Showing papers on "Hartley transform published in 2006"
TL;DR: A new method for image encryption based on Hartley transforms that is a real transform and can be realized by spatially incoherent or coherent illumination is proposed and computer simulations prove it is possible.
Abstract: We propose a new method for image encryption based on Hartley transforms that is a real transform and can be realized by spatially incoherent or coherent illumination. The proposed optical implementation is based on a Michelson-type interferometer in which the pure random intensity is distributed at the Hartley plane in encryption. Computer simulations prove it is possible. A Hartley hologram method is also given and described to resolve the sign ambiguity problem that would be encountered in image reconstruction.
173 citations
TL;DR: The well-known Shannon sampling theorem and previously developed sampling criteria for Fresnel and fractional Fourier transformed signals are shown to be a special cases of the theorem developed here.
170 citations
TL;DR: The sampling and reconstruction formulas are deduced, together with the construction methodology for the multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.
Abstract: As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.
133 citations
TL;DR: This work shows the application of new approach to the 3D HNCO spectrum acquired for protein sample with radial and spiral time domain sampling and enables one to Fourier transform arbitrarily sampled time domain and thus allows for analysis of high dimensionality spectra acquired in a short time.
127 citations
TL;DR: The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms and the double random phase encoding in the MPDFRFT domain significantly enhances data security.
Abstract: The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.
103 citations
TL;DR: An in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented, finding a closed form for the integral kernel may be obtained, leading to further properties, both in the L1 and in theL2 context.
Abstract: Recently several generalizations to higher dimension of the Fourier transform using Clifford algebra have been introduced, including the Clifford-Fourier transform by the authors, defined as an operator exponential with a Clifford algebra-valued kernel.
In this paper an overview is given of all these generalizations and an in depth study of the two-dimensional Clifford-Fourier transform of the authors is presented. In this special two-dimensional case a closed form for the integral kernel may be obtained, leading to further properties, both in the L 1 and in the L 2 context. Furthermore, based on this Clifford-Fourier transform Clifford-Gabor filters are introduced.
91 citations
TL;DR: The algorithm is an extension of the traditional wavelet transform; the mother wavelet and normalization parameter are selected based on the characteristics of optical interferograms and to reduce the processing time, a fast Fourier transform scheme is employed.
Abstract: An advanced continuous wavelet transform algorithm for digital interferogram analysis and processing is proposed. The algorithm is an extension of the traditional wavelet transform; the mother wavelet and normalization parameter are selected based on the characteristics of optical interferograms. To reduce the processing time, a fast Fourier transform scheme is employed to implement the wavelet transform calculation. The algorithm is simple and is a robust tool for interferogram filtering and for whole-field fringe and phase information detection. The concept is verified by computer simulation and actual experimental interferogram analysis.
79 citations
Book•
15 Aug 2006
TL;DR: The Fourier Transform The Laplace Transform The Mellin Transform The Stieltjes Transform The Hilbert Transform Bessel Transforms The Mehler-Fock Transform Finite Integral Transforms OPERATIONAL CALCULUS Introduction The Theorem of Titchmarsh Operators Bases of the Operator Analysis Operators Reducible to Functions Application of Operational Calculus GeneralIZed Functions - Functional Approach Generalized Functions - Sequential Approach Delta Sequences Convergent Sequences Local Properties Irregular Operations Hilbert Transform and Multiplication Forms as discussed by the authors
Abstract: INTEGRAL TRANSFORMS Introduction to Operational Calculus Integral Transforms - Introductory Remarks The Fourier Transform The Laplace Transform The Mellin Transform The Stieltjes Transform The Hilbert Transform Bessel Transforms The Mehler-Fock Transform Finite Integral Transforms OPERATIONAL CALCULUS Introduction The Theorem of Titchmarsh Operators Bases of the Operator Analysis Operators Reducible to Functions Application of Operational Calculus GENERALIZED FUNCTIONS Introduction Generalized Functions - Functional Approach Generalized Functions - Sequential Approach Delta Sequences Convergent Sequences Local Properties Irregular Operations Hilbert Transform and Multiplication Forms
77 citations
Journal Article•
TL;DR: In this article, a new proof of the invariance of the Hankel transform under the binomial transform of a sequence was given, leading to three variations of the k-binomial transform.
