Showing papers on "Hartley transform published in 2011"
TL;DR: A discrete-time wavelet transform for which the Q-factor is easily specified and the transform can be tuned according to the oscillatory behavior of the signal to which it is applied, based on a real-valued scaling factor.
Abstract: This paper describes a discrete-time wavelet transform for which the Q-factor is easily specified. Hence, the transform can be tuned according to the oscillatory behavior of the signal to which it is applied. The transform is based on a real-valued scaling factor (dilation-factor) and is implemented using a perfect reconstruction over-sampled filter bank with real-valued sampling factors. Two forms of the transform are presented. The first form is defined for discrete-time signals defined on all of Z. The second form is defined for discrete-time signals of finite-length and can be implemented efficiently with FFTs. The transform is parameterized by its Q-factor and its oversampling rate (redundancy), with modest oversampling rates (e.g., three to four times overcomplete) being sufficient for the analysis/synthesis functions to be well localized.
500 citations
01 Jan 2011
TL;DR: In this article, a new integral transform namely Elzaki transform was applied to solve linear ODEs with constant coefficients, which was called ELZAKI transform and was used to solve ODE with constant ODE.
Abstract: In this paper a new integral transform namely Elzaki transform was applied to solve linear ordinary differential equations with constant coefficients.
183 citations
TL;DR: The aim of this study is to show how the F-transform technique can be generalized from the cases of constant components to the case of polynomial components.
134 citations
TL;DR: The aim of this monograph is to clarify the role of Fourier Transforms in the development of Functions of Complex Numbers and to propose a procedure called the Radon Transform, which is based on the straightforward transformation of the Tournaisian transform.
Abstract: Series Editor s Preface. Preface. 1 Introduction. 1.1 Signals, Operators, and Imaging Systems. 1.2 The Three Imaging Tasks. 1.3 Examples of Optical Imaging. 1.4 ImagingTasks inMedical Imaging. 2 Operators and Functions. 2.1 Classes of Imaging Operators. 2.2 Continuous and Discrete Functions. Problems. 3 Vectors with Real-Valued Components. 3.1 Scalar Products. 3.2 Matrices. 3.3 Vector Spaces. Problems. 4 Complex Numbers and Functions. 4.1 Arithmetic of Complex Numbers. 4.2 Graphical Representation of Complex Numbers. 4.3 Complex Functions. 4.4 Generalized Spatial Frequency Negative Frequencies. 4.5 Argand Diagrams of Complex-Valued Functions. Problems. 5 Complex-Valued Matrices and Systems. 5.1 Vectors with Complex-Valued Components. 5.2 Matrix Analogues of Shift-Invariant Systems. 5.3 Matrix Formulation of ImagingTasks. 5.4 Continuous Analogues of Vector Operations. Problems. 6 1-D Special Functions. 6.1 Definitions of 1-D Special Functions. 6.2 1-D Dirac Delta Function. 6.3 1-D Complex-Valued Special Functions. 6.4 1-D Stochastic Functions Noise. 6.5 Appendix A: Area of SINC[x] and SINC2[x]. 6.6 Appendix B: Series Solutions for Bessel Functions J0[x] and J1[x]. Problems. 7 2-D Special Functions. 7.1 2-D Separable Functions. 7.2 Definitions of 2-D Special Functions. 7.3 2-D Dirac Delta Function and its Relatives. 7.4 2-D Functions with Circular Symmetry. 7.5 Complex-Valued 2-D Functions. 7.6 Special Functions of Three (orMore) Variables. Problems. 8 Linear Operators. 8.1 Linear Operators. 8.2 Shift-Invariant.Operators. 8.3 Linear Shift-Invariant (LSI) Operators. 8.4 Calculating Convolutions. 8.5 Properties of Convolutions. 8.6 Autocorrelation. 8.7 Crosscorrelation. 8.8 2-DLSIOperations. 8.9 Crosscorrelations of 2-D Functions. 8.10 Autocorrelations of 2-D.Functions. Problems. 9 Fourier Transforms of 1-D Functions. 9.1 Transforms of Continuous-Domain Functions. 9.2 Linear Combinations of Reference Functions. 9.3 Complex-Valued Reference Functions. 9.4 Transforms of Complex-Valued Functions. 