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Showing papers on "Hartley transform published in 2014"


BookDOI
07 Nov 2014
TL;DR: In this article, the authors describe the application of the Fourier Transform in the context of fractional calculus and apply it to the problem of finite differential equations in the complex plane.
Abstract: INTEGRAL TRANSFORMS Brief Historical Introduction Basic Concepts and Definitions FOURIER TRANSFORMS AND THEIR APPLICATIONS Introduction The Fourier Integral Formulas Definition of the Fourier Transform and Examples Fourier Transforms of Generalized Functions Basic Properties of Fourier Transforms Poisson's Summation Formula The Shannon Sampling Theorem Gibbs' Phenomenon Heisenberg's Uncertainty Principle Applications of Fourier Transforms to Ordinary Differential Eqn Solutions of Integral Equations Solutions of Partial Differential Equations Fourier Cosine and Sine Transforms with Examples Properties of Fourier Cosine and Sine Transforms Applications of Fourier Cosine and Sine Transforms to Partial DE Evaluation of Definite Integrals Applications of Fourier Transforms in Mathematical Statistics Multiple Fourier Transforms and Their Applications Exercises LAPLACE TRANSFORMS AND THEIR BASIC PROPERTIES Introduction Definition of the Laplace Transform and Examples Existence Conditions for the Laplace Transform Basic Properties of Laplace Transforms The Convolution Theorem and Properties of Convolution Differentiation and Integration of Laplace Transforms The Inverse Laplace Transform and Examples Tauberian Theorems and Watson's Lemma Exercises APPLICATIONS OF LAPLACE TRANSFORMS Introduction Solutions of Ordinary Differential Equations Partial Differential Equations, Initial and Boundary Value Problems Solutions of Integral Equations Solutions of Boundary Value Problems Evaluation of Definite Integrals Solutions of Difference and Differential-Difference Equations Applications of the Joint Laplace and Fourier Transform Summation of Infinite Series Transfer Function and Impulse Response Function Exercises FRACTIONAL CALCULUS AND ITS APPLICATIONS Introduction Historical Comments Fractional Derivatives and Integrals Applications of Fractional Calculus Exercises APPLICATIONS OF INTEGRAL TRANSFORMS TO FRACTIONAL DIFFERENTIAL EQUATIONS Introduction Laplace Transforms of Fractional Integrals Fractional Ordinary Differential Equations Fractional Integral Equations Initial Value Problems for Fractional Differential Equations Green's Functions of Fractional Differential Equations Fractional Partial Differential Equations Exercises HANKEL TRANSFORMS AND THEIR APPLICATIONS Introduction The Hankel Transform and Examples Operational Properties of the Hankel Transform Applications of Hankel Transforms to Partial Differential Equations Exercises MELLIN TRANSFORMS AND THEIR APPLICATIONS Introduction Definition of the Mellin Transform and Examples Basic Operational Properties Applications of Mellin Transforms Mellin Transforms of the Weyl Fractional Integral and Derivative Application of Mellin Transforms to Summation of Series Generalized Mellin Transforms Exercises HILBERT AND STIELTJES TRANSFORMS Introduction Definition of the Hilbert Transform and Examples Basic Properties of Hilbert Transforms Hilbert Transforms in the Complex Plane Applications of Hilbert Transforms Asymptotic Expansions of One-Sided Hilbert Transforms Definition of the Stieltjes Transform and Examples Basic Operational Properties of Stieltjes Transforms Inversion Theorems for Stieltjes Transforms Applications of Stieltjes Transforms The Generalized Stieltjes Transform Basic Properties of the Generalized Stieltjes Transform Exercises FINITE FOURIER SINE AND COSINE TRANSFORMS Introduction Definitions of the Finite Fourier Sine and Cosine Transforms and Examples Basic Properties of Finite Fourier Sine and Cosine Transforms Applications of Finite Fourier Sine and Cosine Transforms Multiple Finite Fourier Transforms and Their Applications Exercises FINITE LAPLACE TRANSFORMS Introduction Definition of the Finite Laplace Transform and Examples Basic Operational Properties of the Finite Laplace Transform Applications of Finite Laplace Transforms Tauberian Theorems Exercises Z TRANSFORMS Introduction Dynamic Linear Systems and Impulse Response Definition of the Z Transform and Examples Basic Operational Properties The Inverse Z Transform and Examples Applications of Z Transforms to Finite Difference Equations Summation of Infinite Series Exercises FINITE HANKEL TRANSFORMS Introduction Definition of the Finite Hankel Transform and Examples Basic Operational Properties Applications of Finite Hankel Transforms Exercises LEGENDRE TRANSFORMS Introduction Definition of the Legendre Transform and examples Basic Operational Properties of Legendre Transforms Applications of Legendre Transforms to Boundary Value Problems Exercises JACOBI AND GEGENBAUER TRANSFORMS Introduction Definition of the Jacobi Transform and Examples Basic Operational Properties Applications of Jacobi Transforms to the Generalized Heat Conduction Problem The Gegenbauer Transform and its Basic Operational Properties Application of the Gegenbauer Transform LAGUERRE TRANSFORMS Introduction Definition of the Laguerre Transform and Examples Basic Operational Properties Applications of Laguerre Transforms Exercises HERMITE TRANSFORMS Introduction Definition of the Hermite Transform and Examples Basic Operational Properties Exercises THE RADON TRANSFORM AND ITS APPLICATION Introduction Radon Transform Properties of Radon Transform Radon Transform of Derivatives Derivatives of Radon Transform Convolution Theorem for Radon Transform Inverse of Radon Transform Exercises WAVELETS AND WAVELET TRANSFORMS Brief Historical Remarks Continuous Wavelet Transforms The Discrete Wavelet Transform Examples of Orthonormal Wavelets Exercises Appendix A Some Special Functions and Their Properties A-1 Gamma, Beta, and Error Functions A-2 Bessel and Airy Functions A-3 Legendre and Associated Legendre Functions A-4 Jacobi and Gegenbauer Polynomials A-5 Laguerre and Associated Laguerre Functions A-6 Hermite and Weber-Hermite Functions A-7 Hurwitz and Riemann zeta Functions Appendix B Tables of Integral Transforms B-1 Fourier Transforms B-2 Fourier Cosine Transforms B-3 Fourier Sine Transforms B-4 Laplace Transforms B-5 Hankel Transforms B-6 Mellin Transforms B-7 Hilbert Transforms B-8 Stieltjes Transforms B-9 Finite Fourier Cosine Transforms B-10 Finite Fourier Sine Transforms B-11 Finite Laplace Transforms B-12 Z Transforms B-13 Finite Hankel Transforms Answers and Hints to Selected Exercises Bibliography Index

