problems.- 8 Digital Operations.- Smoothing.- Nonconvolutional Smoothing.- Trend Reduction.- Sharpening.- What is a Digital Filter?.- Guard Zone.- Transform Aspect of Smoothing Operator.- Finite Impulse Response (FIR).- Special Filters.- Densifying.- The Arbitrary Operator.- Derivatives.- The Laplacian Operator.- Projection as a Digital Operation.- Moire Patterns.- Functions of an Image.- Digital Representation of Objects.- Filling a Polygon.- Edge Detection and Segmentation.- Discrete Binary Objects.- Operations on Discrete Binary Objects.- Union and Intersection.- Pixel Morphology.- Dilation.- Coding a Binary Matrix.- Granulometry.- Conclusion.- Literature Cited.- Problems.- 9 Rotational Symmetry.- What Is a Bessel Function?.- The Hankel Transform.- The jinc Function.- The Struve Function.- The Abel Transform.- Spin Averaging.- Angular Variation and Chebyshev Polynomials.- Summary.- Table of the jinc Function.- Problems.- 10 Imaging by Convolution.- Mapping by Antenna Beam.- Scanning the Spherical Sky.- Photography.- Microdensitometry.- Video Recording.- Eclipsometry.- The Scanning Acoustic Microscope.- Focusing Underwater Sound.- Literature Cited.- Problems.- 11 Diffraction Theory of Sensors and Radiators.- The Concept of Aperture Distribution.- Source Pair and Wave Pair.- Two-Dimensional Apertures.- Rectangular Aperture.- Example of Circular Aperture.- Duality.- The Thin Lens.- What Happens at a Focus?.- Shadow of a Straight Edge.- Fresnel Diffraction in General.- Literature Cited.- Problems.- 12 Aperture Synthesis and Interferometry.- Image Extraction from a Field.- Incoherent Radiation Source.- Field of Incoherent Source.- Correlation in the Field of an Incoherent Source.- Visibility.- Measurement of Coherence.- Notation.- Interferometers.- Radio Interferometers.- Rationale Behind Two-Element Interferometer.- Aperture Synthesis (Indirect Imaging).- Literature Cited.- Problems.- 13 Restoration.- Restoration by Successive Substitutions.- Running Means.- Eddington's Formula.- Finite Differences.- Finite Difference Formula.- Chord Construction.- The Principal Solution.- Finite Differencing in Two Dimensions.- Restoration in the Presence of Errors.- The Additive Noise Signal.- Determination of the Real Restoring Function.- Determination of the Complex Restoring Function.- Some Practical Remarks.- Artificial Sharpening.- Antidiffusion.- Nonlinear Methods.- Restoring Binary Images.- CLEAN.- Maximum Entropy.- Literature Cited.- Problems.- 14 The Projection-Slice Theorem.- Circular Symmetry Reviewed.- The Abel-Fourier-Hankel Cycle.- The Projection-Slice Theorem.- Literature Cited.- Problems.- 15 Computed Tomography.- Workingfrom Projections.- An X-Ray Scanner.- Fourier Approach to Computed Tomography.- Back-Projection Methods.- The Radon Transform.- The Impulse Response of the Radon Transformation.- Some Radon Transforms.- The Eigenfunctions.- Theorems for the Radon Transform.- The Radon Boundary.- Applications.- Literature Cited.- Problems.- 16 Synthetic-Aperture Radar.- Doppler Radar.- Some History of Radiofrequency Doppler.- Range-Doppler Radar.- Radargrarnmetry.- Literature Cited.- Problems.- 17 Two-Dimensional Noise Images.- Some Types of Random Image.- Gaussian Noise.- The Spatial Spectrum of a Random Scatter.- Autocorrelation of a Random Scatter.- Pseudorandom Scatter.- Random Orientation.- Nonuniform Random Scatter.- Spatially Correlated Noise.- The Familiar Maze.- The Drunkard's Walk.- Fractal Polygons.- Conclusion.- Literature Cited.- Problems.- Appendix A Solutions to Problems.
133 citations
TL;DR: The purpose of this paper is to show that the main results on nonlinear functions can be easily generalized to the case of arbitrary abelian groups if the Walsh-Hadamard transform is replaced by the discrete Fourier transform.
102 citations
TL;DR: The dilating Gabor transformation is introduced by using a changeable window of Gaussian function in a conventional Gabor transform to analyze the optical fringes of 3-D shape measurement, providing more accurate results than Fourier transform and Gabortransform.
Abstract: In order to overcome the limitations of conventional Fourier transform and Gabor transform analyzing nonstationary signals, dilating Gabor transform is applied to analyze the optical fringes of 3-D shape measurement. The dilating Gabor transformation is introduced by using a changeable window of Gaussian function in a conventional Gabor transform. This phase analysis method provides more accurate results than Fourier transform and Gabor transform. Simulation and experimental results are presented that demonstrate the validity of the principle.
44 citations
TL;DR: A windowed Fourier transform method is proposed, with advantages of signal localization and noise filtering, for demodulation of carrier fringes, and an example demonstrates the improved result.
Abstract: The Fourier transform method for demodulation of carrier fringes has been extensively developed and widely used in optical metrology. However, the Fourier transform being a global operation, it has poor ability to localize the signal properties and hence the result of FTM is not ideal. A windowed Fourier transform method is thus proposed, with advantages of signal localization and noise filtering. An example demonstrates the improved result compared to the traditional Fourier transform.
41 citations
Patent•
17 Dec 2004TL;DR: In this article, a 2D transform and its inverse have been implemented as a sequence of lifting steps arranged for reduced computational complexity (i.e., reducing a number of non-trivial operations).
Abstract: A 2D transform and its inverse have an implementation as a sequence of lifting steps arranged for reduced computational complexity (i.e., reducing a number of non-trivial operations). This transform pair has energy compaction properties similar to the discrete cosine transform (DCT), and is also lossless and scale-free. As compared to a separable DCT transform implemented as 1D DCT transforms applied separably to rows and columns of a 2D data block, the transforms operations are re-arranged into a cascade of elementary transforms, including the 2×2 Hadamard transform, and 2×2 transforms incorporating lifting rotations. These elementary transforms have implementations as a sequence of lifting operations.
31 citations
TL;DR: In this article, a unified mathematical model for some classical signal transforms is presented and discussed, and the specific values of the variables and parameter functions of FT, STFT and WT are given.
26 citations
01 Jan 2004
TL;DR: The numerical examples show that the continuous wavelet transform constructed in this paper has higher computing accuracy compared with the classical continuous wavelets transform.
Abstract: In this paper, we propose a new type of continuous wavelet transform. However we discretize the variables of integral a and b, any numerical integral has a high resolution and a does not appear in the denominator of the integrand. Furthermore, we give two discretization methods of the new wavelet transform. For the one-dimensional situation, we give quadrature formula of the discretized inverse wavelet transform. For the multidimensional situation, we develop the commonly wavelet network based on the discretized inverse wavelet transform of the new wavelet transform. Finally, the numerical examples show that the continuous wavelet transform constructed in this paper has higher computing accuracy compared with the classical continuous wavelet transform.
26 citations
11 Dec 2004
TL;DR: A new transform for two-dimensional signal representation and some of its properties are proposed in this paper and a few properties of the transform are studied and the relevant relations are presented.
Abstract: A new transform (MRT) for two-dimensional signal representation and some of its properties are proposed in this paper. The transform helps to do the frequency domain analysis of two-dimensional signals without any complex operations but in terms of real additions. It is obtained by exploiting the periodicity and symmetry of the exponential term in the discrete Fourier transform (DFT) relation, and by grouping related data. Forward and inverse relations of the transform are presented. The transform coefficients show useful redundancies among themselves. These redundancies, which can be used to implement the transform, are studied. A few properties of the transform are studied and the relevant relations are presented.
25 citations
23 Aug 2004
TL;DR: An algorithm for detecting cyclic and dihedral symmetries of an object that is algebraically accurate and uses no interpolations is presented.
Abstract: We present an algorithm for detecting cyclic and dihedral symmetries of an object. Both symmetry types can be detected by the special patterns they generate in the object's Fourier transform. These patterns are effectively detected and analyzed using the "angular difference function" (ADF), which measures the difference in the angular content of images. The ADF is accurately computed by using the pseudo-polar Fourier transform, which rapidly computes the Fourier transform of an object on a near-polar grid. The algorithm detects all the axes of centered and non-centered symmetries. The proposed algorithm is algebraically accurate and uses no interpolations.
24 citations
TL;DR: In this paper, real Paley-Wiener theorems for the inverse Fourier transform on a semisimple Riemannian symmetric space G/K of the noncompact type were proved.
Abstract: We prove real Paley-Wiener theorems for the inverse Fourier transform on a semisimple Riemannian symmetric space G/K of the noncompact type. The functions on G/K whose Fourier transform has compact support are characterised by a L 2 growth condition. We also obtain real Paley-Wiener theorems for the inverse spherical transform.
01 Jan 2004
TL;DR: In this paper, a canonical transform that directly maps the measured field to the impact parameter representation without first carrying out a back-propagation was proposed, which can be expressed as a Fourier transform.
Abstract: In the present work, we study a canonical transform that directly maps the measured field to the impact parameter representation without first carrying out a back-propagation. This canonical transform is determined to first order in a small parameter that measures the deviation of the satellite orbit from a circle. When the parameter is equal to zero, i.e., for circular orbits, our canonical transform reduces to a Fourier transform. In the general case, the form of the generating function is such that it does not directly allow an implementation as an FFT-like algorithm. However, using approximations the direct canonical transform mapping yields fast, efficient numerical implementations.
TL;DR: A novel procedure is proposed that uses localized information and estimates both the unknown frequencies and the phases of the fringe pattern (using an interpolated fast Fourier transform method) to demodulate the fringe patterns without any distortion.
Abstract: Recently a powerful Fourier transform technique was introduced that was able to extract the phase from only one image. However, because the method is based on the two-dimensional Fourier transform, it inherently suffers from leakage effects. A novel procedure is proposed that does not exhibit this distortion. The procedure uses localized information and estimates both the unknown frequencies and the phases of the fringe pattern (using an interpolated fast Fourier transform method). This allows us to demodulate the fringe pattern without any distortion. The proposed technique is validated on both computer simulations and the profile measurements of a tube.
Posted Content•
TL;DR: In this paper, a systematic treatment of problems in analysis and integral geometry related to inversion of the Radon transform on the space of real rectangular matrices is presented, with a special emphasis on new higher rank phenomena, in particular, on possibly minimal conditions under which the radon transform is well defined and can be explicitly inverted.
Abstract: The monograph contains a systematic treatment of a circle of problems in analysis and integral geometry related to inversion of the Radon transform on the space of real rectangular matrices. This transform assigns to a function $f$ on the matrix space the integrals of $f$ over the so-called matrix planes, the linear manifolds determined by the corresponding matrix equations. Different inversion methods are discussed. They rely on close connection between the Radon transform, the Fourier transform, the Garding-Gindikin fractional integrals, and matrix modifications of the Riesz potentials. A special emphasis is made on new higher rank phenomena, in particular, on possibly minimal conditions under which the Radon transform is well defined and can be explicitly inverted. Apart of the space of Schwartz functions, we also employ $L^p$-spaces and the space of continuous functions. Many classical results for the Radon transform on $R^n$ are generalized to the higher rank case.
23 Aug 2004
TL;DR: Capus et al. as discussed by the authors used the fractional Fourier transform (FFT) to process linear chirp coded ultrasound signals and achieved a degree of pulse compression similar to that given by a matched filter.
Abstract: The fractional Fourier transform represents a generalisation of the conventional Fourier transform. Previous work (M. Bennett et al, Proc. IEEE EMBS vol. 1, pp. 882-885, 2003) has shown that the application of the fractional Fourier transform to conventional, un-coded ultrasound signals has little advantage over conventional filtering techniques such as band-pass filtering. However, the fractional Fourier transform can be 'tuned' to be sensitive to signals of a particular chirp rate (C. Capus et al, IEE Seminar on Time-Scale and Time-Freq. Analysis and Appl., 2000) and can achieve levels of pulse compression similar to those obtained using a matched filter. To this end a system was developed which could generate and transmit linear chirp coded ultrasound signals. The fractional Fourier transform was then used to process the signals received from a simple phantom arrangement. When the transform was used with the optimum transform order corresponding to the chirp rate of the signals, the transform domain signals demonstrated a degree of pulse compression similar to that given by a matched filter. Results are also presented which demonstrate that a chirp signal identified in the fractional Fourier domain may be completely recovered in the time domain through the use of the inverse transform. Matched filtering was found to give a greater degree of pulse compression, but the fraction Fourier method can be applied without a-priori knowledge of the transmitted signal. Further work will be carried out to determine the best way of extracting useful information from the fractional domain signals.