9.5 Fourier Analysis of Dirac Delta Functions. 9.6 Inverse Fourier Transform. 9.7 Fourier Transforms of 1-D Special Functions. 9.8 Theorems of the Fourier Transform. 9.9 Appendix: Spectrum of Gaussian via Path Integral. Problems. 10 Multidimensional Fourier Transforms. 10.1 2-D Fourier Transforms. 10.2 Spectra of Separable 2-D Functions. 10.3 Theorems of 2-D Fourier Transforms. Problems. 11 Spectra of Circular Functions. 11.1 The Hankel Transform. 11.2 Inverse Hankel Transform. 11.3 Theorems of Hankel Transforms. 11.4 Hankel Transforms of Special Functions. 11.5 Appendix: Derivations of Equations (11.12) and (11.14). Problems. 12 The Radon Transform. 12.1 Line-Integral Projections onto Radial Axes. 12.2 Radon Transforms of Special Functions. 12.3 Theorems of the Radon Transform. 12.4 Inverse Radon Transform. 12.5 Central-Slice Transform. 12.6 Three Transforms of Four Functions. 12.7 Fourier and Radon Transforms of Images. Problems. 13 Approximations to Fourier Transforms. 13.1 Moment Theorem. 13.2 1-D Spectra via Method of Stationary Phase. 13.3 Central-Limit Theorem. 13.4 Width Metrics and Uncertainty Relations. Problems. 14 Discrete Systems, Sampling, and Quantization. 14.1 Ideal Sampling. 14.2 Ideal Sampling of Special Functions. 14.3 Interpolation of Sampled Functions. 14.4 Whittaker Shannon Sampling Theorem. 14.5 Aliasingand Interpolation. 14.6 Prefiltering to Prevent Aliasing. 14.7 Realistic Sampling. 14.8 Realistic Interpolation. 14.9 Quantization. 14.10 Discrete Convolution. Problems. 15 Discrete Fourier Transforms. 15.1 Inverse of the Infinite-Support DFT. 15.2 DFT over Finite Interval. 15.3 Fourier Series Derived from Fourier Transform. 15.4 Efficient Evaluation of the Finite DFT. 15.5 Practical Considerations for DFT and FFT. 15.6 FFTs of 2-D Arrays. 15.7 Discrete Cosine Transform. Problems. 16 Magnitude Filtering. 16.1 Classes of Filters. 16.2 Eigenfunctions of Convolution. 16.3 Power Transmission of Filters. 16.4 Lowpass Filters. 16.5 Highpass Filters. 16.6 Bandpass Filters. 16.7 Fourier Transform as a Bandpass Filter. 16.8 Bandboost and Bandstop Filters. 16.9 Wavelet Transform. Problems. 17 Allpass (Phase) Filters. 17.1 Power-Series Expansion for Allpass Filters. 17.2 Constant-Phase Allpass Filter. 17.3 Linear-Phase Allpass Filter. 17.4 Quadratic-Phase Filter. 17.5 Allpass Filters with Higher-Order Phase. 17.6 Allpass Random-Phase Filter. 17.7 Relative Importance of Magnitude and Phase. 17.8 Imaging of Phase Objects. 17.9 Chirp Fourier Transform. Problems. 18 Magnitude Phase Filters. 18.1 Transfer Functions of Three Operations. 18.2 Fourier Transform of Ramp Function. 18.3 Causal Filters. 18.4 Damped Harmonic Oscillator. 18.5 Mixed Filters with Linear or Random Phase. 18.6 Mixed Filter with Quadratic Phase. Problems. 19 Applications of Linear Filters. 19.1 Linear Filters for the Imaging Tasks. 19.2 Deconvolution Inverse Filtering . 19.3 Optimum Estimators for Signals in Noise. 19.4 Detection of Known Signals Matched Filter. 19.5 Analogies of Inverse and Matched Filters. 19.6 Approximations to Reciprocal Filters. 19.7 Inverse Filtering of Shift-Variant Blur. Problems. 20 Filtering in Discrete Systems. 20.1 Translation, Leakage, and Interpolation. 20.2 Averaging Operators Lowpass Filters. 20.3 Differencing Operators Highpass Filters. 20.4 Discrete Sharpening Operators. 20.5 2-DGradient. 20.6 Pattern Matching. 20.7 Approximate Discrete Reciprocal Filters. Problems. 21 Optical Imaging in Monochromatic Light. 21.1 Imaging Systems Based on Ray Optics Model. 21.2 Mathematical Model of Light Propagation. 21.3 Fraunhofer Diffraction. 21.4 Imaging System based on Fraunhofer Diffraction. 21.5 Transmissive Optical Elements. 21.6 Monochromatic Optical Systems. 21.7 Shift-Variant Imaging Systems. Problems. 22 Incoherent Optical Imaging Systems. 22.1 Coherence. 22.2 Polychromatic Source Temporal Coherence. 22.3 Imaging in Incoherent Light. 22.4 System Function in Incoherent Light. Problems. 23 Holography. 23.1 Fraunhofer Holography. 23.2 Holography in Fresnel Diffraction Region. 23.3 Computer-Generated Holography. 23.4 Matched Filtering with Cell-Type CGH. 23.5 Synthetic-Aperture Radar (SAR). Problems. References. Index.