805 citations


Journal ArticleDOI
01 Aug 2014-Optik
TL;DR: The generalized wavelet transform (GWT) as discussed by the authors is a time-frequency transformation tool based on the idea of the linear canonical transform (LCT) and is capable of representing signals in the time-fractional frequency plane.

46 citations


Journal ArticleDOI
TL;DR: A cost-effective and efficient modulation scheme for intensity-modulated and direct-detection (IM/DD) optical orthogonal frequency division multiplexing (O-OFDM) systems, which combines complex-to-real transform (C2RT) and fast Hartley transform (FHT), named as fast-fast Fourier transform (FFT).
Abstract: We propose a cost-effective and efficient modulation scheme for intensity-modulated and direct-detection (IM/DD) optical orthogonal frequency division multiplexing (O-OFDM) systems, which combines complex-to-real transform (C2RT) and fast Hartley transform (FHT), named as fast-fast Fourier transform (FFT). The proposed scheme can modulate the complex constellation by the real-valued operations. Compared with the FFT method, the same OFDM signal can also be generated by fast-FFT, but the computational complexity nearly halved. Meanwhile, compared with the FHT scheme, fast-FFT can modulate the complex constellations by adding a simple C2RT module for a wide applicable range. The transmission experiment of over 50-km standard single-mode fiber (SSMF) has been implemented to verify the feasibility of fast-FFT-based IM/DD O-OFDM systems, including asymmetrically clipping and DC-bias O-OFDM systems. It reveals that fast-FFT shares the same bit-error-rate (BER) performance as FFT, but fast-FFT shows superiority on computational complexity.

31 citations


Journal ArticleDOI
TL;DR: This novel fractional Fourier transform has removed the restriction on the dimension of transform order and highly enhances the security of image encryption scheme proposed in this paper without increasing the computational complexity and hardware cost.

