Showing papers on "Hartley transform published in 2007"
TL;DR: Hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images, and the properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms.
Abstract: Fourier transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed
535 citations
Book•
13 Apr 2007
Abstract: Mathematics of Multidimensional Fourier Transform AlgorithmsThe DFTThe Discrete Fourier TransformComputational Frameworks for the Fast Fourier TransformFourier Analysis on Finite Groups and ApplicationsSimpson's Discrete Fourier TransformThe DFTFoundations of Signal ProcessingThe Discrete Fourier TransformFast Fourier TransformsAn Introduction to Laplace Transforms and Fourier SeriesAlgorithms for Discrete Fourier Transform and ConvolutionFourier Series, Fourier Transform and Their Applications to Mathematical PhysicsFourier TransformsMusic Through Fourier SpaceFourier TransformsDiscrete and Continuous Fourier TransformsAnalysis of Boolean FunctionsDiscrete Fourier Analysis and WaveletsPrinciples of Fourier AnalysisDiscrete Fourier AnalysisDiscrete Harmonic AnalysisThe Scientist and Engineer's Guide to Digital Signal ProcessingDiscrete Fourier Analysis and WaveletsMastering the Discrete Fourier Transform in One, Two or Several DimensionsThe XFT Quadrature in Discrete Fourier AnalysisMathematics of the Discrete Fourier Transform (DFT)Signal ProcessingDiscrete Fourier Transforms and their Applications,Harmonic AnalysisLectures on the Fourier Transform and Its ApplicationsAn Introduction to Fourier AnalysisDiscrete Fourier AnalysisTheory of Discrete and Continuous Fourier AnalysisThe Fast Fourier Transform and Its ApplicationsDiscrete Fourier and Wavelet TransformsIntroduction to Digital FiltersFourier Analysis—A Signal Processing ApproachMathematics of the Discrete Fourier Transform (DFT) with Music and Audio ApplicationsMathematics of the Discrete Fourier Transform (DFT)
308 citations
TL;DR: New sampling formulae for reconstructing signals that are band-limited or time-limited in the linear canonical transform sense have been proposed and well-known sampling theorems in Fourier domain or fractional Fourierdomain are shown to be special cases of the achieved results.
143 citations
TL;DR: A novel random fractional Fourier transform is proposed by randomizing the transform kernel function of the conventional fractional fourier transform, which can be directly used in optical image encryption and decryption.
Abstract: We propose a novel random fractional Fourier transform by randomizing the transform kernel function of the conventional fractional Fourier transform. The random fractional Fourier transform inherits the excellent mathematical properties from the fractional Fourier transform and can be easily implemented in optics. As a primary application the random fractional Fourier transform can be directly used in optical image encryption and decryption. The double phase encoding image encryption schemes can thus be modeled with cascaded random fractional Fourier transformers.
126 citations
TL;DR: The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.
Abstract: The offset linear canonical transform (OLCT) is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms in engineering such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform (FRST), frequency modulation, time shifting, time scaling, chirping and others. Therefore the OLCT provides a unified framework for studying the behavior of many practical transforms and system responses. In this paper the sampling theorem for OLCT is considered. The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.
100 citations
TL;DR: State variables from brain potentials are time series that are either recorded and digitized or derived from recordings by use of the Fourier and Hilbert transforms.
Abstract: State variables from brain potentials are time series that are either recorded and digitized or derived from recordings by use of the Fourier and Hilbert transforms; they provide the primary raw materials by which models of brain dynamics are constructed and evaluated.
77 citations
31 May 2007
TL;DR: This paper considers functions f which are simultaneously bent and negabent, i.e. which have optimum periodic and negaperiodic properties and several constructions and classifications are presented.
Abstract: Bent functions f : F2m → F2 achieve largest distance to all linear functions. Equivalently, their spectrum with respect to the Hadamard-Walsh transform is flat (i.e. all spectral values have the same absolute value). That is equivalent to saying that the function f has optimum periodic autocorrelation properties. Negaperiodic correlation properties of f are related to another unitary transform called the nega-Hadamard transform. A function is called negabent if the spectrum under the nega-Hadamard transform is flat. In this paper, we consider functions f which are simultaneously bent and negabent, i.e. which have optimum periodic and negaperiodic properties. Several constructions and classifications are presented.
67 citations
TL;DR: In this article, the authors proposed a transform-domain active noise control (ANC) algorithm using only convolution machines, which is different from conventional block least-mean-square (LMS) algorithms that use both convolution and cross-correlation machines.
