Showing papers on "Hartley transform published in 2005"

The Design and Implementation of FFTW3

Abstract: FFTW is an implementation of the discrete Fourier transform (DFT) that adapts to the hardware in order to maximize performance. This paper shows that such an approach can yield an implementation that is competitive with hand-optimized libraries, and describes the software structure that makes our current FFTW3 version flexible and adaptive. We further discuss a new algorithm for real-data DFTs of prime size, a new way of implementing DFTs by means of machine-specific single-instruction, multiple-data (SIMD) instructions, and how a special-purpose compiler can derive optimized implementations of the discrete cosine and sine transforms automatically from a DFT algorithm.

The Convolution Transform

Abstract: Introduction. The material I am reporting on here was prepared in collaboration with I. I. Hirschman. It will presently appear in book form in the Princeton Mathematical Series. I wish also to refer at once to the researches of I. J. Schoenberg and his students. Their work has been closely related to ours and has supplemented it in certain respects. Let me call attention especially to an article of Schoenberg [5, p. 199] in this Bulletin where the whole field is outlined and the historical development is traced. In view of the existence of this paper I shall t ry to avoid any parallel development here. Let me take rather a heuristic point of view and concentrate chiefly on trying to entertain you with what seems to me a fascinating subject.

FPGA implementations of fast Fourier transforms for real-time signal and image processing

Abstract: Applications based on the fast Fourier transform (FFT), such as signal and image processing, require high computational power, plus the ability to experiment with algorithms. Reconfigurable hardware devices in the form of field programmable gate arrays (FPGAs) have been proposed as a way of obtaining high performance at an economical price. However, users must program FPGAs at a very low level and have a detailed knowledge of the architecture of the device being used. They do not therefore facilitate easy development of, or experimentation with, signal/image processing algorithms. To try to reconcile the dual requirements of high performance and ease of development, this paper reports on the design and realisation of a high level framework for the implementation of 1-D and 2-D FFTs for real-time applications. A wide range of FFT algorithms, including radix-2, radix-4, split-radix and fast Hartley transform (FHT) have been implemented under a common framework in order to enable the system designers to meet different system requirements. Results show that the parallel implementation of 2-D FFT achieves linear speed-up and real-time performance for large matrix sizes. Finally, an FPGA-based parametrisable environment based on 2-D FFT is presented as a solution for frequency-domain image filtering application.

Fractional Fourier transform: A novel tool for signal processing

Abstract: The fractional Fourier transform (FRFT) is the generalization of the classical Fourier transform. It depends on a parameter ? (= a ?/2) and can be interpreted as a rotation by an angle ? in the time-frequency plane or decomposition of the signal in terms of chirps. This paper discusses discrete FRFT (DFRFT), time-frequency distributions related to FRFT, optimal filter and beamformer in FRFT domain, filtering using window functions and other fractional transforms along with simulation results.

New inversion methods for the Lorentz Integral Transform

Abstract: The Lorentz Integral Transform approach allows microscopic calculations of electromagnetic reaction cross-sections without explicit knowledge of final-state wave functions. The necessary inversion of the transform has to be treated with great care, since it constitutes a so-called ill-posed problem. In this work new inversion techniques for the Lorentz Integral Transform are introduced. It is shown that they all contain a regularization scheme, which is necessary to overcome the ill-posed problem. In addition, it is illustrated that the new techniques have a much broader range of application than the present standard inversion method of the Lorentz Integral Transform.

The discrete Pascal transform and its applications

Abstract: We introduce a new discrete polynomial transform constructed from the rows of Pascal's triangle. The forward and inverse transforms are computed the same way in both the one- and two-dimensional cases, and the transform matrix can be factored into binary matrices for efficient hardware implementation. We conclude by discussing applications of the transform in digital image processing, such as bump and edge detection.

Generalized prolate spheroidal wave functions for optical finite fractional Fourier and linear canonical transforms

Abstract: Prolate spheroidal wave functions (PSWFs) are known to be useful for analyzing the properties of the finite-extension Fourier transform (fi-FT). We extend the theory of PSWFs for the finite-extension fractional Fourier transform, the finite-extension linear canonical transform, and the finite-extension offset linear canonical transform. These finite transforms are more flexible than the fi-FT and can model much more generalized optical systems. We also illustrate how to use the generalized prolate spheroidal functions we derive to analyze the energy-preservation ratio, the self-imaging phenomenon, and the resonance phenomenon of the finite-sized one-stage or multiple-stage optical systems.

Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line

Abstract: We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L^2-functions and prove identities in the spirit of Bang for L^p-functions. The proofs seem to be new also in the special case of the Fourier transform.

Fourier transform representation by frequency-time wavelets

Abstract: A new concept of the A-wavelet transform is introduced, and the representation of the Fourier transform by the A-wavelet transform is described. Such a wavelet transform uses a fully scalable modulated window but not all possible shifts. A geometrical locus of frequency-time points for the A-wavelet transform is derived, and examples are given. The locus is considered "optimal" for the Fourier transform when a signal can be recovered by using only values of its wavelet transform defined on the locus. The inverse Fourier transform is also represented by the A/sup */-wavelet transform defined on specific points in the time-frequency plane. The concept of the A-wavelet transform can be extended for representation of other unitary transforms. Such an example for the Hartley transform is described, and the reconstruction formula is given.

A double exponential formula for the Fourier transforms

Abstract: In this paper, we propose a new and efficient method that is applicable for the computation of the Fourier transform of a function which may possess a singular point or slowly converge at infinity. The proposed method is based on a generalization of the method of the double exponential (DE) formula; the DE formula is a powerful numerical quadrature proposed by H. Takahasi and M. Mori in 1974 [1]. Although it is a widely applicable formula, it is not effective in computing the Fourier transform of a slowly decreasing function. Actually it is not very efficient even if one wants to compute the value of a Fourier transform at a particular point, i.e., a Fourier-type integral. To conquer this weakness at least for a Fourier-type integral, M. Mori and the author proposed a new DE formula in 1991 [2]. See also [3] for a further improvement. The method proposed there is effective for Fourier-type integrals, but it is still weak in the computation of the Fourier transform. Here we propose another DE formula which is applicable to the computation of the Fourier transform. One point in the new method proposed here is that it makes use of fixed sampling points even if we change the point where the Fourier integral is evaluated. In this paper we propose the new method and illustrate the efficiency of the new method in several concrete examples through the comparison with the older methods.

An Explanation for the Logarithmic Connection between Linear and Morphological System Theory

Abstract: Dorst/van den Boomgaard and Maragos introduced the slope transform as the morphological equivalent of the Fourier transform. Generalising the conjugacy operation from convex analysis it formed the basis of a morphological system theory that bears an almost logarithmic relation to linear system theory; a connection that has not been fully understood so far. Our article provides an explanation by disclosing that morphology in essence is linear system theory in specific algebras. While linear system theory uses the standard plus-prod algebra, morphological system theory is based on the max-plus algebra and the min-plus algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions in the latter algebras. The logarithmic Laplace transform makes a natural appearance as it corresponds to the conjugacy operation in the max-plus algebra. Its conjugate is given by the so-called Cramer transform. Originating from stochastics, the Cramer transform maps Gaussians to quadratic functions and relates standard convolution to erosion. This fundamental transform relies on the logarithm and constitutes the direct link between linear and morphological system theory. Many numerical examples are presented that illustrate the convexifying and smoothing properties of the Cramer transform.

Method of flow graph simplification for the 16-point discrete Fourier transform

Abstract: An efficient method for the realization of the paired algorithm for calculation of the one-dimensional (1-D) discrete Fourier transform (DFT), by simplifying the signal-flow graph of the transform, is described. The signal-flow graph is modified by separating the calculation for real and imaginary parts of all inputs and outputs in the signal-flow graph and using properties of the transform. The examples for calculation of the eight- and 16-point DFTs are considered in detail. The calculation of the 16-point DFT of real data requires 12 real multiplications and 58 additions. Two multiplications and 20 additions are used for the eight-point DFT.

Transform-based methods for indexing and retrieval of 3D objects

Abstract: We compare two transform-based indexing methods for retrieval of 3D objects. We apply 3D discrete Fourier transform (DFT) and 3D radial cosine transform (RCT) to the voxelized data of 3D objects. Rotation invariant features are derived from the coefficients of these transforms. Furthermore we compare two different voxel representations, namely, binary denoting object and background space, and continuous after distance transformation. In the binary voxel representation the voxel values are simply set to 1 on the surface of the object and 0 elsewhere. In the continuous-valued representation the space is filled with a function of distance transform. The rotation invariance properties of the DFT and RCT schemes are analyzed. We have conducted retrieval experiments on the Princeton Shape Benchmark and investigated the retrieval performance of the methods using several quality measures.

Fourier transform for the spatial quincunx lattice

Abstract: We derive a new, two-dimensional nonseparable signal transform for computing the spectrum of spatial signals residing on a finite quincunx lattice. The derivation uses the connection between transforms and polynomial algebras, which has long been known for the discrete Fourier transform (DFT), and was extended to other transforms in recent research. We also show that the new transform can be computed with O(n/sup 2/ log(n)) operations, which puts it in the same complexity class as its separable counterparts.

A linear Euclidean distance transform algorithm based on the linear-time Legendre transform

Abstract: We introduce a new exact Euclidean distance transform algorithm for binary images based on the linear-time Legendre Transform algorithm. The three-step algorithm uses dimension reduction and convex analysis results on the Legendre-Fenchel transform to achieve linear-time complexity. First, computation on a grid (the image) is reduced to computation on a line, then the convex envelope is computed, and finally the squared Euclidean distance transform is obtained. Examples and an extension to non-binary images are provided.

Fourier transform for the directed quincunx lattice

Abstract: We introduce a new signal transform for computing the spectrum of a signal given on a two-dimensional directional quincunx lattice. The transform is non-separable, but closely related to a two-dimensional (separable) discrete Fourier transform. We derive the transform using recently discovered connections between signal transforms and polynomial algebras. These connections also yield several important properties of the new transform.