27 citations


Proceedings ArticleDOI
TL;DR: A novel method for color image enhancement based on the discrete quaternion Fourier transform that not only provides true color fidelity for poor quality images but also averages the color components to gray value for balancing colors.
Abstract: This paper presents a novel method for color image enhancement based on the discrete quaternion Fourier transform. We choose the quaternion Fourier transform, because it well-suited for color image processing applications, it processes all 3 color components (R,G,B) simultaneously, it capture the inherent correlation between the components, it does not generate color artifacts or blending , finally it does not need an additional color restoration process. Also we introduce a new CEME measure to evaluate the quality of the enhanced color images. Preliminary results show that the α-rooting based on the quaternion Fourier transform enhancement method out-performs other enhancement methods such as the Fourier transform based α-rooting algorithm and the Multi scale Retinex. On top, the new method not only provides true color fidelity for poor quality images but also averages the color components to gray value for balancing colors. It can be used to enhance edge information and sharp features in images, as well as for enhancing even low contrast images. The proposed algorithms are simple to apply and design, which makes them very practical in image enhancement.

26 citations


Journal ArticleDOI
01 Sep 2014
TL;DR: The ordinary convolution theorem and some of its existing extensions related to the FRFT are shown to be special cases of thederived results, and some applications of the derived results are presented.
Abstract: The fractional Fourier transform FRFT-a generalization of the well-known Fourier transform FT-is a comparatively new and powerful mathematical tool for signal processing. Many results in Fourier analysis have currently been extended to the FRFT, including the ordinary convolution theorem. However, the extension of the ordinary convolution theorem associated with the FRFT has been developed differently and is still not having a widely accepted closed-form expression. In this paper, a generalized convolution theorem for the FRFT is proposed, and the dual of it is also presented. The ordinary convolution theorem and some of its existing extensions related to the FRFT are shown to be special cases of the derived results. Moreover, some applications of the derived results are presented. Copyright © 2012 John Wiley & Sons, Ltd.

24 citations


Journal ArticleDOI
TL;DR: In this paper, the mean transform of bounded linear operators acting on a complex Hilbert space was introduced and compared with the Aluthge transform, and the mean transformation of weighted shifts was explored.

23 citations


Journal ArticleDOI
TL;DR: Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractionsal cosine transform is better in encryption of image and video.
Abstract: The mathematical transforms such as Fourier transform, wavelet transform and fractional Fourier transform have long been influential mathematical tools in information processing. These transforms process signal from time to frequency domain or in joint time–frequency domain. In this paper, with the aim to review a concise and self-reliant course, the discrete fractional transforms have been comprehensively and systematically treated from the signal processing point of view. Beginning from the definitions of fractional transforms, discrete fractional Fourier transforms, discrete fractional Cosine transforms and discrete fractional Hartley transforms, the paper discusses their applications in image and video compression and encryption. The significant features of discrete fractional transforms benefit from their extra degree of freedom that is provided by fractional orders. Comparison of performance states that discrete fractional Fourier transform is superior in compression, while discrete fractional cosine transform is better in encryption of image and video. Mean square error and peak signal-to-noise ratio with optimum fractional order are considered quality check parameters in image and video.

21 citations


Journal ArticleDOI
TL;DR: The watermark embedding and detecting techniques are proposed and discussed based on the discrete linear canonical transform, and the results show that the watermark cannot be detected when the parameters of thelinear canonical transform used in the detection are not all the same as the parameters in the embedding progress.
Abstract: The linear canonical transform, which can be looked at the generalization of the fractional Fourier transform and the Fourier transform, has received much interest and proved to be one of the most powerful tools in fractional signal processing community. A novel watermarking method associated with the linear canonical transform is proposed in this paper. Firstly, the watermark embedding and detecting techniques are proposed and discussed based on the discrete linear canonical transform. Then the Lena image has been used to test this watermarking technique. The simulation results demonstrate that the proposed schemes are robust to several signal processing methods, including addition of Gaussian noise and resizing. Furthermore, the sensitivity of the single and double parameters of the linear canonical transform is also discussed, and the results show that the watermark cannot be detected when the parameters of the linear canonical transform used in the detection are not all the same as the parameters used in the embedding progress.

20 citations


Journal ArticleDOI
TL;DR: It is proposed to symmetrically clip the transmitted signal and apply low complexity (LC) distortionless PAPR reduction schemes able to mitigate, at the same time, P APR, quantization and clipping noise.