80 citations
TL;DR: Fast Fourier transform (FFT) algorithm can be introduced into the calculation of convolution format of gyrator transform in the discrete case by using convolution operation.
65 citations
TL;DR: A novel multi-image encryption and decryption algorithm based on Fourier transform and fractional Fourier transforms that has features of enhancement in decryption accuracy and high optical efficiency is presented.
47 citations
TL;DR: In this article, Brackx et al. studied the Fourier transform of hypercomplex signals and their Fourier transforms from general principles, using four different yet equivalent definitions of the classical Fourier Transform.
Abstract: Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of the classical Fourier transform. This is applied to the so-called Clifford-Fourier transform (see Brackx et al., J. Fourier Anal. Appl. 11:669–681, 2005). The integral kernel of this transform is a particular solution of a system of PDEs in a Clifford algebra, but is, contrary to the classical Fourier transform, not the unique solution. Here we determine an entire class of solutions of this system of PDEs, under certain constraints. For each solution, series expressions in terms of Gegenbauer polynomials and Bessel functions are obtained. This allows to compute explicitly the eigenvalues of the associated integral transforms. In the even-dimensional case, this also yields the inverse transform for each of the solutions. Finally, several properties of the entire class of solutions are proven.
42 citations
TL;DR: In this article, a method which determines a distribution from the knowledge of its q-Fourier transform and some supplementary information is presented, which conveniently extends the inverse of the standard Fourier transform, and is therefore expected to be very useful in many complex systems.
42 citations
TL;DR: This paper shows that the Discrete Fuzzy Transform is invariant with respect to the interpolating and least-squares approximation, and defines the geometry- and confidence-driven Discrete fuzzy Transforms, which take into account the intrinsic geometry and the confidence weights associated to the data.
37 citations
TL;DR: A new reciprocal-orthogonal parametric discrete Fourier transform (DFT) is proposed by appropriately replacing some specific twiddle factors in the kernel of the classical DFT by independent parameters that can be chosen arbitrarily from the complex plane.
Abstract: In this paper, we propose a new reciprocal-orthogonal parametric discrete Fourier transform (DFT) by appropriately replacing some specific twiddle factors in the kernel of the classical DFT by independent parameters that can be chosen arbitrarily from the complex plane. A new class of parametric unitary transforms can be obtained from the proposed transform by choosing all its independent parameters from the unit circle. One of the special cases of this class is then exploited for developing a new one-parameter involutory discrete Hartley transform (DHT). The proposed parametric DFT and DHT can be computed using the existing fast algorithms of the DFT and DHT, respectively, with computational complexities similar to those of the latter. Indeed, for some special cases, the proposed transforms require less number of operations. In view of the fact that the transforms of small sizes are used in some image and video compression techniques and employed as building blocks for larger size transform algorithms, we develop new algorithms for the proposed small-size transforms. The proposed parametric DFT and DHT, in view of the introduction of the independent parameters, offer more flexibility in achieving better performance compared to the classical DFT and DHT. As examples of possible applications of the proposed transforms, image compression, Wiener filtering, and spectral analysis are considered.
