The Experts below are selected from a list of 33966 Experts worldwide ranked by ideXlab platform
Stephen J Sangwine - One of the best experts on this subject based on the ideXlab platform.
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quaternion Fourier Transforms for signal and image processing
2014Co-Authors: Todd A Ell, Nicolas Le Bihan, Stephen J SangwineAbstract:Based on updates to signal and image processing technology made in the last two decades, this text examines the most recent research results pertaining to Quaternion Fourier Transforms. QFT is a central component of processing color images and complex valued signals. The books attention to mathematical concepts, imaging applications, and Matlab compatibility render it an irreplaceable resource for students, scientists, researchers, and engineers.
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complex and hypercomplex discrete Fourier Transforms based on matrix exponential form of euler s formula
Applied Mathematics and Computation, 2012Co-Authors: Stephen J SangwineAbstract:We show that the discrete complex, and numerous hypercomplex, Fourier Transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler’s formula ejθ=cosθ+jsinθ, and a matrix root of -1 isomorphic to the imaginary root j. The Transforms thus defined can be computed using standard matrix multiplications and additions with no hypercomplex code, the complex or hypercomplex algebra being represented by the form of the matrix root of -1, so that the matrix multiplications are equivalent to multiplications in the appropriate algebra. We present examples from the complex, quaternion and biquaternion algebras, and from Clifford algebras Cl1,1 and Cl2,0. The significance of this result is both in the theoretical unification, which permits comparisons between Transforms in different hypercomplex algebras, and also in the scope it affords for insight into the structure of the various Transforms, since the formulation is such a simple generalization of the classic complex case. It also shows that hypercomplex discrete Fourier Transforms may be computed using standard matrix arithmetic packages without the need for a hypercomplex library, which is of importance in providing a reference implementation for verifying faster implementations based on hypercomplex code.
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hypercomplex Fourier Transforms of color images
IEEE Transactions on Image Processing, 2007Co-Authors: Todd A Ell, Stephen J SangwineAbstract:Fourier Transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier Transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed
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decomposition of 2d hypercomplex Fourier Transforms into pairs of complex Fourier Transforms
European Signal Processing Conference, 2000Co-Authors: Todd A Ell, Stephen J SangwineAbstract:Hypercomplex 2D Fourier Transforms have been proposed by several authors with applications in image processing of both grayscale and colour images. Previously published works on hypercomplex Fourier Transforms have utilized direct evaluation of a Fast Fourier transform using hypercomplex arithmetic. This paper shows that such Transforms may be implemented by decomposition into two independent complex Fourier Transforms and may thus be implemented by building upon existing complex code. This is a significant step because it makes available to researchers using hypercomplex Fourier Transforms all the investment made by others in efficient complex fft implementations, and requires substantially less effort than coding hypercomplex versions of existing code.
Todd A Ell - One of the best experts on this subject based on the ideXlab platform.
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quaternion Fourier Transforms for signal and image processing
2014Co-Authors: Todd A Ell, Nicolas Le Bihan, Stephen J SangwineAbstract:Based on updates to signal and image processing technology made in the last two decades, this text examines the most recent research results pertaining to Quaternion Fourier Transforms. QFT is a central component of processing color images and complex valued signals. The books attention to mathematical concepts, imaging applications, and Matlab compatibility render it an irreplaceable resource for students, scientists, researchers, and engineers.
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hypercomplex Fourier Transforms of color images
IEEE Transactions on Image Processing, 2007Co-Authors: Todd A Ell, Stephen J SangwineAbstract:Fourier Transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier Transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed
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decomposition of 2d hypercomplex Fourier Transforms into pairs of complex Fourier Transforms
European Signal Processing Conference, 2000Co-Authors: Todd A Ell, Stephen J SangwineAbstract:Hypercomplex 2D Fourier Transforms have been proposed by several authors with applications in image processing of both grayscale and colour images. Previously published works on hypercomplex Fourier Transforms have utilized direct evaluation of a Fast Fourier transform using hypercomplex arithmetic. This paper shows that such Transforms may be implemented by decomposition into two independent complex Fourier Transforms and may thus be implemented by building upon existing complex code. This is a significant step because it makes available to researchers using hypercomplex Fourier Transforms all the investment made by others in efficient complex fft implementations, and requires substantially less effort than coding hypercomplex versions of existing code.