18 citations


Journal ArticleDOI
TL;DR: In this article, the Boas theorem is used to describe the CFT of Clifford-valued functions that vanish on a neighborhood of the origin, which is a real Paley-Wiener type theorem.
Abstract: Associated with the Dirac operator and partial derivatives, this paper establishes some real Paley-Wiener type theorems to characterize the Clifford-valued functions whose Clifford Fourier transform (CFT) has compact support. Based on the Riemann-Lebesgue theorem for the CFT, the Boas theorem is provided to describe the CFT of Clifford-valued functions that vanish on a neighborhood of the origin.

Journal ArticleDOI
TL;DR: In this article, a hybrid wavelet transform matrix is formed using two component orthogonal transforms, one is base transform which contributes to global features of an image and another transform contributes to local features.
Abstract: In this paper image compression using hybrid wavelet transform is proposed. Hybrid wavelet transform matrix is formed using two component orthogonal transforms. One is base transform which contributes to global features of an image and another transform contributes to local features. Here base transform is varied to observe its effect on image quality at different compression ratios. Different transforms like Discrete Kekre Transform (DKT), Walsh, Real-DFT, Sine, Hartley and Slant transform are chosen as base transforms. They are combined with Discrete Cosine Transform (DCT) that contributes to local features of an image. Sizes of component orthogonal transforms are varied as 16-16, 32-8 and 64-4 to generate hybrid wavelet transform of size 256x256. Results of different combinations are compared and it has been observed that, DKT as a base transform combined with DCT gives better results for size 16x16 of both component transforms.

Journal ArticleDOI
TL;DR: In this article, the Fourier transform on the Schwartz space on the Heisenberg group was studied and a similar characterisation was obtained for the transform on Rn under the same assumption.
Abstract: A recent theorem of S. Alesker, S. Artstein-Avidan and V. Milman characterises the Fourier transform on Rn as essentially the only transform on the space of tempered distributions which interchanges convolutions and pointwise products. In this note we study the image of the Schwartz space on the Heisenberg group under the Fourier transform and obtain a similar characterisation for the Fourier transform on the Heisenberg group.

Proceedings ArticleDOI
TL;DR: The method of filtering the frequency components of the signals and images, by using the discrete signal-induced heap transforms (DsiHT), which are composed by elementary rotations or Givens transformations, are described and compared with the known method of the Fourier transform.
Abstract: In this paper, we describe the method of filtering the frequency components of the signals and images, by using the discrete signal-induced heap transforms (DsiHT), which are composed by elementary rotations or Givens transformations. The transforms are fast, because of a simple form of decomposition of their matrices, and they can be applied for signals of any length. Fast algorithms of calculation of the direct and inverse heap transforms do not depend on the length of the processed signals. Due to construction of the heap transform, if the input signal contains an additive component which is similar to the generator, this component is eliminated in the transform of this signal, while preserving the remaining components of the signal. The energy of this component is preserved in the first point, only. In particular case, when such component is the wave of a given frequency, this wave is eliminated in the heap transform. Different examples of the filtration over signals and images by the DsiHT are described and compared with the known method of the Fourier transform.

Proceedings ArticleDOI
10 Dec 2014
TL;DR: In this article, the Sumudu transform is compared to other well known transforms, such as the Fourier, Laplace, and Mellin transform, and it is shown that problems solved by any of the three transforms can be equally and perhaps at times more easily treated by the SumUDu transform.
Abstract: In this paper, we study connections of the Sumudu transform to other well known transforms, such as the Fourier, Laplace, and Mellin transforms. We afford the Mellin transform a particular attention. Due to its properties, we find that the Sumudu transform may be looked as the source of the other transforms. The sense of this comparison expressed in the paper takes root in the Sumudu-Laplace and Sumudu-Fourier dualities established by Belgacem et al, in the last decade. Our work means that almost in all cases, problems solved any of the three transforms can be equally and perhaps at times more easily treated by the Sumudu transform.

Patent
Xinmiao Zhang1, Ying Yu Tai1
22 Apr 2014
TL;DR: In this paper, low complexity partial parallel architectures for performing a Fourier transform and an inverse-fourier transform over subfields of a finite field are described, which have simplified multipliers and/or computational units.
Abstract: Low complexity partial parallel architectures for performing a Fourier transform and an inverse Fourier transform over subfields of a finite field are described. For example, circuits to perform the Fourier transforms and the inverse Fourier transform as described herein may have architectures that have simplified multipliers and/or computational units as compared to traditional Fourier transform circuits and traditional inverse Fourier transform circuits that have partial parallel designs. In a particular embodiment, a method includes, in a data storage device including a controller and a non-volatile memory, the controller includes an inverse Fourier transform circuit having a first number of inputs coupled to multipliers, receiving elements of an input vector and providing the elements to the multipliers. The multipliers are configured to perform calculations associated with an inverse Fourier transform operation. The first number is less than a number of inverse Fourier transform results corresponding to the inverse Fourier transform operation.

Journal ArticleDOI
TL;DR: A new conjugate symmetric discrete orthogonal transform (CS-DOT-generating method) that has a radix-2 fast algorithm so that it is suitable for hardware implementation and easy to switch the behaviors between these transforms.
Abstract: A new conjugate symmetric discrete orthogonal transform (CS-DOT)-generating method is proposed. The spectra of the CS-DOT for real input signals are conjugate symmetric so that we only need half memory size to store data. Meanwhile, the proposed CS-DOT also has a radix-2 fast algorithm so that it is suitable for hardware implementation. The CS-DOT generalized the existing transforms such that the discrete Fourier transform (DFT) and the conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) are special cases of the CS-DOT. The CS-DOT-generating method is more systematic and generalized than that of the original CS-SCHT. We can use the same implementation structure but only adjust the twiddle factors to construct the CS-SCHT and DFT so that it is easy to switch the behaviors between these transforms.

Journal ArticleDOI
TL;DR: The improved Flip-OFDM scheme is proposed for IM/DD optical systems, where the modulation/demodulation processing takes advantage of the fast Hartley transform (FHT) algorithm and has the same BER performance with conventional scheme, but great superiority on complexity.
Abstract: In this paper, an improved Flip-OFDM scheme is proposed for IM/DD optical systems, where the modulation/demodulation processing takes advantage of the fast Hartley transform (FHT) algorithm. We realize the improved scheme in one symbol period while conventional Flip-OFDM scheme based on fast Fourier transform (FFT) in two consecutive symbol periods. So the complexity of many operations in improved scheme is half of that in conventional scheme, such as CP operation, polarity inversion and symbol delay. Compared to FFT with complex input constellation, the complexity of FHT with real input constellation is halved. The transmission experiment over 50-km SSMF has been realized to verify the feasibility of improved scheme. In conclusion, the improved scheme has the same BER performance with conventional scheme, but great superiority on complexity.

Book
01 Jan 2014
TL;DR: The Fourier transform is very useful in solving a variety of linear constant coefficient ordinary and partial differential equations describing processes which take place over an infinite interval, −∞ < x < ∞.
Abstract: The Fourier transform is very useful in solving a variety of linear constant coefficient ordinary and partial differential equations describing processes which take place over an infinite interval, −∞ < x < ∞. We will provide a number of examples of this sort of application in the present section. Our first example involves a simple, time independent, equilibrium process. Example 1 We consider a stretched string, or cord, with small transverse displacement y(x), subject to an external transverse force f (x) and a transverse restoring force −κ y(x), maintained at tension τ > 0 over the interval −∞ < x < ∞ and constrained so that lim |x| → ∞ y(x) = 0. It can then be shown that y(x) satisfies τ d 2 y dx 2 − κ y(x) + f (x) = 0. Taking a 2 = κ/τ and applying the Fourier transform to both sides of this equation, using the differentiation property (twice) we have − ξ 2 + a 2 ˆ y(ξ) + 1 τ ˆ f (ξ) = 0 ⇒ ˆ y(ξ) = 1 τ ˆ f (ξ) ξ 2 + a 2. Using the convolution property of the Fourier transform we obtain y(x) = 1 τ ∞ −∞ F −1 1 ξ 2 + a 2 (r) f (x − r) dr as the solution. To make any further progress on this we need to find the function K(x) such thatˆK(ξ) ≡ (F K) (ξ) = 1 ξ 2 +a 2. It turns out that this function is K(x) =        e −ax 2a , x > 0, e ax 2a , x < 0.