01 Jan 2004
TL;DR: In this article, a generalization of the Radon transform to the infinite dimensional setting, using Gauss measure instead of Lebesgue measure, is presented. But this generalization is restricted to a single hyperplane in an infinite dimensional Hilbert space.
Abstract: Gaussian measure is constructed for any given hyperplane in an infinite dimensional Hilbert space, and this is used to define a generalization of the Radon transform to the infinite dimensional setting, using Gauss measure instead of Lebesgue measure. An inversion formula is obtained and a support theorem proved.
Patent•
02 Nov 2004
TL;DR: In this article, the authors proposed a method for image processing utilizing numerical calculation of fractional exponential powers of a diagonalizable numerical transform operator for use in an iterative or other larger computational environments.
Abstract: Image processing utilizing numerical calculation of fractional exponential powers of a diagonalizable numerical transform operator for use in an iterative or other larger computational environments. In one implementation, a computation involving a similarity transformation is partitioned so that one part remains fixed and may be reused in subsequent iterations. The numerical transform operator may be a discrete Fourier transform operator, discrete fractional Fourier transform operator, centered discrete fractional Fourier transform operator, and other operators, modeling propagation through physical media. Such iterative environments for these types of numerical calculations are useful in correcting the focus of misfocused images which may originate from optical processes involving light (for example, with a lens or lens system) or from particle beams (for example, in electron microscopy or ion lithography).
16 Aug 2004
TL;DR: A novel digital image-watermarking scheme that is invariant to rotation, scale and translation (RST) attacks is presented, using an invariant centroid as the origin of Radon transform and 2-D Fourier transform.
Abstract: Based on Radon transform and 2-D Fourier transform, this paper presents a novel digital image-watermarking scheme that is invariant to rotation, scale and translation (RST) attacks. By use of an invariant centroid as the origin of Radon transform, the watermark is embedded into a domain obtained by taking Radon transform of a circular area selected from the image, and then extracting 3-D Fourier magnitude of the Radon transformed image. Furthermore, to prevent the watermarked image from degrading due to inverse Radon transform, only watermark signal is inverse Radon transformed and then add to the original image. Experimental results demonstrate that the proposed scheme is robust to RST attacks.
TL;DR: A Fourier transform representation of the gamma functions is presented, which leads naturally to a distributional representation for them, and new identities for the integrals of gamma functions multiplied by other functions are presented.
Abstract: We present a Fourier transform representation of the gamma functions, which leads naturally to a distributional representation for them. Both of these representations lead to new identities for the integrals of gamma functions multiplied by other functions, which are also presented here.
01 Jan 2004
TL;DR: This paper presents a multidimensional lifting approach for reducing approximation error in the frequency domain of lifting-based integer transforms, where large parts of the transform are calculated without rounding operations, only the output is rounded and added.
Abstract: Recently lifting-based integer transforms have received much attention, especially in the area of lossless audio and image coding. The usual approach is to apply the lifting scheme to each Givens rotation. Especially in the case of long transform sizes in audio coding applications, this leads to a considerable approximation error in the frequency domain. This paper presents a multidimensional lifting approach for reducing this approximation error. In this approach, large parts of the transform are calculated without rounding operations, only the output is rounded and added. The new approach is applied and evaluated for both the Integer Modified Discrete Cosine Transform (IntMDCT) and the Integer Fast Fourier Transform (IntFFT).
TL;DR: Several generalizations of the spherical Fourier transform on Riemannian symmetric spaces have emerged in the last decades as mentioned in this paper, and these generalizations can be viewed as special instances of the Θ-spherical transform.
Abstract: Several generalizations of the spherical Fourier transform on Riemannian symmetric spaces have emerged in the last decades. All these integral transforms can be viewed as special instances of the Θ-spherical Fourier transform. One of the main questions related to any integral transform is the description of the image of classical function spaces. In this article we discuss several examples of Paley–Wiener type theorems, leading up to and motivating the Paley–Wiener theorem for the Θ-spherical transform.
17 May 2004
TL;DR: This paper proposes a methodology for obtaining and obtain variants of the discrete fractional cosine (sine) transform which share real-valuedness as well as most of the properties required for a fractional transform matrix.
