Results 1  10
of
13
Fast Fourier transforms for nonequispaced data: A tutorial
, 2000
"... In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity o ..."
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Cited by 110 (32 self)
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In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTalgorithms with respect to roundoff errors and apply NDFTalgorithms for the fast computation of Bessel transforms.
Using NFFT 3  a software library for various nonequispaced fast Fourier transforms
, 2008
"... NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and ..."
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Cited by 12 (8 self)
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NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.
Fast errorbounded surfaces and derivatives computation for volumetric particle data
, 2005
"... Volumetric smooth particle data arise as atomic coordinates with electron density kernels for molecular structures, as well as fluid particle coordinates with a smoothing kernel in hydrodynamic flow simulations. In each case there is the need for efficiently computing approximations of relevant surf ..."
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Cited by 11 (4 self)
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Volumetric smooth particle data arise as atomic coordinates with electron density kernels for molecular structures, as well as fluid particle coordinates with a smoothing kernel in hydrodynamic flow simulations. In each case there is the need for efficiently computing approximations of relevant surfaces (molecular surfaces, material interfaces, shock waves, etc), along with surface and volume derivatives (normals, curvatures, etc.), from the irregularly spaced smooth particles. Additionally, molecular properties (charge density, polar potentials), as well as field variables from numerical simulations are often evaluated on these computed surfaces. In this paper we show how all the above problems can be reduced to a fast summation of irregularly spaced smooth kernel functions. For a scattered smooth particle system of M smooth kernels in R 3, where the Fourier coefficients have a decay of the type 1/ω 3, we present an O(M + n 3 log n + N) time, Fourier based algorithm to compute N approximate, irregular samples of a level set surface and its derivatives within a relative L2 error norm ǫ, where n is O(M 1/3 ǫ 1/3). Specifically, a truncated Gaussian of the form e −bx2 has the above decay, and n grows as √ b. In the case when the N output points are samples on a uniform grid, the back transform can be done exactly using a Fast Fourier transform algorithm, giving us an algorithm with O(M + n 3 log n + N log N) time complexity, where n is now approximately half its previously estimated value.
New Fourier reconstruction algorithms for computerized tomography
"... In this paper, we propose two new algorithms for high quality Fourier reconstructions of digital N × N images from their Radon transform. Both algorithms are based on fast Fourier transforms for nonequispaced data (NFFT) and require only O(N²log N) arithmetic operations. While the rst alg ..."
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Cited by 7 (3 self)
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In this paper, we propose two new algorithms for high quality Fourier reconstructions of digital N × N images from their Radon transform. Both algorithms are based on fast Fourier transforms for nonequispaced data (NFFT) and require only O(N²log N) arithmetic operations. While the rst algorithm includes a bivariate NFFT on the polar grid, the second algorithm consists of several univariate NFFTs on the socalled linogram.
Fast evaluation of trigonometric polynomials from hyperbolic crosses
"... The discrete Fourier transform in d dimensions with equispaced knots in space and
frequency domain can be computed by the fast Fourier transform (FFT) in O(N
d
log N)
arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multi
variate approximation, interpolations on sparse ..."
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Cited by 6 (4 self)
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The discrete Fourier transform in d dimensions with equispaced knots in space and
frequency domain can be computed by the fast Fourier transform (FFT) in O(N
d
log N)
arithmetic operations. In order to circumvent the ‘curse of dimensionality’ in multi
variate approximation, interpolations on sparse grids were introduced. In particular, for
frequencies chosen from an hyperbolic cross and spatial knots on a sparse grid fast Fourier
transforms that need only O(N log
d
N) arithmetic operations were developed. Recently,
the FFT was generalised to nonequispaced spatial knots by the so called NFFT.
In this paper, we propose an algorithm for the fast Fourier transform on hyperbolic
cross points for nonequispaced spatial knots in two and three dimensions. We call this
algorithm sparse NFFT (SNFFT). Our new algorithm is based on the NFFT and an ap
propriate partitioning of the hyperbolic cross. Numerical examples confirm our theoretical
results.
NONLINEAR APPROXIMATION BY SUMS OF EXPONENTIALS AND TRANSLATES
"... In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let h be a linear combination of exponentials with real frequencies. Determine all frequ ..."
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Cited by 4 (1 self)
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In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands, if finitely many perturbed, uniformly sampled data of h are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let ϕ be a given 1–periodic window function as defined in Section 4. Further let f be a linear combination of translates of ϕ. Determine all shift parameters, all coefficients, and the number of translates, if finitely many perturbed, uniformly sampled data of f are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.
Numerical stability of nonequispaced fast Fourier transforms
"... Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case ..."
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Cited by 3 (1 self)
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Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case study, we show that both approximation error and roundoff error have a strong influence on the numerical stability of NFFT. Numerical tests confirm the theoretical estimates of numerical stability.
Particle Simulation Based on Nonequispaced Fast Fourier Transforms
"... The fast calculation of longrange interactions is a demanding problem in particle simulation. The main focus of our approach is the decomposition of the problem in building blocks and present efficient numerical realizations for these blocks. For that reason we recapitulate the fast Fourier transfo ..."
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Cited by 3 (3 self)
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The fast calculation of longrange interactions is a demanding problem in particle simulation. The main focus of our approach is the decomposition of the problem in building blocks and present efficient numerical realizations for these blocks. For that reason we recapitulate the fast Fourier transform at nonequispaced nodes and the fast summation method. We describe the application of these algorithms to the evaluation of longrange potentials and compare our methods with the existing fast multipole method.
3d image registration using fast fourier transform, with potential applications to geoinformatics and bioinformatics
 In Proceedings of the International Conference on Information Processing and Management of Uncertainty in KnowledgeBased Systems IPMU’06
, 2006
"... In many practical situations, we do not know the relative orientation of the two images. In such situations, it is desirable to register these images, i.e., to find the rotation and the shift after which the images match as much as possible. A similar problem occurs when we have the images of two di ..."
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Cited by 1 (1 self)
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In many practical situations, we do not know the relative orientation of the two images. In such situations, it is desirable to register these images, i.e., to find the rotation and the shift after which the images match as much as possible. A similar problem occurs when we have the images of two different objects whose shapes should match. For example, we may have images of two bioactive molecules. We know that in vivo, these molecules interact because one of these molecules "docks " to the other one, i.e., gets into a position where their surfaces match. In such situations, it is also important to find orientation and shift corresponding to this match. Comment. Sometimes, the images also differ in lighting conditions, as a result of which we may have I2(~x) ss C \Delta I1( * \Delta R~x + ~a) for some unknown factor C.