Abstract: New block formulations for an active noise control (ANC) system using only convolution machines are presented. The proposed approaches are different from conventional block least-mean-square (LMS) algorithms that use both convolution and cross-correlation machines. The block implementation is also applied to the filtering of the reference signal by the secondary-path estimate. In addition to the use of the fast Fourier transform (FFT), the fast Hartley transform (FHT) is used to develop transform-domain ANC structures for reducing computational complexity. In the proposed approach, some FFT and FHT blocks are removed to obtain an additional reduction of the computational burden resulting in the reduced-structure of FFT-based block filtered-X LMS (FBFXLMS) and FHT-based block filtered-X LMS (HBFXLMS) algorithms. The computational complexities of these new ANC structures are evaluated.
60 citations
TL;DR: The numerical simulation and experiment have proved the validity of the multiscale windowed Fourier transform for phase extraction of fringe patterns and makes the extracted phase more precise than other methods.
Abstract: A multiscale windowed Fourier transform for phase extraction of fringe patterns is presented. A local stationary length of signal is used to control the window width of a windowed Fourier transform automatically, which is measured by an instantaneous frequency gradient. The instantaneous frequency of the fringe pattern is obtained by detecting the ridge of the wavelet transform. The numerical simulation and experiment have proved the validity of this method. The combination of the windowed Fourier transform and the wavelet transform makes the extracted phase more precise than other methods.
52 citations
TL;DR: In this article, the Stockwell transform of a signal is defined as a hybrid of the Gabor transform and the wavelet transform, and continuous and discrete reconstruction formulas for a signal from its Stockwell Transform are established.
Abstract: The Stockwell transform of a signal, i.e., a function in L 2(ℝ), is defined as a hybrid of the Gabor transform and the wavelet transform. The continuous and discrete reconstruction formulas for a signal from its Stockwell transform are established.
37 citations
01 Jan 2007
TL;DR: In this paper, the double Sumudu transform of functions expressible as polynomials or convergent infinite series is derived, and the relationship of the former to the double Laplace transform is obtained.
Abstract: The double Sumudu transform of functions expressible as polynomials or convergent infinite series are derived. The relationship of the former to the double Laplace transform is obtained, and it turns out that they are theoretical dual to each other. The applicability of this relatively new transform is demonstrated using some special functions, which arise in the solution of evolution equations of population dynamics as well as partial differential equations.
Posted Content•
TL;DR: In this article, the authors considered a $q$-analogue of the Dunkl transform on the Riemann-Liouville and Weyl transform and established an inversion formula and proved a Plancherel theorem.
Abstract: In this paper, we consider a $q$-analogue of the Dunkl operator on $\mathbb{R}$, we define and study its associated Fourier transform which is a $q$-analogue of the Dunkl transform. In addition to several properties, we establish an inversion formula and prove a Plancherel theorem for this $q$-Dunkl transform. Next, we study the $q$-Dunkl intertwining operator and its dual via the $q$-analogues of the Riemann-Liouville and Weyl transforms. Using this dual intertwining operator, we provide a relation between the $q$-Dunkl transform and the $q^2$-analogue Fourier transform introduced and studied by R. Rubin.
TL;DR: This letter presents a new class of jacket transform not derived from either the WHT or the CWHT, which can be applied to any even length vector, is applicable to finite fields, and is useful for constructing error control codes.
Abstract: Jacket transforms are a class of transforms that are simple to calculate, easily inverted, and size-flexible. Previously reported jacket transforms were generalizations of the well-known Walsh-Hadamard transform (WHT) and the center-weighted Hadamard transform (CWHT). In this letter, we present a new class of jacket transform not derived from either the WHT or the CWHT. This class of transform can be applied to any even length vector, is applicable to finite fields, and is useful for constructing error control codes
TL;DR: In this article, a direct, effective and concise method is adopted to find out the Green's functions for infinite anisotropic piezoelectric media, where partial differential equations satisfied by the Greens functions turn into a set of inhomogeneous algebraic equations after using Fourier transform.
TL;DR: Fast algorithms for computing generalized discrete Hartley transforms in a sliding window are proposed, based on second-order recursive relations between subsequent local transform spectra.
Abstract: Fast algorithms for computing generalized discrete Hartley transforms in a sliding window are proposed. The algorithms are based on second-order recursive relations between subsequent local transform spectra. New efficient inverse algorithms for signal processing in a sliding window are also presented. The computational complexity of the algorithms is compared with that of known fast discrete Hartley transforms and running recursive algorithms
TL;DR: The computational complexity of the proposed method is lower than that of the traditional approach for length Nges8, and the arithmetic operations can be saved from 16% to 24% for N varying from 16 to 64.
