Showing papers on "Hartley transform published in 2013"
TL;DR: A systematic method for developing a binary version of a given transform by using the Walsh-Hadamard transform (WHT) is proposed and it is shown that the resulting BDCT corresponds to the well-known sequency-ordered WHT, whereas the BDHT can be considered as a new Hartley-ordering WHT.
Abstract: In this paper, a systematic method for developing a binary version of a given transform by using the Walsh-Hadamard transform (WHT) is proposed. The resulting transform approximates the underlying transform very well, while maintaining all the advantages and properties of WHT. The method is successfully applied for developing a binary discrete cosine transform (BDCT) and a binary discrete Hartley transform (BDHT). It is shown that the resulting BDCT corresponds to the well-known sequency-ordered WHT, whereas the BDHT can be considered as a new Hartley-ordered WHT. Specifically, the properties of the proposed Hartley-ordering are discussed and a shift-copy scheme is proposed for a simple and direct generation of the Hartley-ordering functions. For software and hardware implementation purposes, a unified structure for the computation of the WHT, BDCT, and BDHT is proposed by establishing an elegant relationship between the three transform matrices. In addition, a spiral-ordering is proposed to graphically obtain the BDHT from the BDCT and vice versa. The application of these binary transforms in image compression, encryption and spectral analysis clearly shows the ability of the BDCT (BDHT) in approximating the DCT (DHT) very well.
83 citations
TL;DR: A novel color image encryption algorithm based on Arnold- and discrete Hartley transform in gyrator transform domain is proposed, which can be well protected under chosen- and known plaintext attacks.
72 citations
TL;DR: A color image encryption algorithm by using chaotic mapping and Hartley transform and an electro-optical encryption structure is designed, constituted by two selected color components of output in real number domain.
64 citations
TL;DR: In this article, the Segal-Bargmann transform has a meaningful limit Gs,t as N → ∞, which can be identified as an operator on the space of complex Laurent polynomials.
50 citations
TL;DR: In this paper, a directional short-time Fourier transform (DSTFT) is introduced, which uses functions in L ∞ (R ) as window and is related to the celebrated Radon transform.
50 citations
01 Jan 2013
41 citations
TL;DR: In this article, a fast butterfly algorithm for the hyperbolic Radon transform is proposed, which reformulates the transform as an oscillatory integral operator and constructs a blockwise low-rank approximation of the kernel function.
Abstract: Generalized Radon transforms, such as the hyperbolic Radon transform, cannot be implemented as efficiently in the frequency domain as convolutions, thus limiting their use in seismic data processing. We have devised a fast butterfly algorithm for the hyperbolic Radon transform. The basic idea is to reformulate the transform as an oscillatory integral operator and to construct a blockwise low-rank approximation of the kernel function. The overall structure follows the Fourier integral operator butterfly algorithm. For 2D data, the algorithm runs in complexity O(N2 log N), where N depends on the maximum frequency and offset in the data set and the range of parameters (intercept time and slowness) in the model space. From a series of studies, we found that this algorithm can be significantly more efficient than the conventional time-domain integration.
37 citations
TL;DR: In this paper, the authors introduce a single straightforward definition of a general geometric Fourier transform covering most versions in the literature and provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context.
Abstract: The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straightforward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features such as linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.
36 citations
TL;DR: In this paper, the Hartley transform is extended and obtained as a well-defined continuous mapping with respect to the convergence for which certain theorems have been proved, and the article is ended up in defining the inverse Hartley transformation and discussing some of its properties in detail.
Abstract: Abstract. The idea of the construction of Boehmians was initiated by the concept of regular operators. The construction of Boehmians is similar to the construction of the field of quotients and, in some cases, it just gives the field of quotients. In this article we consider two spaces of Boehmians. The strong space of Boehmians is continuously viewed in the space of general Boehmians. The Hartley transform is extended and obtained as a well-defined continuous mapping with respect to the convergence for which certain theorems have been proved. The article is ended up in defining the inverse Hartley transform and discussing some of its properties in detail.
35 citations
TL;DR: This paper illustrates systematic approaches to designing useful functions in beamforming by evaluating the properties of the function and its transform, jointly, to optimize the system.
