Results 1  10
of
32
Fast Discrete Curvelet Transforms
, 2005
"... This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequallyspaced fast Fourier transforms (USFFT) while the second is based on the wrap ..."
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Cited by 120 (9 self)
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This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform [12, 10] in two and three dimensions. The first digital transformation is based on unequallyspaced fast Fourier transforms (USFFT) while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n 2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at
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 112 (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.
Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
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Cited by 84 (13 self)
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The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes
 IMA Journal of Numerical Analysis
, 2003
"... An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models wit ..."
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Cited by 34 (13 self)
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An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. For typical model parameters, it is shown that the fixed point iteration reduces the error by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed for avoiding wraparound effects. Numerical tests of convergence for a variety of options are presented.
Fast and accurate Polar Fourier transform
 Appl. Comput. Harmon. Anal.
, 2006
"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."
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Cited by 17 (1 self)
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In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudoPolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudoPolar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudoPolar FFT plays the role of a halfway point—a nearlyPolar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesianbased unequallysampled FFT method to ours, both algorithms using a smallsupport interpolation and no precompensating, and show marked advantage to the use of the pseudoPolar initial grid.
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.
Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT
 IEEE Trans. Medical Imaging
, 2002
"... We show an iterative reconstruction framework for diffraction ultrasound tomography. The use of broadband illumination allows significant reduction of the number of projections compared to straight ray tomography. The proposed algorithm makes use of forward nonuniform fast Fourier transform (NUFFT) ..."
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Cited by 9 (0 self)
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We show an iterative reconstruction framework for diffraction ultrasound tomography. The use of broadband illumination allows significant reduction of the number of projections compared to straight ray tomography. The proposed algorithm makes use of forward nonuniform fast Fourier transform (NUFFT) for iterative Fourier inversion. Incorporation of total variation regularization allows the reduction of noise and Gibbs phenomena while preserving the edges. The complexity of the NUFFTbased reconstruction is comparable to the frequencydomain interpolation (gridding) algorithm, whereas the reconstruction accuracy (in sense of the and the norm) is better. Index TermsAcoustic diffraction tomography, image reconstruction, nonuniform fast Fourier transform (NUFFT).
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.
FOURIER VOLUME RENDERING OF IRREGULAR DATA SETS
, 2002
"... Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We p ..."
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Cited by 6 (0 self)
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Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We present an algorithm to create a regularly sampled data set from an irregular one. This new data set is not only an approximation to the original, but allows the original points to be accurately recovered, while still remaining relatively small. This result is accompanied by an efficient ‘zooming ’ operation that allows the user to increase the resolution while gaining new details, all without regridding the data. The technique is presented in Ndimensions, but is particularly well suited to Fourier Volume Rendering, which is the fastest known method of direct volume rendering. Together, these techniques allow accurate and efficient, multiresolution exploration of volume data.
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.