Abstract: We give a new proof of the invariance of the Hankel transform under the binomial transform of a sequence. Our method of proof leads to three variations of the binomial transform; we call these the k-binomial transforms. We give a simple means of constructing these transforms via a triangle of numbers. We show how the exponential generating function of a sequence changes after our transforms are applied, and we use this to prove that several sequences in the On-Line Encyclopedia of Integer Sequencesarerelatedviaourtransforms. Intheprocess,weprovethreeconjecturesin theOEIS.AddressingaquestionofLayman,wethenshowthattheHankeltransform of a sequence is invariant under one of our transforms, and we show how the Hankel transform changes after the other two transforms are applied. Finally, we use these results to determine the Hankel transforms of severalinteger sequences.
59 citations
16 Oct 2006
TL;DR: The linear canonical transform (LCT) as mentioned in this paper provides a unified framework for studying the behavior of many practical transforms and system responses in optics and engineering in general, and provides a powerful tool for design and analysis of the characteristics of optical systems.
Abstract: The linear canonical transform (LCT), is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms and operators in engineering and physics such as the Fourier transform, fractional Fourier transform (FRFT), Fresnel transform (FRST), time scaling, chirping, and others. Therefore the LCT provides a unified framework for studying the behavior of many practical transforms and system responses in optics and engineering in general. From the system‐engineering point of view the LCT provides a powerful tool for design and analysis of the characteristics of optical systems. Despite this fact only few authors take advantage of the powerful and general LCT theory for analysis and design of optical systems. In this paper we review some important properties about the continuous LCT and we present some new results regarding the discretization and computation of the LCT.
48 citations
01 Jan 2006
TL;DR: The definition of LCT and some special cases are given at first, followed by its properties as listed, and the discrete linear canonical transform is introduced.
Abstract: As an emerging tool for signal processing,the linear canonical transform(LCT) proves itself to be more general and flexible than the Fourier transform as well as the fractional Fourier transform.So it can slove problems that can't be dealt with well by the latter.In this paper,the definition of LCT and some special cases are given at first,followed by its properties as listed.Besides,the discrete linear canonical transform is introduced.The implication of LCT is illustrated finally,displaying(LCT's) potentials and capabilities in the field of signal processing.
TL;DR: The Hilbert Huang transform (HHT) is introduced as an efficient means for analysis of bioacoustical signals and shows that HHT is a viable alternative to the wavelet transform.
Abstract: While marine mammals emit variant signals (in time and frequency), the Fourier spectrogram appears to be the most widely used spectral estimator. In certain cases, this approach is suboptimal, particularly for odontocete click analysis and when the signal-to-noise ratio varies during the continuous recordings. We introduce the Hilbert Huang transform (HHT) as an efficient means for analysis of bioacoustical signals. To evaluate this method, we compare results obtained from three time-frequency representations: the Fourier spectrogram, the wavelet transform, and the Hilbert Huang transform. The results show that HHT is a viable alternative to the wavelet transform. The chosen examples illustrate certain advantages. (1) This method requires the calculation of the Hilbert transform; the time-frequency resolution is not restricted by the uncertainty principle; the frequency resolution is finer than with the Fourier spectrogram. (2) The original signal decomposition into successive modes is complete. If we were to multiply some of these modes, this would contribute to attenuate the presence of noise in the original signal and to being able to select pertinent information. (3) Frequency evolution for each mode can be analyzed as one-dimensional (1D) signal. We not need a complex 2D post-treatment as is usually required for feature extraction.
TL;DR: This letter looks into efficient DHT schemes yielding the same throughput as the DFT counterparts and provides simulations assuming perfect channel state information (CSI)
Abstract: Orthogonal-frequency-division-multiplex (OFDM) and single-carrier (SC) frequency domain structures are simple equalization schemes that make use of the discrete Fourier transform (DFT) diagonalization properties. This letter approaches these problems within a well-known matrix algebra that allows for straightforward extensions of recent results on real trigonometric transforms, namely, the discrete cosine (DCT), sine (DST), as well as new Hartley transform (DHT) schemes. They are especially useful for real modulations and can outperform the corresponding DFT-based schemes when the channel memory is smaller than the introduced redundancy. We further look into efficient DHT schemes yielding the same throughput as the DFT counterparts and provide simulations assuming perfect channel state information (CSI)
TL;DR: 3D discrete Hartley transform is applied for the compression of two medical modalities, namely, magnetic resonance images and X-ray angiograms and the performance results are compared with those of 3-D discrete cosine and Fourier transforms using the parameters such as PSNR and bit rate.