Abstract: The unitarity property of transforms is useful in many applications (source compression, transmission, watermarking, to name a few). In many cases, when a transform is applied on real-valued data, it is very useful to obtain real-valued coefficients (i.e. a reality-preserving transform). In most applications, the decorrelation property of the transform is of importance and it would be very useful to control it under some transform parameter (e.g. in joint source-channel coding). This paper focuses on fractional transforms, as tools for obtaining such properties. We propose a methodology for obtaining them and obtain variants of the discrete fractional cosine (sine) transform which share real-valuedness as well as most of the properties required for a fractional transform matrix. As shown in (I. Venturini et al. IEEE Trans. Signal Proc.), such matrices cannot be symmetric.
TL;DR: In this article, the Stechkin problem of the approximation of derivatives by bounded linear functionals was studied, and exact Kolmogorov-type inequalities for derivatives corresponding to these problems were obtained.
Abstract: The problems of the optimal recovery of the derivatives of functions from inaccurate information about the Fourier transforms of these functions on a finite interval or the entire number line are considered. The Stechkin problem of the approximation of derivatives by bounded linear functionals, which is closely connected to this range of problems, is also studied. Precise Kolmogorov-type inequalities for derivatives corresponding to these problems are obtained.
TL;DR: This correspondence describes a new bit-reversal permutation algorithm based on a trivial symmetry that has not been exploited until now that outperforms the fastest algorithms known to the author.
Abstract: This correspondence describes a new bit-reversal permutation algorithm based on a trivial symmetry that has not been exploited until now. According to timing experiments, this algorithm outperforms the fastest algorithms known to the author. This is of interest for applications using intensive fast Fourier transforms (or fast Hartley transforms) of constant length, such as transform domain adaptive filtering.
TL;DR: An efficient split algorithm for calculating the one-dimensional discrete Hartley transforms, by using a special partitioning in the frequency domain, is introduced.
Abstract: An efficient split algorithm for calculating the one-dimensional discrete Hartley transforms, by using a special partitioning in the frequency domain, is introduced. The partition determines a fast paired transform that splits the 2/sup r/-point unitary Hartley transform into a set of 2/sup r-n/-point odd-frequency Hartley transforms, n=1:r. A proposed method of calculation of the 2/sup r/-point Hartley transform requires 2/sup r-1/(r-3)+2 multiplications and 2/sup r-1/(r+9)-r/sup 2/-3r-6 additions.
Journal Article•
TL;DR: In this article, the Fourier transform associated with the spherical mean operator R is used to characterize the range of some spaces of functions by Fourier transformation associated with R. The authors also give a new description of Schwartz spaces.
Abstract: We characterize the range of some spaces of functions by the Fourier transform associated with the spherical mean operator R and we give a new description of the Schwartz spaces. Next, we prove a Paley-Wiener and a Paley-Wiener-Schawrtz theorems. Mathematics Subject Classification: 42B35, 43A32, 35S30
TL;DR: In this article, the Fourier transform is restricted to a certain k-dimensional quadratic submanifold of n ≥ 2.5.1. The Fourier transformation is then restricted to n ≥ 3.
Abstract: We obtain estimates for the restriction of the Fourier transform to a certain k-dimensional quadratic submanifold of n
.
TL;DR: In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the DHT of an arbitrary length N = q*2/sup m/, where q is an odd integer, which reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table.
Abstract: In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length N=q*2/sup m/, where q is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient indexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- q*2/sup m/ DHTs. Since the proposed algorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case.
01 Jan 2004
TL;DR: It is shown that the arithmetic complexity (multiplications additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- DHTs.
Abstract: In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length , where is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient in- dexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- DHTs. Since the proposed al- gorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case. Index Terms—Discrete Hartley transform (DHT), fast Hartley transform (FHT) algorithms, mixed radix, split radix.
TL;DR: Using Type-2 theory of effectivity, computability notions on the spaces of Lebesgue-integrable functions on the real line that are based on two natural approaches to integrability from measure theory are defined.
Abstract: Using Type-2 theory of effectivity, we define computability notions on the spaces of Lebesgue-integrable functions on the real line that are based on two natural approaches to integrability from measure theory. We show that Fourier transform and convolution on these spaces are computable operators with respect to these representations. By means of the orthonormal basis of Hermite functions in L2, we show the existence of a linear complexity bound for the Fourier transform. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)