Results 11  20
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24
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.
Sparse Fourier transform via butterfly algorithm
, 2008
"... We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is that the interaction between a frequency region and a spatia ..."
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Cited by 4 (4 self)
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We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their radii are bounded by the maximum frequency. Based on this property, equivalent sources located at Cartesian grids are used to speed up the computation of the interaction between these two regions. The overall structure of our algorithm follows the recentlyintroduced butterfly algorithm. The computation is further accelerated by exploiting the tensorproduct property of the Fourier kernel in two and three dimensions. The proposed algorithm is accurate and has an O(N log N) complexity. Finally, we present numerical results in both two and three dimensions.
A FAST BUTTERFLY ALGORITHM FOR THE COMPUTATION OF FOURIER INTEGRAL OPERATORS
, 2009
"... This paper is concerned with the fast computation of Fourier integral operators of the general form ∫ Rd e2πıΦ(x,k) f(k)dk, wherekisafrequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest, for such fundamental computation ..."
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Cited by 3 (2 self)
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This paper is concerned with the fast computation of Fourier integral operators of the general form ∫ Rd e2πıΦ(x,k) f(k)dk, wherekisafrequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest, for such fundamental computations are connected with the problem of finding numerical solutions to wave equations and also frequently arise in many applications including reflection seismology, curvilinear tomography, and others. In two dimensions, when the input and output are sampled on N × N Cartesian grids, a direct evaluation requires O(N 4) operations, which is often times prohibitively expensive. This paper introduces a novel algorithm running in O(N 2 log N) time, i.e., with nearoptimal computational complexity, and whose overall structure follows that of the butterfly algorithm. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel e2πıΦ(x,k) to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel is approximately lowrank; we propose constructing such lowrank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part, and finally remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.
Accurate and Fast Discrete Polar Fourier Transform
 in Proc. 37th Asilomar Conf. Signals, Systems & Computers
, 2003
"... 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 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. ..."
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Cited by 2 (0 self)
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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 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 approach is the pseudopolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. The pseudopolar FFT plays the role of a halfway point  a nearlypolar system from which conversion to Polar Coordinates uses processes relying purely on interpolation operations. We describe the conversion process, and compare accuracy results obtained by unequallysampled FFT methods to ours and show marked advantage to our approach.
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.
Simplified MRI Signal Model
, 2002
"... MRI rosette kspace trajectory MRI Application ky kx 2 Ignoring lots of things: ..."
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MRI rosette kspace trajectory MRI Application ky kx 2 Ignoring lots of things:
© DIGITAL VISION A Tutorial on Fast Fourier Sampling [How to apply it to problems]
"... Suppose that x is a discretetime signal of length N that can be expressed with only m digital frequencies where m ≪ N: x[t] = 1 N m∑ ..."
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Suppose that x is a discretetime signal of length N that can be expressed with only m digital frequencies where m ≪ N: x[t] = 1 N m∑
A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators
, 2008
"... This paper is concerned with the fast computation of Fourier integral operators of the general form ∫ Rd e2πıΦ(x,k) f(k)dk, where k is a frequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest for such fundamental computat ..."
Abstract
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This paper is concerned with the fast computation of Fourier integral operators of the general form ∫ Rd e2πıΦ(x,k) f(k)dk, where k is a frequency variable, Φ(x, k) is a phase function obeying a standard homogeneity condition, and f is a given input. This is of interest for such fundamental computations are connected with the problem of finding numerical solutions to wave equations, and also frequently arise in many applications including reflection seismology, curvilinear tomography and others. In two dimensions, when the input and output are sampled on N × N Cartesian grids, a direct evaluation requires O(N 4) operations, which is often times prohibitively expensive. This paper introduces a novel algorithm running in O(N 2 log N) time, i. e. with nearoptimal computational complexity, and whose overall structure follows that of the butterfly algorithm [30]. Underlying this algorithm is a mathematical insight concerning the restriction of the kernel e 2πıΦ(x,k) to subsets of the time and frequency domains. Whenever these subsets obey a simple geometric condition, the restricted kernel has approximately lowrank; we propose constructing such lowrank approximations using a special interpolation scheme, which prefactors the oscillatory component, interpolates the remaining nonoscillatory part and, lastly, remodulates the outcome. A byproduct of this scheme is that the whole algorithm is highly efficient in terms of memory requirement. Numerical results demonstrate the performance and illustrate the empirical properties of this algorithm.
Rafael de la Llave, Supervisor Oscar Gonzalez
"... certifies that this is the approved version of the following dissertation: Existence and persistence of invariant objects in dynamical systems and mathematical physics Committee: ..."
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certifies that this is the approved version of the following dissertation: Existence and persistence of invariant objects in dynamical systems and mathematical physics Committee: