A New Class of Analysis-Based Fast Transforms

Rokhlin, Vladimir; O’Neil, Michael

doi:10.21236/ada471850

Cited by 19 publications

(19 citation statements)

References 25 publications

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“…We begin by offering a general description of the butterfly structure and then provide several concrete examples. This general structure was originally introduced in [30], and later generalized in [31].…”

Section: The Butterfly Algorithmmentioning

confidence: 99%

A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

Candès¹,

Demanet²,

Ying³

2009

Multiscale Model. Simul.

131

229

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This paper is concerned with the fast computation of Fourier integral operators of the general form R d e 2πıΦ(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 near-optimal 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 low-rank; we propose constructing such low-rank 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.

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Section: The Butterfly Algorithmmentioning

confidence: 99%

A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

Candès¹,

Demanet²,

Ying³

2009

Multiscale Model. Simul.

131

229

View full text Add to dashboard Cite

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“…To realize the above idea, the butterfly algorithm (Michielssen and Boag, 1996;O'Neil and Rokhlin, 2007) turns out to be an appropriate tool. The main data structure underlying the algorithm is a pair of dyadic trees T X and T K .…”

Section: Butterfly Structurementioning

confidence: 99%

A fast butterfly algorithm for the hyperbolic Radon transform

Fomel

Demanet

et al. 2012

SEG Technical Program Expanded Abstracts 2012

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SUMMARYWe introduce a fast butterfly algorithm for the hyperbolic Radon transform commonly used in seismic data processing. For two-dimensional data, the algorithm runs in complexity O (N 2 log N), where N is representative of the number of points in either dimension of data space or model space. Using a series of examples, we show that the proposed algorithm is significantly more efficient than conventional integration.

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“…The butterfly algorithm originates from work of Michielssen and Boag, 21 and has recently been popularized in a string of papers by Rokhlin and O'Neil, 23 Ying, 33 Candès, Demanet and Ying 10 and Tygert. 29 Note that the algorithmic variant presented in our earlier work 10 is particularly well suited for the application to SAR imaging: unlike ONeil and Rokhlin 23 it does not have a precomputation step.…”

Section: The Butterfly Algorithmmentioning

confidence: 99%

A butterfly algorithm for synthetic aperture radar

et al. 2011

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It is not currently known if it is possible to accurately form a synthetic aperture radar image from N data points in provable near-linear complexity, where accuracy is defined as the 2 error between the full O(N 2 ) backprojection image and the approximate image. To bridge this gap, we present a backprojection algorithm with complexity O(log(1/ )N log N), with the tunable pixelwise accuracy. It is based on the butterfly scheme, which works for vastly more general oscillatory integrals than the discrete Fourier transform. Unlike previous methods this algorithm allows the user to directly choose the amount of acceptable image error based on a well-defined metric. Additionally, the algorithm does not invoke the far-field approximation or place restrictions on the antenna flight path, nor does it impose the frequency-independent beampattern approximation required by time-domain backprojection techniques.

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A New Class of Analysis-Based Fast Transforms

Cited by 19 publications

References 25 publications

A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

A Fast Butterfly Algorithm for the Computation of Fourier Integral Operators

A fast butterfly algorithm for the hyperbolic Radon transform

A butterfly algorithm for synthetic aperture radar

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