Journal Article•
TL;DR: In this paper, it is shown that it is possible to obtain a good quality approximate inverse to the Constant Q transform provided that the signal to be inverted has a sparse representation in the Discrete Fourier Transform domain.
Abstract: The Constant Q transform has found use in the analysis of musical signals due to its logarithmic frequency resolution. Unfortunately, a considerable drawback of the Constant Q transform is that there is no inverse transform. Here we show it is possible to obtain a good quality approximate inverse to the Constant Q transform provided that the signal to be inverted has a sparse representation in the Discrete Fourier Transform domain. This inverse is obtained through the use of `0 and `1 minimisation approaches to project the signal from the constant Q domain back to the Discrete Fourier Transform domain. Once the signal has been projected back to the Discrete Fourier Transform domain, the signal can be recovered by performing an inverse Discrete Fourier Transform. 1. THE CONSTANT Q TRANSFORM The Constant Q transform (CQT) was derived by Brown as a means of creating a log-frequency resolution spectrogram [1]. This has considerable advantages for the analysis of musical signals, as the frequency resolution can be set to match that of the equal tempered scale used in western music, where the frequencies are geometrically spaced, as opposed to the linear spacing that occurs in the discrete Fourier transform (DFT). The frequency components of the CQT have a constant ratio of center frequency to resolution, as opposed to the constant frequency difference and constant resolution of the DFT. This constant ratio results in a constant pattern for the spectral components making up notes played on a given instrument, and this has been used to attempt sound source separation of pitched instruments from both single channel and multi-channel mixtures of instruments[2],[3]. Given an inital minimum frequency f0 for the CQT, the center frequencies for each band can be obtained from: fk = f02 k b (k = 0, 1, ...) (1) where b is the number of bins per octave. The fixed ratio of center frequency to bandwidth is then given by Q = ( 2 1 b − 1 )−1 (2) The desired bandwidth of each frequency band is FitzGerald et al. Towards an ICQT then obtained by choosing a window of length
01 Oct 2006
TL;DR: The proposed transforms smartly solve two inherent problems of the Radon transform in rotation estimation in digital images, i.e., direction-dependency and nonhomogeneity, that come from the different numbers of pixels projected on a line for different directions and/or coordinates of a direction.
Abstract: This paper presents two novel transforms based on the discrete Radon transform. The proposed transforms smartly solve two inherent problems of the Radon transform in rotation estimation in digital images, i.e., direction-dependency and nonhomogeneity, that come from the different numbers of pixels projected on a line for different directions and/or coordinates of a direction. While the first transform considers the sample mean operator on the same sets of pixels for a direction instead of summation in the discrete Radon transform, the second transform uses the mean operator on sets of pixels with the equal number of elements. In order to show the efficiency of the proposed transforms, we apply them on image collections from the Brodatz album for estimating the directional information. Experimental results show a significant increase in correct estimation as well as in the processing time compared to the conventional Radon transform.
TL;DR: An adaptive windowed Fourier transform method in 3-D measurement based on a wavelet transform is proposed, in which, by applying aWavelet ridge, a series of scaling factors are calculated to determine the series of prime windows needed in the windowed Fresnel transform method.
Abstract: An adaptive windowed Fourier transform method in 3-D measurement based on a wavelet transform is proposed, in which, by applying a wavelet ridge, a series of scaling factors are calculated to determine the series of prime windows needed in the windowed Fourier transform method. Because the spectrum of each local fringe is simpler than that of the whole fringe, even though there is frequency aliasing as far as the whole fringe is concerned, the fundamental spectrum may separate into components in each local fringe. It is easy to filter out one of the fundamental frequency components from the local spectra. Adding these local fundamental components, the full fundamental component can be obtained correctly. The advantage of the method is that it not only eliminates the frequency aliasing, but also obtains the modulation distribution function to guide phase unwrapping.