Abstract: In beamforming, a critical design issue is the choice of the window or apodization function Because the transverse beam pattern at focal depth is related to the Fourier transform of the apodization function, designers must evaluate the properties of the function and its transform, jointly, to optimize the system This paper illustrates systematic approaches to designing useful functions
33 citations
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TL;DR: A Hilbert transform method for pricing Bermudan options in Levy process models is presented and the corresponding optimal stopping problem can be solved using a backward induction.
Abstract: This paper presents a Hilbert transform method for pricing Bermudan options in Levy process models. The corresponding optimal stopping problem can be solved using a backward induction, where a sequence of inverse Fourier and Hilbert transforms need to be evaluated. Using results from a sinc expansion based approximation theory for analytic functions, the inverse Fourier and Hilbert transforms can be approximated using very simple rules. The approximation errors decay exponentially with the number of terms used to evaluate the transforms for many popular Levy process models. The resulting discrete approximations can be efficiently implemented using the fast Fourier transform. The early exercise boundary is obtained at the same time as the price. Accurate American option prices can be obtained by using Richardson extrapolation.
01 May 2013
TL;DR: A convolution method for calculating the Hough transform for finding circles of arbitrary radius by performing a three-dimensional convolution of the input image with an appropriate Hough kernel is described.
Abstract: The Hough transform is a well-established family of algorithms for locating and describing geometric figures in an image. However, the computational complexity of the algorithm used to calculate the transform is high when used to target complex objects. As a result, the use of the Hough transform to find objects more complex than lines is uncommon in real-time applications. We describe a convolution method for calculating the Hough transform for finding circles of arbitrary radius. The algorithm operates by performing a three-dimensional convolution of the input image with an appropriate Hough kernel. The use of the fast Fourier transform to calculate the convolution results in a Hough transform algorithm with reduced computational complexity and thus increased speed. Edge detection and other convolution-based image processing operations can be incorporated as part of the transform, which removes the need to perform them with a separate pre-processing or post-processing step. As the Discrete Fourier Transform implements circular convolution rather than linear convolution, consideration must be given to padding the input image before forming the Hough transform.
01 Mar 2013
TL;DR: The advantage of discrete fractional Fourier transform (DFrFT) as compared to other transforms for steganography in image processing is illustrated.
Abstract: The Fractional Fourier transform (FrFT), as a generalization of the classical Fourier transform, was introduced many years ago in mathematics literature. For the enhanced computation of fractional Fourier transform, discrete version of FrFT came into existence i.e. DFrFT. This paper illustrates the advantage of discrete fractional Fourier transform (DFrFT) as compared to other transforms for steganography in image processing. The simulation result shows same PSNR in both domain (time and frequency) but DFrFT gives an advantage of additional stego key i.e. order parameter of this transform.
TL;DR: The q-Sumudu transform is the theoritical dual of the Laplace transform as discussed by the authors, and it has many applications in sciences and engineering for its special fundamental properties.
Abstract: Although Sumudu transform is the theoritical dual of the Laplace transform, it has many applications in sciences and engineering for its special fundamental properties. In a previous paper (3), we studied q-analogues of the Sumudu transform and derived some fundamental properties. This paper follows the previous paper and aims to provide some applications of the q-Sumudu transform. The authors give q-Sumudu transforms of some q-polynomials and q-functions. Also, we evaluated the q-Sumudu transform of basic analogue of Fox's H-function.
01 Jan 2013
TL;DR: This chapter explores the evolution of QFT definitions as a framework from which to solve specific problems in vector-image and vector-signal processing.
Abstract: Quaternion Fourier transforms (QFT’s) provide expressive power and elegance in the analysis of higher-dimensional linear invariant systems. But, this power comes at a cost – an overwhelming number of choices in the QFT definition, each with consequences. This chapter explores the evolution of QFT definitions as a framework from which to solve specific problems in vector-image and vector-signal processing.
TL;DR: The Heisenberg Uncertainty principles for the two-dimensional nonseparable linear canonical transform (2-D NSLCT) are derived and the functions that can achieve the lower bound of the inequality are found.
01 Jan 2013
TL;DR: In this paper, a suitable Boehmian space is constructed to extend the distributional Mellin transform, which is defined as a quotient of analytic functions, and it is shown that the generalized Mellin transformation has all its usual properties.