Abstract: We present in this letter an efficient direct method for the computation of a length-N type-II generalized discrete Hartley transform (GDHT) when given two adjacent length-N/2 GDHT coefficients. The computational complexity of the proposed method is lower than that of the traditional approach for length Nges8. The arithmetic operations can be saved from 16% to 24% for N varying from 16 to 64. Furthermore, the new approach can be easily implemented
TL;DR: In this article, the authors characterize the support of the Fourier transform of the band-limited scaling function and give an approach to the construction of the scaling function based on the relations of translates of three point sets.
Journal Article•
TL;DR: In this paper, it was shown that the generalized backscattering transform (so outgoing angle ω = Sθ in terms of the incoming angle with S orthogonal and Id−S invertible) may be further restricted to give an entire, globally Fredholm operator on appropriate Sobolev spaces of potentials with compact support.
Abstract: Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle ω = Sθ in terms of the incoming angle with S orthogonal and Id−S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the modified backscattering map is a local isomorphism near elements of a generic set of potentials.
TL;DR: In this article, the authors proposed a new method for the exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane, which can be expressed as a convolution on the motion group.
Abstract: This paper presents a new method for the exponential Radon transform inversion based on the harmonic analysis of the Euclidean motion group of the plane. The proposed inversion method is based on the obser- vation that the exponential Radon transform can be modified to obtain a new transform, defined as the modified exponential Radon transform, that can be expressed as a convolution on the Euclidean motion group. The con- volution representation of the modified exponential Radon transform is block diagonalized in the Euclidean motion group Fourier domain. Further analysis of the block diagonal representation provides a class of relationships between the spherical harmonic decompositions of the Fourier transforms of the func- tion and its exponential Radon transform. These relationships and the block diagonalization lead to three new reconstruction algorithms. The proposed algorithms are implemented using the fast implementation of the Euclidean motion group Fourier transform and their performances are demonstrated in numerical simulations. Our study shows that convolution representation and harmonic analysis over groups motivates novel solutions for the inversion of the exponential Radon transform.
TL;DR: The proposed transform is compared with the classical multi-frequency HOS based on the Fourier transform and it is shown that the proposedtransform is more effective for processing of transient processes in comparison with the Classical transform.
Abstract: A new multidimensional time/multi-frequency higher order spectral(HOS) transform is proposed for transient signals with nonlinear polynomial variation of instantaneous frequency: the short time higher order chirp spectra (HOCS) based on the higher order chirp-Fourier transform and time-domain windowing technique. The proposed transform is compared with the classical multi-frequency HOS based on the Fourier transform. It is shown that the proposed transform is more effective for processing of transient processes in comparison with the classical transform.
TL;DR: The resulting area-efficient and scalable single-PE architecture is shown to yield a generic solution to the real-data FFT which is capable of achieving the throughput of the most advanced commercially available solutions for just a fraction of the silicon area and, for new applications, at negligible re-design cost.
Abstract: A highly parallel hardware solution to the regularised fast Hartley transform (FHT), an algorithm developed only recently for the efficient computation of the discrete Hartley transform (DHT) and hence of the real-data fast Fourier transform (FFT), is discussed. A drawback of conventional FHT algorithms lies in the loss of regularity arising from the need for two types of 'butterfly' for catering for fixed-radix formulations. A generic radix-4 double butterfly was therefore developed for the regularised FHT which overcame the problem in an elegant fashion. An architecture for the parallel computation of the generic double butterfly and the resulting regularised FHT is now discussed in some detail; this exploits a single high-performance processing element (PE) which yields an attractive solution, particularly when implemented with field-programmable gate array (FPGA) technology, which is both area-efficient and scalable in terms of transform length. High performance is achieved by having the PE able to process the input/ output data sets to the double butterfly in parallel, this in turn implying the need to be able to access simultaneously, and without conflict, both multiple data and multiple twiddle factors, or coefficients, from their respective memories. The resulting area-efficient and scalable single-PE architecture is shown to yield a generic solution to the real-data FFT which is capable of achieving the throughput of the most advanced commercially available solutions for just a fraction of the silicon area and, for new applications, at negligible re-design cost.
TL;DR: In this article, the deformation of a homogenous electric field caused by a long, thin inclined bed, which is of considerable importance in the exploration of ground water and minerals, is presented using the Hartley transform.
Abstract: Spectral analysis of the deformation of a homogenous electric field caused by a long, thin inclined bed, which is of considerable importance in the exploration of ground water and minerals, is presented using the Hartley transform. The Hartley transform is an alternative and real replacement for the well-known complex Fourier transform in the field of spectral analysis. The thickness of the bed and the inclination are given as functions of frequency by simple expressions. A theoretical example illustrates the method while the applicability is demonstrated by the field examples from the fractured crystalline basement complex in Burkina Faso, Africa and the Precambrian limestones of the Cuddapah basin, Andhra Pradesh, India. The results obtained by this method agree well with those of the drilling.