TL;DR: In this article, a real color fractional Fourier transform hologram (FLFTH) was proposed for anti-counterfeiting, which is based on the FFTH.
TL;DR: In this article, the q-wavelets and the qwavelet transforms were studied for a fixed fix q ∈]0, 1[] 0, 1] 1.
Abstract: This paper aims to study the q-wavelets and the q-wavelet transforms, using only the q-Jackson integrals and the q-cosine Fourier transform, for a fix q ∈]0, 1[. For this purpose, we shall attempt to extend the classical theory by giving their q-analogues.
TL;DR: This algorithm offers substantial reductions in the numbers of multiplications, additions, data transfers, and twiddle factor evaluations or accesses to the look-up table, without a significant increase in the structural complexity compared to that of the existing 3-D vector radix FHT algorithm.
Abstract: In this paper, we propose a three-dimensional (3-D) split vector-radix fast Hartley transform (FHT) algorithm. The main idea behind the proposed algorithm is that the radix-2/4 approach is introduced in the decomposition of the 3-D discrete Hartley transform by using an appropriate index mapping and the Kronecker product. This provides an algorithm based on a mixture of radix-(2times2times2) and radix-(4times4times4) index maps and has a butterfly that is characterized by simple closed-form expressions. This algorithm offers substantial reductions in the numbers of multiplications, additions, data transfers, and twiddle factor evaluations or accesses to the look-up table, without a significant increase in the structural complexity compared to that of the existing 3-D vector radix FHT algorithm
16 Jan 2006
TL;DR: A generic double butterfly is developed for the radix-4 FHT which overcomes the need for two sizes of ‘butterfly’ for efficient fixed-radix implementations and lends itself naturally to parallelisation and to mapping onto a regular computational structure for implementation with algorithmically specialised hardware.
Abstract: The paper describes the design and parallel computation of a regularised fast Hartley transform (FHT), to be used for computation of the discrete Fourier transform (DFT) of real-valued data. For the processing of such data, the FHT has attractions over the fast Fourier transform (FFT) in terms of reduced arithmetic operation counts and reduced memory requirement, whilst its bilateral property means it may be straightforwardly applied to both forward and inverse DFTs. A drawback, however, of conventional FHT algorithms lies in the loss of regularity arising from the need for two sizes of ‘butterfly’ for efficient fixed-radix implementations. A generic double butterfly is therefore developed for the radix-4 FHT which overcomes the problem in an elegant fashion. The result is a recursive single-butterfly solution, referred to as the regularised FHT, which lends itself naturally to parallelisation and to mapping onto a regular computational structure for implementation with algorithmically specialised hardware.
Dissertation•
23 Nov 2006
TL;DR: This paper presents a probabilistic procedure that can be used to estimate the intensity of the response of the immune system to carbon monoxide poisoning.
Abstract: Thapar Institute of Engineering & Technology, Department of Electronics and Communication Engineering
TL;DR: This paper describes periodic functions for the one- and two-dimensional Fourier transforms and shows that by processing only one splitting-signal, one can achieve an enhancement that may exceed results of traditional methods of image enhancement.
Abstract: The analysis of the mathematical structure of the integral Fourier transform shows that the transform can be split and represented by certain sets of frequencies as coefficients of Fourier series of periodic functions in the interval $$[0,2\pi)$$ . In this paper we describe such periodic functions for the one- and two-dimensional Fourier transforms. The approximation of the inverse Fourier transform by periodic functions is described. The application of the new representation is considered for the discrete Fourier transform, when the transform is split into a set of short and separable 1-D transforms, and the discrete signal is represented as a set of short signals. Properties of such representation, which is called the paired representation, are considered and the basis paired functions are described. An effective application of new forms of representation of a two-dimensional image by splitting-signals is described for image enhancement. It is shown that by processing only one splitting-signal, one can achieve an enhancement that may exceed results of traditional methods of image enhancement.
TL;DR: This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.