33 citations
TL;DR: The computer simulation results show that the proposed image encryption algorithm is feasible, secure and robust to noise attack and occlusion.
TL;DR: An O(NlogN) algorithm to compute the LCT is obtained by using a chirp-FFT-chirp transformation yielded by a convergent quadrature formula for the fractional Fourier transform to give a unitary discrete LCT in closed form.
01 Jan 2011
TL;DR: A good starting point is the finite Fourier transform that underpins the contents of the first thirteen chapters of the book as mentioned in this paper, which is the basis for the present paper's analysis.
Abstract: A good starting point is the finite Fourier transform that underpins the contents of the first thirteen chapters of the book.
TL;DR: In this paper, the Fourier-Bessel transform has been shown to have support of finite measure and the dilation of a non-zero function and its Fourier Bessel transform cannot both be linearly independent.
01 Jan 2011
TL;DR: Kekre transform surpasses all other discussed transforms in performance with highest precision and recall values for fractional coefficients and computation is lowered by 94.08% as compared to Cosine or Sine or Hartlay transforms.
Abstract: The desire of better and faster retrieval techniques has always fuelled to the research in content based image retrieval (CBIR). The extended comparison of innovative content based image retrieval (CBIR) techniques based on feature vectors as fractional coefficients of transformed images using various orthogonal transforms is presented in the paper. Here the fairly large numbers of popular transforms are considered along with newly introduced transform. The used transforms are Discrete Cosine, Walsh, Haar, Kekre, Discrete Sine, Slant and Discrete Hartley transforms. The benefit of energy compaction of transforms in higher coefficients is taken to reduce the feature vector size per image by taking fractional coefficients of transformed image. Smaller feature vector size results in less time for comparison of feature vectors resulting in faster retrieval of images. The feature vectors are extracted in fourteen different ways from the transformed image, with the first being all the coefficients of transformed image considered and then fourteen reduced coefficients sets are considered as feature vectors (as 50%, 25%, 12.5%, 6.25%, 3.125%, 1.5625% ,0.7813%, 0.39%, 0.195%, 0.097%, 0.048%, 0.024%, 0.012% and 0.06% of complete transformed image coefficients). To extract Gray and RGB feature sets the seven image transforms are applied on gray image equivalents and the color components of images. Then these fourteen reduced coefficients sets for gray as well as RGB feature vectors are used instead of using all coefficients of transformed images as feature vector for image retrieval, resulting into better performance and lower computations. The Wang image database of 1000 images spread across 11 categories is used to test the performance of proposed CBIR techniques. 55 queries (5 per category) are fired on the database o find net average precision and recall values for all feature sets per transform for each proposed CBIR technique. The results have shown performance improvement (higher precision and recall values) with fractional coefficients compared to complete transform of image at reduced computations resulting in faster retrieval. Finally Kekre transform surpasses all other discussed transforms in performance with highest precision and recall values for fractional coefficients (6.25% and 3.125% of all coefficients) and computation are lowered by 94.08% as compared to Cosine or Sine or Hartlay transforms.
TL;DR: This work reconsiders the continuous curvelet transform from a signal processing point of view and shows that the analyzing elements of the curvelets can be understood as analytic signals in the sense of the partial Hilbert transform, which yields a well-interpretable amplitude/phase decomposition of the transform coefficients over all scales.
Abstract: We reconsider the continuous curvelet transform from a signal processing point of view. We show that the analyzing elements of the curvelet transform, the curvelets, can be understood as analytic signals in the sense of the partial Hilbert transform. We then generalize the usual curvelets by the monogenic curvelets, which are analytic signals in the sense of the Riesz transform. They yield a new transform, called the monogenic curvelet transform. This transform has the useful property that it behaves at the fine scales like the usual curvelet transform and at the coarse scales like the monogenic wavelet transform. In particular, the new transform is highly anisotropic at the fine scales and yields a well-interpretable amplitude/phase decomposition of the transform coefficients over all scales. We illustrate the advantage of this new directional multiscale amplitude/phase decomposition for the estimation of directional regularity.