Christian Wulker - One of the best experts on this subject based on the ideXlab platform.
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fast sgl Fourier Transforms for scattered data
Applied and Computational Harmonic Analysis, 2020Co-Authors: Christian WulkerAbstract:Abstract Spherical Gauss-Laguerre (SGL) basis functions, i.e., normalized functions of the type L n − l − 1 ( l + 1 / 2 ) ( r 2 ) r l Y l m ( ϑ , φ ) , | m | ≤ l n ∈ N , L n − l − 1 ( l + 1 / 2 ) being a generalized Laguerre polynomial, Y l m a spherical harmonic, constitute an orthonormal polynomial basis of the space L 2 on R 3 with radial Gaussian (multivariate Hermite) weight exp ( − r 2 ) . We have recently described fast Fourier Transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in R 3 . In this paper, we present fast SGL Fourier Transforms for scattered data. The idea is to employ well-known basal fast algorithms to determine a three-dimensional trigonometric polynomial that coincides with the bandlimited function of interest where the latter is to be evaluated. This trigonometric polynomial can then be evaluated efficiently using the well-known non-equispaced FFT (NFFT). We prove an error estimate for our algorithms and validate their practical suitability in extensive numerical experiments.
Jonatan Lenells - One of the best experts on this subject based on the ideXlab platform.
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nonlinear Fourier Transforms for the sine gordon equation in the quarter plane
Journal of Differential Equations, 2018Co-Authors: Lin Huang, Jonatan LenellsAbstract:Abstract Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions a , b , A , B . The functions a ( k ) and b ( k ) are defined via a nonlinear Fourier transform of the initial data, whereas A ( k ) and B ( k ) are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier Transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.
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nonlinear Fourier Transforms for the sine gordon equation in the quarter plane
arXiv: Analysis of PDEs, 2017Co-Authors: Lin Huang, Jonatan LenellsAbstract:The solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann-Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier Transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.
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nonlinear Fourier Transforms and the mkdv equation in the quarter plane
Studies in Applied Mathematics, 2016Co-Authors: Jonatan LenellsAbstract:The unified transform method introduced by Fokas can be used to analyze initial-boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier Transforms and the formulation of a Riemann-Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier Transforms. Using the theory of L2-RH problems, we consider the construction of quarter plane solutions which are C1 in time and C3 in space.
Nicolas Tremblay - One of the best experts on this subject based on the ideXlab platform.
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approximate fast graph Fourier Transforms via multilayer sparse approximations
IEEE Transactions on Signal and Information Processing over Networks, 2018Co-Authors: Luc Le Magoarou, Remi Gribonval, Nicolas TremblayAbstract:The fast Fourier transform is an algorithm of paramount importance in signal processing as it allows to apply the Fourier transform in $\mathcal {O}(n \log n)$ instead of $\mathcal {O}(n^2)$ arithmetic operations. Graph signal processing is a recent research domain that generalizes classical signal processing tools, such as the Fourier transform, to situations where the signal domain is given by any arbitrary graph instead of a regular grid. Today, there is no method to rapidly apply graph Fourier Transforms. In this paper, we propose a method to obtain approximate graph Fourier Transforms that can be applied rapidly and stored efficiently. It is based on a greedy approximate diagonalization of the graph Laplacian matrix, carried out using a modified version of the famous Jacobi eigenvalues algorithm. The method is described and analyzed in detail, and then applied to both synthetic and real graphs, showing its potential.