Journal ArticleDOI
25 Jun 2014
TL;DR: In this paper, the Fourier transform and the Hilbert transform are discussed in terms of their adjacency and interplay in a specic interesting manner, and some relations of that kind are new, while in other cases well-known formulas are considered in a different setting.
Abstract: Well-known and recently observed situations where the two main transforms in harmonic analysis, the Fourier transform and the Hilbert transform, show their adjacency and interplay in a specic interesting manner are overviewed. Some relations of that kind are new, while in other cases well-known formulas are considered in a different setting.

Journal ArticleDOI
TL;DR: In this article, the generalized Parseval equality for the Mellin transform and elementary trigonometric formulas was employed for the iterated Hartley transform on the nonnegative half-axis (the iterated half-Hartley transform).
Abstract: Employing the generalized Parseval equality for the Mellin transform and elementary trigonometric formulas, the iterated Hartley transform on the nonnegative half-axis (the iterated half-Hartley transform) is investigated in $L_2$. Mapping and inversion properties are discussed, its relationship with the iterated Stieltjes transform is established. Various compositions with the Fourier cosine and sine transforms are obtained. The results are applied to the uniqueness and universality of the closed form solutions for certain new singular integral and integro-functional equations. \bigskip

Journal ArticleDOI
TL;DR: Initially, linear transforms like short time Fourier transform, continuous wavelet transforms, s-transform etc. are revisited and the application of these transforms to normal and abnormal ECG signals is illustrated.
Abstract: The diagnostic analysis of non-stationary multi component signals such as electrocardiogram (ECG) involves the use of time–frequency transforms. So, the application of time–frequency transforms to an ECG signal is an important problem of research. In this paper, initially, linear transforms like short time Fourier transform, continuous wavelet transforms, s-transform etc. are revisited. Then the application of these transforms to normal and abnormal ECG signals is illustrated. It has been observed that s-transform provides better time and frequency resolution compared to other linear transforms. The fractional Fourier transform provides rotation to the spectrogram representation.

Proceedings ArticleDOI
18 Sep 2014
TL;DR: Since outputs of proposed new transforms are random, they can be applied in image encryptions and have required good properties to be fractional transforms.
Abstract: In this paper, two new real fractional transforms with many parameters are constructed. They are the real discrete fractional Fourier transform (RDFRFT) and the real discrete fractional Hartley transform (RDFRHT). The eigenvectors of these two new transforms are all random, and they both have only two distinct eigenvalues: 1 or -1. Real eigenvectors of both two transforms are constructed from random DFT-commuting matrices. We also propose an alternative definition of RDFRHT based on a diagonal-like matrix. All of the proposed new transforms have required good properties to be fractional transforms. Finally, since outputs of proposed new transforms are random, they can be applied in image encryptions.

Journal ArticleDOI
TL;DR: In this article, the authors extend Natterer's results to the attenuated Radon transform with imaginary coefficients and derive explicit inversion formulas with two different methods, which are closely related to Novikov's inversion formula.
Abstract: In this paper, we successfully extend Natterer’s results to the attenuated Radon transform with imaginary coefficients And by means of the theory of complex analysis and integral transform, we study the inversion of two-dimensional weighted Radon transform and derive similar explicit inversion formulas with two different methods, which are closely related to Novikov’s inversion formula

Proceedings ArticleDOI
01 Dec 2014
TL;DR: The theme of work presented in this paper is a novel Iris recognition technique using partial energies of transformed iris image to generate feature vectors using the concept of energy compaction of transforms in higher coefficients.
Abstract: The theme of work presented in this paper is a novel Iris recognition technique using partial energies of transformed iris image. To generate transformed iris images, various transforms like Cosine, Walsh, Haar, Kekre, Hartley transforms and their wavelet transforms are applied on the iris images. Feature vectors are then generated from these transformed Iris images using the concept of energy compaction of transforms in higher coefficients. 5 different ways are used to generate the feature vectors from the transformed iris images. First way considers all the higher energy coefficients of the transformed iris image while the rest considers 99%, 98%, 97%, and 96% of the higher energy coefficients for generating the feature vector. Considering partial energies reduces the feature vector size thus lowering the number of computations and results shows that this gives better performance. To test the performance of the proposed techniques, Genuine Acceptance Rate (GAR) is used as a metric. Better Performance in terms of Speed and Accuracy is obtained by considering Partial Energies. Among all the Transforms and Wavelet Transforms, Walsh Transform and Walsh Wavelet Transform gives highest GAR value. Results show that most wavelet transforms outperforms other transforms. Also, using Partial Energy gives better performance as compared to using 100% energies. The proposed technique is tested on Palacky University Dataset.