TL;DR: The proposed new optical architecture of Hartley transform is based on the Fresnel diffraction which requires no Fourier transform lenses and the main advantages are that it uses fewer optical devices and the decryption scheme is straightforward and more secure than the previous works.
TL;DR: Using a new formulation of Graf's addition formula related to the third Jackson q-Bessel function, the authors studied the positivity of the generalized q-translation operator associated with the q-Hankel transform.
Abstract: Using a new formulation of Graf’s addition formula related to the third Jackson q-Bessel function, we study the positivity of the generalized q-translation operator associated with the q-Hankel transform.
26 Jun 2011
TL;DR: In this article, the performance of selective mapping (SLM), interleaving and partial transmit sequence (PTS) techniques for transmitting real-valued OFDM signals is analyzed for optical orthogonal frequency division multiplexing (O-OFDM) systems.
Abstract: In this paper, we present different peak-to-average power ratio (PAPR) reduction techniques, for optical orthogonal frequency division multiplexing (O-OFDM) systems based on both the fast Fourier transform (FFT) and the fast Hartley transform (FHT). We analyze the performance of selective mapping (SLM), interleaving and partial transmit sequence (PTS) techniques for transmitting real-valued OFDM signals. In O-OFDM systems based on FFT, a direct implementation of PTS and interleaving techniques destroy the Hermitian symmetry. In FHT-based O-OFDM, all the proposed techniques are implemented and compared, showing that PTS provides the highest PAPR reduction. Furthermore, we demonstrate that the proposed solutions considerably improve the bit error rate (BER), by reducing the clipping noise in intensity-modulated direct-detection O-OFDM systems.
TL;DR: In this article, Chen and Wang used the Bedrosian identity for the decomposition of a lower harmonic from a signal composition and presented a new interpretation for the formula using the BER for overlapping signals.
01 Jan 2011
TL;DR: A new method for image encryption is introduced on the basis of two-dimensional (2-D) generalization of 1-D fractional Hartley transform that has been redefined recently in search of its inverse transform.
Abstract: A new method for image encryption is introduced on the basis of two-dimensional (2-D) generalization of 1-D fractional Hartley transform that has been redefined recently in search of its inverse transform We encrypt the image by two fractional orders and random phase codes. It has an advantage over Hartley transform, for its fractional orders can also be used as addictional keys, and that, of course, strengthens image security. Only when all of these keys are correct, can the image be well decrypted. Computer simulations are also perfomed to confirm the possibilty of proposed method.
TL;DR: A generalized interpolated Fourier transform, hereafter called GIFT, is proposed to speed up the Radon transform, and a methodology that can detect straight lines from a gray scale image without any pre-processing is implemented.
TL;DR: In this paper, the authors used hyperbolic geometry to establish a diffeomorphism between the circular average transform and the Funk transform, which is used for the inversion of the circular averages transform.
Abstract: The integral of a function defined on the half-plane along the semi-circles centered on the boundary of the half-plane is known as the circular averages transform. Circular averages transform arises in many tomographic image reconstruction problems. In particular, in synthetic aperture radar (SAR) when the transmitting and receiving antennas are colocated, the received signal is modeled as the integral of the ground reflectivity function of the illuminated scene over the intersection of spheres centered at the antenna location and the surface topography. When the surface topography is flat the received signal becomes the circular averages transform of the ground reflectivity function. Thus, SAR image formation requires inversion of the circular averages transform. Apart from SAR, circular averages transform also arises in thermo-acoustic tomography and sonar inverse problems. In this paper, we present a new inversion method for the circular averages transform using the Funk transform. For a function defined on the unit sphere, its Funk transform is given by the integrals of the function along the great circles. We used hyperbolic geometry to establish a diffeomorphism between the circular averages transform, hyperbolic x-ray and Funk transforms. The method is exact and numerically efficient when fast Fourier transforms over the sphere are used. We present numerical simulations to demonstrate the performance of the inversion method.Dedicated to Dennis Healy, a friend of Applied Mathematics and Engineering.