Modified ICA algorithms for finding optimal transforms in transform coding

Abstract: In this paper we present two new algorithms that compute the linear optimal transform in high-rate transform coding, for non Gaussian data. One algorithm computes the optimal orthogonal transform, and the other the optimal linear transform. Comparison of the performances in high-rate transform coding between the classical Karhunen-Loeve Transform (KLT) and the transforms returned by the new algorithms are given. On synthetic data, the transforms given by the new algorithms perform significantly better that the KLT, however on real data all the transforms, included KLT, give roughly the same coding gain.

An analysis of real-Fourier domain-based adaptive algorithms implemented with the Hartley transform using cosine-sine symmetries

Abstract: The least mean squared (LMS) algorithm and its variants have been the most often used algorithms in adaptive signal processing. However the LMS algorithm suffers from a high computational complexity, especially with large filter lengths. The Fourier transform-based block normalized LMS (FBNLMS) reduces the computation count by using the discrete Fourier transform (DFT) and exploiting the fast algorithms for implementing the DFT. Even though the savings achieved with the FBNLMS over the direct-LMS implementation are significant, the computational requirements of FBNLMS are still very high, rendering many real-time applications, like audio and video estimation, infeasible. The Hartley transform-based BNLMS (HBNLMS) is found to have a computational complexity much less than, and a memory requirement almost of the same order as, that of the FBNLMS. This paper is based on the cosine and sine symmetric implementation of the discrete Hartley transform (DHT), which is the key in reducing the computational complexity of the FBNLMS by 33% asymptotically (with respect to multiplications). The parallel implementation of the discrete cosine transform (DCT) in turn can lead to more efficient implementations of the HBNLMS.

On unified architectures for synthesizing and implementation of fast parametric transforms

Abstract: In this paper a methodology to design VLSI architectures for parametric transform families is proposed. A parametric transform family consists of discrete orthogonal transforms such that they all may be computed with a fast algorithm of similar structure where parameters defining the transform within the family are used. In our previous work, an algorithm to synthesize transforms with predefined basis functions was introduced and efficiently applied to image compression. The methodology proposed in this paper allows of designing VLSI architectures that may not only switch from one transform of a family to another by setting parameters, but also to actually set these parameters so that the matrix of the resulting transform has predefined basis functions

Complex horospherical transform on real sphere

Abstract: We define a new integral transform on the real sphere which is invariant relative to the orthogonal group and similar to the horospherical Radon transform for the hyperbolic space. This transform involves complex geometry associated with the sphere.

Quantum circuit design of discrete Hartley transform using recursive decomposition formula

Abstract: In this paper, the quantum circuit design of a discrete Hartley transform (DHT) is investigated. The proposed design procedure can be divided into the following two steps. First, the N-point transform matrix is decomposed into the combination of N/2-point transform matrices and several sparse matrices such that any 2/sup n/-point transform can be computed recursively from the 2-point transform. Second, starting from the 2-point Hadamard transform, the quantum circuit of 2/sup n/-point transform can be designed recursively by using elementary quantum gates. The design results are compared with those of the conventional factorization method to demonstrate the effectiveness of proposed method.

Multiresolution of the Fourier transform

Abstract: In this paper, the integral Fourier transform is described as a specific wavelet-like transform with a fully scalable modulated window, but not all possible translations. The transform is defined by sinusoidal waves of half periods. A geometrical locus of frequency-time points for the proposed wavelet-like transform is derived and the function coverage is described and compared with the short-time Fourier transform as well as with the wavelet transform.

Data transform method and apparatus

Abstract: A data transform method comprises a linear transform step of acquiring 16 linear Hadamard transform coefficients by multiplying the 16 pieces of data by an Hadamard transform matrix, an offset step of classifying the 16 linear Hadamard transform coefficients into four groups, each group containing an odd number of coefficients, and adding a predetermined offset value to linear Hadamard transform coefficients in every group, and an integral step of acquiring the lossless Hadamard transform coefficients by truncating decimal fractions down from a decimal point from the linear Hadamard transform coefficients added said predetermined offset value. With this configuration, a 16-point lossless Hadamard transform is implemented using only one rounding process.

On the Fourier transform of SO( d )-finite measures on the unit sphere

Abstract: The paper presents a simple new approach to the problem of computing Fourier transforms of SO(d)-finite measures on the unit sphere in the euclidean space. Representing such measures as restrictions of homogeneous polynomials we use the canonical decomposition of homogeneous polynomials together with the plane wave expansion to derive a formula expressing such transforms under two forms, one of which was established previously by F. J. Gonzalez Vieli. We showthat equivalence of these two forms is related to a certain multi-step recurrence relation for Bessel functions, which encompasses several classical identities satisfied by Bessel functions. We show it leads further to a certain periodicity relation for the Hankel transform, related to the Bochner- Coifman periodicity relation for the Fourier transform. The purported novelty of this approach rests on the systematic use of the detailed form of the canonical decomposition of homogeneous polynomials, which replaces the more traditional approach based on integral identities related to the Funk-Hecke theorem. In fact, in the companion paper the present authors were able to deduce this way a fairly general expansion theorem for zonal functions, which includes the plane wave expansion used here as a special case.