01 Jul 2007
TL;DR: A rotation-invariant texture analysis technique using Radon and Fourier transforms, which uses Radon transform to convert rotation to translation, then utilizes Fourier transform and takes the moduli of the Fouriertransform of these functions to make the translation invariant.
Abstract: Texture analysis is a basic issue in image processing and computer vision, and how to attain the Rotation-invariant texture characterization is a key problem. This paper proposes a rotation-invariant texture analysis technique using Radon and Fourier transform. This method uses Radon transform to convert rotation to translation, then utilizes the Fourier transform and takes the modules of the Fourier transform of these functions to make the translation invariant. A k-nearest-neighbor rule is employed to classify textures images. The proposed method is robust to additive white noise as a result of summing pixel values to generate projections in the Radon transform step. To test and evaluate the method, several different kinds of experiments are employed. Experiments results show the feasibility of the proposed method and its robustness to additive white noise.
TL;DR: By reconfiguring the (decimation-in-frequency)-fast Fourier transform (DIF-FFT) framework to adopt different internal calculations, the efficient calculation of radix-2 multiresolution Fouriertransform (MFT) is discussed, which can also be regarded as a collection of short-time Fourier transforms (STFTs) with multiple 2-based window sizes.
TL;DR: It is numerically demonstrate that the image encryption algorithm with intensity random filtering in the Hartley domain proposed in a recent Letter by Chen and Zhao has the problem of low security.
Abstract: We numerically demonstrate that the image encryption algorithm with intensity random filtering in the Hartley domain proposed in a recent Letter by Chen and Zhao [Opt. Lett.31, 3438 (2006)] has the problem of low security. Original image information can be visually revealed only by an inverse Hartley transform without any decryption key.
Patent•
04 Dec 2007TL;DR: The order-16 transform method as discussed by the authors is used in video coding, which involves expanding an order-8 transform by generating an order16 integer matrix and a scaling matrix, which is used for video coding.
Abstract: The invention is used in video coding. Systems, apparatuses and methods for processing an order-16 integer transform from an order-8 transform are provided. The order-16 transform method involves expanding an order-8 transform by generating an order-16 integer matrix and a scaling matrix.
01 Oct 2007
TL;DR: In this paper, the DHT-based interpolation formula is applied to design fractional delay FIR filter by using suitable index mapping, and the filter coefficients are easily computed because closed-form design is obtained.
Abstract: In this paper, the design of fractional delay FIR filter is investigated. First, the interpolation formula of discrete-time sequence is derived by using discrete Hartley transform (DHT). Then, the DHT-based interpolation formula is applied to design fractional delay FIR filter by using suitable index mapping. The filter coefficients are easily computed because closed-form design is obtained. Finally, design examples are demonstrated to show the proposed method has smaller design error than the conventional Lagrange fractional delay FIR filter using the same design parameters
TL;DR: A comparison of digital Fourier and Walsh versions of the named holographic scheme is conducted by means of their associative properties and the retrieving of information stored in the hologram using parts of different amounts of the original data is considered.
Abstract: A digital implementation of the Fourier holography scheme with identical reference and signal beams is considered. Investigations of the properties of this scheme were reported on in a number of papers. At the same time a concept of virtual optics is widely used in data encryption and data processing, whereas digital implementation of the Fourier holography scheme may be provided using not only discrete Fourier transform. Another discrete digital transform may be used as soon as it has several properties similar to the Fourier transform. The Walsh transform is the subject to consider in this work. A comparison of digital Fourier and Walsh versions of the named holographic scheme is conducted by means of their associative properties. Considered is the retrieving of information stored in the hologram using parts of different amounts of the original data. Comparison is provided with the two types of data stored: a raster image representing visual information and an image representing a set of data bits. Comparable parameters are the mean image contrast for visual data image and the bit detection accuracy for a set of data bits.
TL;DR: Methods for the parallel computation of a multidimensional hypercomplex discrete Fourier transform (HDFT) are considered and additional possibilities for increasing the efficiency of the algorithm are provided by the natural parallelism of the multiddimensional Cooley-Tukey scheme.
Abstract: Methods for the parallel computation of a multidimensional hypercomplex discrete Fourier transform (HDFT) are considered. The basic idea consists in the application of the properties of the hypercomplex algebra in which this transform is performed. Additional possibilities for increasing the efficiency of the algorithm are provided by the natural parallelism of the multidimensional Cooley-Tukey scheme.
TL;DR: In this paper, the authors investigated various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous and studied the relationships involving this transform and its first variation.
Abstract: Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebra S of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous. Also we study the relationships involving this transform and the first variation.