Abstract: A suitable Boehmian space is constructed to extend the distributional Mellin transform. Mellin transform of a Boehmian is defined as a quotient of analytic functions. We prove that the generalized Mellin transform has all its usual properties. We also discuss the relation between the Mellin transform and the Laplace transform in the context of Boehmians.
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TL;DR: The definition of convolution on the ball is studied in this context, showing explicitly how translation on the radial line may be viewed as convolution with a shifted Dirac delta function.
Abstract: We review the Fourier-Laguerre transform, an alternative harmonic analysis on the three-dimensional ball to the usual Fourier-Bessel transform. The Fourier-Laguerre transform exhibits an exact quadrature rule and thus leads to a sampling theorem on the ball. We study the definition of convolution on the ball in this context, showing explicitly how translation on the radial line may be viewed as convolution with a shifted Dirac delta function. We review the exact Fourier-Laguerre wavelet transform on the ball, coined flaglets, and show that flaglets constitute a tight frame.
TL;DR: A new spinor Fourier transform is introduced for both gray-level and color image processing by means of so-called spin characters and may be used to perform frequency filtering that takes into account the Riemannian geometry of the image.
Abstract: We propose in this paper to introduce a new spinor Fourier transform for both gray-level and color image processing. Our approach relies on the three following considerations: mathematically speaking, defining a Fourier transform requires to deal with group actions; vectors of the acquisition space can be considered as generalized numbers when embedded in a Clifford algebra; the tangent space of the image surface appears to be a natural parameter of the transform we define by means of so-called spin characters. The resulting spinor Fourier transform may be used to perform frequency filtering that takes into account the Riemannian geometry of the image. We give examples of low-pass filtering interpreted as diffusion process. When applied to color images, the entire color information is involved in a really non marginal process.
TL;DR: In this article, the ridgelet transform of (Lizorkin) distributions is defined and studied, and connections with the Radon and wavelet transforms are established with respect to wavelet transform.
Abstract: We define and study the ridgelet transform of (Lizorkin) distributions. We establish connections with the Radon and wavelet transforms.
TL;DR: It has been observed that full hybrid wavelet transform obtained by combining Real Fourier Transform and DCT gives best performance of all, and is compared with DCT Full Wavelet Transform.
Abstract: This paper proposes new image compression technique that uses Real Fourier Transform. Discrete Fourier Transform (DFT) contains complex exponentials. It contains both cosine and sine functions. It gives complex values in the output of Fourier Transform. To avoid these complex values in the output, complex terms in Fourier Transform are eliminated. This can be done by using coefficients of Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST). DCT as well as DST are orthogonal even after sampling and both are equivalent to FFT of data sequence of twice the length. DCT uses real and even functions and DST uses real and odd functions which are equivalent to imaginary part in Fourier Transform. Since coefficients of both DCT and DST contain only real values, Fourier Transform obtained using DCT and DST coefficients also contain only real values. This transform called Real Fourier Transform is applied on colour images. RMSE values are computed for column, Row and Full Real Fourier Transform. Wavelet transform of size N2xN2 is generated using NxN Real Fourier Transform. Also Hybrid Wavelet Transform is generated by combining Real Fourier transform with Discrete Cosine Transform. Performance of these three transforms is compared using RMSE as a performance measure. It has been observed that full hybrid wavelet transform obtained by combining Real Fourier Transform and DCT gives best performance of all. It is compared with DCT Full Wavelet Transform. It beats the performance of Full DCT Wavelet transform. Reconstructed image quality obtained in Real Fourier-DCT Full Hybrid Wavelet Transform is superior to one obtained in DCT, DCT Wavelet and DCT Hybrid Wavelet Transform.
TL;DR: This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which is compared with the existing convolution theorem and is found to be a better and befitting proposition.
Abstract: The Linear Canonical Transform LCT is a four parameter class of integral transform which plays an important role in many fields of signal processing. Well-known transforms such as the Fourier Transform FT, the FRactional Fourier Transform FRFT, and the FreSnel Transform FST can be seen as special cases of the linear canonical transform. Many properties of the LCT are currently known but the extension of FRFTs and FTs still needs more attention. This paper presents a modified convolution and product theorem in the LCT domain derived by a representation transformation in quantum mechanics, which seems a convenient and concise method. It is compared with the existing convolution theorem for the LCT and is found to be a better and befitting proposition. Further, an application of filtering is presented by using the derived results.
