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96
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.
On Approximation of Functions by Exponential Sums
, 2005
"... We introduce a new approach, and associated algorithms, for the efficient approximation of functions and sequences by short linear combinations of exponential functions with complexvalued exponents and coefficients. These approximations are obtained for a finite but arbitrary accuracy and typically ..."
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Cited by 15 (1 self)
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We introduce a new approach, and associated algorithms, for the efficient approximation of functions and sequences by short linear combinations of exponential functions with complexvalued exponents and coefficients. These approximations are obtained for a finite but arbitrary accuracy and typically have significantly fewer terms than Fourier representations. We present several examples of these approximations and discuss applications to fast algorithms. In particular, we show how to obtain a short separated representation (sum of products of onedimensional functions) of certain multidimensional Green’s functions.
FAST COMPUTATION OF FOURIER INTEGRAL OPERATORS
, 2007
"... We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a socalled Fourier integral operat ..."
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Cited by 14 (6 self)
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We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation, general hyperbolic equations, and curvilinear tomography. The problem is to numerically evaluate a socalled Fourier integral operator (FIO) of the form ∫ e2πiΦ(x,ξ) a(x, ξ) ˆ f(ξ)dξ at points given on a Cartesian grid. Here, ξ is a frequency variable, ˆ f(ξ) is the Fourier transform of the input f, a(x, ξ) isan amplitude, and Φ(x, ξ) is a phase function, which is typically as large as ξ; hence the integral is highly oscillatory. Because a FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size N by N would require O(N 4) operations. This paper develops a new numerical algorithm which requires O(N 2.5 log N) operations and as low as O ( √ N) in storage space (the constants in front of these estimates are small). It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel e2πiΦ(x,ξ) a(x, ξ) into two components: (1) a diffeomorphism which is handled by means of a nonuniform FFT and (2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.
Convolution Generated Motion and Generalized Huygens' Principles for Interface Motion
, 1998
"... A physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry. Accordingly, there is great interest in algorithms that generate such geometric interface motion. In this paper we unify and generalize two simple algorithms for constant and mean curv ..."
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Cited by 13 (6 self)
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A physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry. Accordingly, there is great interest in algorithms that generate such geometric interface motion. In this paper we unify and generalize two simple algorithms for constant and mean curvature based interface motion: the classical Huygens' principle, and diffusiongenerated motion. We show that resulting generalization can be viewed both geometrically as a type of Huygens' principle, and algebraically as a convolution generated motion. Using the geometricalgebraic duality from the unification, we construct specific convolution generated motion algorithms for a common class of anisotropic, curvaturedependent motion laws. We validate these algorithms with numerical experiments, and show that they can be implemented accurately and efficiently with adaptive resolution and fast Fourier transform techniques. Department of Mathematics, University of California at Los Angeles. (...
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 12 (5 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.
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).
On the computation of the polar FFT
 Appl. Comput. Harmon. Anal
, 2007
"... We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT. ..."
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Cited by 9 (7 self)
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We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT.
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 ..."
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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 &times; N images from their Radon transform. Both algorithms are based on fast Fourier transforms for nonequispaced data (NFFT) and require only O(N&sup2;log 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.