Abstract: This article presents an overview of a transform method for solving linear and integrable nonlinear partial differential equations. This new transform method, proposed by Fokas, yields a generalization and unification of various fundamental mathematical techniques and, in particular, it yields an extension of the Fourier transform method.
TL;DR: In this article, the authors investigated the behavior of Fourier transformation on the W-spaces of W-type spaces and applied it to the theory of partial differential equations, where the Fourier transform was applied to the problem of W -spaces.
Abstract: The spaces of W-type were studied by I.M. Gel’fand and G.E. Shilov [22]. They investigated the behaviour of Fourier transformation on theW-spaces. AlsoW-spaces are applied to the theory of partial differential equations.
25 Sep 2006
TL;DR: The use of the Hilbert transform (HT) in the area of electrocardiogram analysis is investigated and four different approaches, including the Time-Domain approach, the Frequency-Domain approaches, the Boche approach and the Remez filter approach for calculating the Hilbert Transform of an ECG wave are discussed.
Abstract: I agree that the Library, University of Saskatchewan, may make this thesis freely available for inspection. I further agree that permission for copying of this thesis for scholarly purpose may be granted to the professor or professors who supervised the thesis work recorded herein or, in their absence, by the Head of the Department or the Dean of the College in which the thesis work was done. It is understood that due recognition will be given to me and to the University of Saskatchewan in any use of the material in this thesis. Copying or publication or any other use of this thesis for financial gain without approval by the University of Saskatchewan and my written permission is prohibited. ii ACKNOWLEDGEMENTS I wish to express my gratitude to the following people who not only made this thesis possible but also an enjoyable experience: Dr. Ronald J. Bolton: my supervisor, for his valuable guidance, criticisms and consistent encouragement throughout the course of this research work. My husband, Zhanghai Wang: for his love and encouragement. My parents, my sister Xiangrong Wang and brother-in-law Xianggang Yu: for the support they provided to me. Xing: for making me feel welcome. They will always be special friends in my life. The Department of Electrical Engineering: for supplying the opportunity to study in Canada and the necessary facilities with which to work. iii ABSTRACT Many people have abnormal heartbeats from time to time. A Holter monitor is a device used to record the electrical impulses of the heart when people do ordinary activities. Holter monitoring systems that can record heart rate and rhythm when you feel chest pain or symptoms of an irregular heartbeat (called an arrhythmia) and automatically perform electrocardiogram (ECG) signal analysis are desirable. The use of the Hilbert transform (HT) in the area of electrocardiogram analysis is investigated. A property of the Hilbert transform, i.e., to form the analytic signal, was used in this thesis. Subsequently pattern recognition can be used to analyse the ECG data and lossless compression techniques can be used to reduce the ECG data for storage. The thesis discusses one part of the Holter Monitoring System, Input processing. Four different approaches, including the Time-Domain approach, the Frequency-Domain approach, the Boche approach and the Remez filter approach for calculating the Hilbert transform of an ECG wave are discussed in this thesis. By comparing them from the running time and the …
01 Jan 2006
TL;DR: In this article, the authors defined the Hilbert transform in the linear canonical transform domain (LCT), which is the generalized form of the Fourier transform and the fractional transform.
Abstract: Hilbert transform(HT) is an important signal processing tool that is used in many applications such as modulations,edge detection and filter design in signal processingAn analytic signal and its original signal are connected by the HTAt the same time,the linear canonical transform(LCT) is the generalized form of Fourier transform and the fractional Fourier transform,it can be considered as a general linear transformThe definition and properties of Hilbert transform in linear canonical transform domain were proposedSome important properties of this kind of Hilbert transform were given,and it is shown that the newly defined Hilbert transform has some similar properties with the Hilbert transform in the traditional Fourier transform domain
TL;DR: The authors introduce the exponential integral transform and the complementary error transform, then it is shown that the third iterate of the Laplace transform is the exponential Integral transform and a modified third iterates of theLaplace transform are the complementaryerror transform.
TL;DR: This paper studies some of the practical limits introduced by using a converging spherical lens of finite aperture to produce the illuminating field in the implementation of the SOFT and shows that the worst-case errors in the resulting SOFT can be quantified and avoided.