18 Nov 2013
TL;DR: A new method for double image encryption based on the fractional Hartley transform (FrHT) and the Arnold transform (AT) is proposed in this work and digital results that confirm the approach are presented.
Abstract: A new method for double image encryption based on the fractional Hartley transform (FrHT) and the Arnoldtransform (AT) is proposed in this work. The encryption method encodes the rst input image in amplitudeand the second input image is encoded in phase, in order to de ne a complex image. This complex imageis successively four times transformed using FrHT and AT, and the resulting complex image represents theencrypted image. The decryption method is the same method as the encryption method applied in the inversesense. The AT is a process of image shearing and stitching in which pixels of the image are rearranged. This ATis used in the encryption method with the purpose of spreading the information content of the two input imagesonto the encrypted image and to increase the security of the encrypted image. The fractional orders of the FrHTsand the parameters of the ATs correspond to the keys of the encryption-decryption method. Only when all ofthose keys are correct in the decryption method, the two original images can be recovered. We present digitalresults that con rm our approach.Keywords: Encryption-decryption method, fractional Hartley transform, Arnold transform.
TL;DR: In this paper, the authors established a real Paley-Wiener theorem to characterize the quaternion-valued functions whose Fourier transform has compact support by the partial derivative and also a Boas theorem to describe the functions that vanish on a neighborhood of the origin by an integral operator.
Abstract: This paper establishes a real Paley-Wiener theorem to characterize the quaternion-valued functions whose quaternion Fourier transform has compact support by the partial derivative and also a Boas theorem to describe the quaternion Fourier transform of these functions that vanish on a neighborhood of the origin by an integral operator.
TL;DR: In this paper, the authors considered integral equations of convolution type with the Toeplitz plus Hankel kernels firstly posed by Tsitsiklis and Levy (1981) and obtained a necessary and sufficient condition for the solvability and unique explicit L 2 -solution.
TL;DR: In this article, the correlation theorem for the quaternion Fourier transform (QFT) of the two quaternions functions was derived using properties of convolution, and the correlation was further extended to the other functions.
Abstract: In this paper we introduce convolution theorem for the Fourier transform (FT) of two complex functions. We show that the correlation theorem for the FT can be derived using properties of convolution. We develop this idea to derive the correlation theorem for the quaternion Fourier transform (QFT) of the two quaternion functions.
TL;DR: This paper proposes a novel method of image encryption using discrete fractional Fourier transform (DFRFT) using exponential random phase mask, which makes it almost impossible to retrieve the image without using both the right keys.
01 Aug 2013
TL;DR: Theoretical analysis and experimental results demonstrate that the algorithm is favorable, and the security of the proposed algorithm depends on the transformation algorithm, sensitivity to the randomness of phase mask and the orders of FRFT.
Abstract: In order to transmit image data in open network, a novel image encryption algorithm based on fractional Fourier transform and block-based transformation is proposed in this paper. The image encryption process includes two steps: the original image was divided into blocks, which were rearranged into a transformed image using a transformation algorithm, and then the transformed image was encrypted using the fractional Fourier transform (FRFT) algorithm. The security of the proposed algorithm depends on the transformation algorithm, sensitivity to the randomness of phase mask and the orders of FRFT. Theoretical analysis and experimental results demonstrate that the algorithm is favorable.
TL;DR: In this article, it was shown that the restriction map f→f|Ω is essentially invertible on PWα(b) if and only if Ω is sufficiently dense.
Abstract: The aim of this paper is to establish an analogue of Logvinenko–Sereda's theorem for the Fourier–Bessel transform (or Hankel transform) ℱα of order α>−½. Roughly speaking, if we denote by PWα(b) the Paley–Wiener space of L 2-functions with the Fourier–Bessel transform supported in [0, b], then we show that the restriction map f→f|Ω is essentially invertible on PWα(b) if and only if Ω is sufficiently dense. Moreover, we give an estimate of the norm of the inverse map. As a side result, we prove a Bernstein-type inequality for the Fourier–Bessel transform.