Journal ArticleDOI
TL;DR: In this paper, the reciprocal inverse operator in the weighted L2-space related to the Hilbert transform on the nonnegative half-axis is derived. And the corresponding convolution and Titchmarsh's theorems for the half-Hilbert transform are proved.
Abstract: While exploiting the generalized Parseval equality for the Mellin transform, we derive the reciprocal inverse operator in the weighted L2-space related to the Hilbert transform on the nonnegative half-axis. Moreover, employing the convolution method, which is based on the Mellin–Barnes integrals, we prove the corresponding convolution and Titchmarsh's theorems for the half-Hilbert transform. Some applications to the solvability of a new class of singular integral equations are demonstrated. Our technique does not require the use of methods of the Riemann–Hilbert boundary value problems for analytic functions. The same approach is applied recently to invert the half-Hartley transform and to establish its convolution theorem.

01 Jan 2014
TL;DR: This paper presents a novel Iris feature extraction technique using fractional energies of transformed iris image, which gives better performance as compared to using 100% energies.
Abstract: This paper presents a novel Iris feature extraction technique using fractional energies of transformed iris image. To generate image transforms various transforms like Cosine, Walsh, Haar, Kekre and Hartley transforms are used. The above transforms are applied on the iris images to obtain transformed iris images. From these transformed Iris images, feature vectors are extracted by taking the advantage of energy compaction of transforms in higher coefficients. Due to this the size of feature vector reduces greatly. Feature vectors are extracted in 5 different ways from the transformed iris images. First way considers all the higher energy coefficients of the transformed iris image while the rest considers 99%, 98%, 97%, and 96% of the higher energy coefficients for generating the feature vector. Considering fractional energies lowers the computations and gives better performance. Performance comparison among various proposed techniques of feature extraction is done using Genuine Acceptance Rate (GAR). Better Performance in terms of Speed and Accuracy is obtained by considering Fractional Energies. Among all the Transforms, Cosine and Walsh Transform gives good GAR value of 85% by considering 99% of Fractional Energy. Thus, using Fractional Energy gives better performance as compared to using 100% energies. The proposed technique is tested on Palacky University Dataset.

Journal ArticleDOI
TL;DR: In this article, a generalized convolution of functions f,g for the Hartley (H1,H2) and the Fourier sine (Fs) integral transforms is proposed.
Abstract: In this paper, we construct and study a new generalized convolution (f * g)(x) of functions f,g for the Hartley (H1,H2) and the Fourier sine (Fs) integral transforms. We will show that these generalized convolutions satisfy the following factorization equalities: We prove the existence of this generalized convolution on different function spaces, such as . As examples, applications to solve a type of integral equations and a type of systems of integral equations are presented. Copyright © 2013 John Wiley & Sons, Ltd.

Journal ArticleDOI
TL;DR: For all but finitely many complex values of $\alpha$ except one, this article showed that the transformation is a composition of the Radon transform with an explicitly written O(n)-invariant differential operator.
Abstract: The goal of this paper is to describe the $\alpha$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many complex $\alpha$ the $\alpha$-cosine transform is a composition of the $(\alpha+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $\alpha$ except one we interpret the $\alpha$-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value $\alpha$, which is $-(min\{i,n-i\}+1)$, is still an open problem.

01 Jan 2014
TL;DR: In this article, the correspondence between the phase-only correlation (POC) function obtained by means of FFT and by FHT was analyzed and the Hartley transform was used for pattern matching and image registration.
Abstract: In image processing or pattern recognition, Fourier Transform is widely used for frequency-domain analysis. In particular, the Phase Only Correlation (POC) method demonstrates high robustness and accuracy in the pattern matching and the image registration. However, there is a disadvantage in required memory machine because of the calculation of 2D-FFT. In this case, Hartley transform can be a very good substitute for more commonly used Fourier transform when the real input data are concerned. The Hartley transform is similar to the Fourier transform, but it is free from the need to process complex numbers. It also has some distinctive features that make it an interesting choice when a greater efficiency in memory requirements is needed. In this paper we show the correspondence between the Phase-Only Correlation (POC) function obtained by means of FFT and by FHT.