26 Jun 2011
TL;DR: The authors discuss the feasibility of a constant envelope optical OFDM based on fast Hartley transform, showing a back-to-back sensitivity of −34.6 dBm for a 10−3 BER.
Abstract: In this paper the authors discuss the feasibility of a constant envelope optical OFDM based on fast Hartley transform. The digital signal processing output drives an optical phase modulator for obtaining a 0 dB PAPR signal at the transmission side, whereas the use of a Hartley transform increases power efficiency and relaxes the level of the constellation used. Additionally, a set of simulations of the system are presented, showing a back-to-back sensitivity of −34.6 dBm for a 10−3 BER.
TL;DR: This paper proposes transceivers with practical zero-forcing (ZF) and minimum mean-squared error (MMSE) receivers using DHT, diagonal, and antidiagonal matrices, and the resulting systems are asymptotically as simple as orthogonal frequency-division multiplex and single-carrier with frequency-domain equalization Transceivers.
01 Mar 2011
TL;DR: This implementation of a two-phase implementation of the filters used in the computation of the fractional Fourier transform speeds up the classical code by an average factor from 2 to 4.
Abstract: We describe the implementation of a two-phase implementation of the filters used in the computation of the fractional Fourier transform. This implementation speeds up the classical code by an average factor from 2 to 4.
24 May 2011
TL;DR: In this paper, a discrete Hartley transform (DHT) was used as an alternative to replace the conventional complex valued and mature discrete Fourier transform (DFT) as OFDM modulator and demodulator.
Abstract: The investigation on discrete Hartley transform (DHT) as an alternative to replace the conventional complex valued and mature discrete Fourier transform (DFT) as OFDM modulator and demodulator was carried out in this research. The random binary data was generated and transmitted via the dispersive channel with using additive white Gaussian noise(AWGN) channel model. The performance of the system was evaluated by calculating the number of bit errors for several value of signal to noise ratio (SNR). The measurements of parameters were repeated five times to obtain and to calculate the accuracy of the new system. The plotted graph of average BER verses SNR showed that the BER was decreased with the increasing of given value SNR. The analysis of system performance was also evaluated for several numbers of sub carriers used and the results showcased some advantages. The modification of DHT algorithm is expected to reduce the complexity and time consuming for long-length of computational on the performing of OFDM modulator and demodulator based on DHT.
12 Dec 2011
TL;DR: In this article, it was shown that the linear canonical transform (LCT) is a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula, sampling theorems, convolution theorem and Hilbert transform can be deduced from those of the Fourier Transform by a simple change of variable.
Abstract: Linear canonical transform (LCT) is a four-parameter (a,b,c,d) class of linear integral transform. It has been the focus of many research papers. In this paper, we show that the linear canonical transform is nothing more than a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula, sampling theorems, convolution theorems and Hilbert transform can be deduced from those of the Fourier transform by a simple change of variable. Finally, An example of the application of the LCT is also given.
TL;DR: In this article, the windowed Fourier transform (CWFT) was studied in the framework of Clifford analysis and several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel were derived.
Abstract: We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier transform (CWFT). Based on the spectral representation of the Clifford Fourier transform (CFT), we derive several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an example to show the differences between the classical windowed Fourier transform (WFT) and the CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWFT.
26 Jul 2011
TL;DR: The experimental results show that the positions of the pixels are strongly irregularized using the proposed transform and it is shown that the new transform provides a high level of image scrambling and is robust under common attacks and noise.
Abstract: A new linear transform for scrambling images is proposed in this paper. The forward transform scrambles the image and the inverse transform unscrambles the image. We define transformation matrices for both scalar and blocked cases. Recursive and non-recursive algorithms based on the new transform are also proposed. The degree of scrambling or unscrambling can be determined by the user. The experimental results show that the positions of the pixels are strongly irregularized using the proposed transform. Unscrambling using a wrong key fails and results in an unintelligible image which cannot be recognized. We also show that the new transform provides a high level of image scrambling and is robust under common attacks and noise. The proposed linear transform is very simple and its implementation as triple-matrix multiplication is straightforward.