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14
Combinatorial sublineartime fourier algorithms,” Submitted. Available at http://www.ima.umn.edu/∼iwen/index.html
, 2008
"... We study the problem of estimating the best k term Fourier representation for a given frequencysparse signal (i.e., vector) A of length N ≫ k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomia ..."
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Cited by 16 (5 self)
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We study the problem of estimating the best k term Fourier representation for a given frequencysparse signal (i.e., vector) A of length N ≫ k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomial(k, log N) time. Randomized sublinear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem [24, 25]. In this paper we develop the first known deterministic sublinear time sparse Fourier Transform algorithm which is guaranteed to produce accurate results. As an added bonus, a simple relaxation of our deterministic Fourier result leads to a new Monte Carlo Fourier algorithm with similar runtime/sampling bounds to the current best randomized Fourier method [25]. Finally, the Fourier algorithm we develop here implies a simpler optimized version of the deterministic compressed sensing method previously developed in [30]. 1
A deterministic sublinear time sparse fourier algorithm via nonadaptive compressed sensing methods
 in Proceedings of the 19th Symposium on Discrete Algorithms (SODA
, 2008
"... We study the problem of estimating the best B term Fourier representation for a given frequencysparse signal (i.e., vector) A of length N≫B. More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomial( ..."
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Cited by 15 (5 self)
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We study the problem of estimating the best B term Fourier representation for a given frequencysparse signal (i.e., vector) A of length N≫B. More explicitly, we investigate how to deterministically identify B of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomial(B, log N) time. Randomized sublinear time algorithms which have a small (controllable) probability of failure for each processed signal exist for solving this problem. However, for failure intolerant applications such as those involving missioncritical hardware designed to process many signals over a long lifetime, deterministic algorithms with no probability of failure are highly desirable. In this paper we build on the deterministic Compressed Sensing results of Cormode and Muthukrishnan (CM) [26, 6, 7] in order to develop the first known deterministic sublinear time sparse Fourier Transform algorithm suitable for failure intolerant applications. Furthermore, in the process of developing our new Fourier algorithm, we present a simplified deterministic Compressed Sensing algorithm which improves on CM’s algebraic compressibility results while simultaneously maintaining their results concerning exponential decay. 1
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.
Improved bounds for a deterministic sublineartime Sparse Fourier Algorithm
 In 42nd Annual Conference on Information Sciences and Systems (CISS
, 2008
"... Abstract—This paper improves on the bestknown runtime and measurement bounds for a recently proposed Deterministic sublineartime Sparse Fourier Transform algorithm (hereafter called DSFT). In [1], [2], it is shown that DSFT can exactly reconstruct the Fourier transform (FT) of an Nbandwidth signa ..."
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Cited by 7 (5 self)
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Abstract—This paper improves on the bestknown runtime and measurement bounds for a recently proposed Deterministic sublineartime Sparse Fourier Transform algorithm (hereafter called DSFT). In [1], [2], it is shown that DSFT can exactly reconstruct the Fourier transform (FT) of an Nbandwidth signal f, consisting of B ≪ N nonzero frequencies, using O(B 2 ·polylog(N)) time and O(B 2 · polylog(N)) fsamples. DSFT works by taking advantage of natural aliasing phenomena to hash a frequencysparse signal’s FT information modulo O(B·polylog(N)) pairwise coprime numbers via O(B · polylog(N)) small Discrete Fourier Transforms. Number theoretic arguments then guarantee the original DFT frequencies/coefficients can be recovered via the Chinese Remainder Theorem. DSFT’s usage of primes makes its runtime and signal sample requirements highly dependent on the sizes of sums and products of small primes. Our new bounds utilize analytic number theoretic techniques to generate improved (asymptotic) bounds for DSFT. As a result, we provide better bounds for the sampling complexity/number of lowrate analogtodigital converters (ADCs) required to deterministically recover frequencysparse wideband signals via DSFT in signal processing applications [3], [4]. Index Terms—Fourier transforms, Discrete Fourier transforms, Algorithms, Number theory, Signal processing
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.
THE FAST SINC TRANSFORM AND IMAGE RECONSTRUCTION FROM NONUNIFORM SAMPLES IN kSPACE
"... A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an a ..."
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A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an algorithm which computes the convolution of arbitrarily spaced data with the sinc kernel in O.N log N / operations, where N denotes the number of data points. We briefly discuss its application to the construction of optimal density compensation weights for Fourier reconstruction and to the iterative approximation of the pseudoinverse of the signal equation in MRI. 1.
1 Field Inhomogeneity Correction based on Gridding Reconstruction for Magnetic Resonance Imaging
"... Abstract — Spatial variations of the main field give rise to artifacts in magnetic resonance images if disregarded in reconstruction. With nonCartesian kspace sampling, they often lead to unacceptable blurring. Data from such acquisitions are usually reconstructed with gridding methods and optiona ..."
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Abstract — Spatial variations of the main field give rise to artifacts in magnetic resonance images if disregarded in reconstruction. With nonCartesian kspace sampling, they often lead to unacceptable blurring. Data from such acquisitions are usually reconstructed with gridding methods and optionally restored with various correction methods. Both types of methods essentially face the same basic problem of adequately approximating an exponential function to enable an efficient processing with Fast Fourier Transforms. Nevertheless, they have commonly addressed it differently so far. In the present work, a unified approach is pursued. The principle behind gridding methods is first generalized to nonequispaced sampling in both domains and then applied to field inhomogeneity correction. Three new iterative algorithms are derived in this way from a straightforward embedding of the data into a higher dimensional space. Their evaluation in simulations and phantom experiments with spiral kspace sampling shows that one of them promises to provide a favorable compromise between fidelity and complexity compared with existing algorithms. Moreover, it allows a simple choice of key parameters involved in approximating an exponential function and a balance between the accuracy of reconstruction and correction. Index Terms — Magnetic resonance imaging, image reconstruction, gridding, field inhomogeneity, offresonance correction, conjugate phase reconstruction, iterative reconstruction, spiral imaging I.
Domain Decomposition Based Hybrid Methods for Solving RealLife Electromagnetic Scattering and Radiation Problems
, 2012
"... opportunity to work with him and supporting my studies at the University of Michigan. I would like to thank to the members of my dissertation committee Kamal Sarabandi, Mona Jarrahi, and Christopher Ruf for their time reading this dissertation and their valuable comments. Also, I would like to menti ..."
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opportunity to work with him and supporting my studies at the University of Michigan. I would like to thank to the members of my dissertation committee Kamal Sarabandi, Mona Jarrahi, and Christopher Ruf for their time reading this dissertation and their valuable comments. Also, I would like to mention Professors Fawwaz Ulaby and Anthony Grbic who were in my preliminary exam. I would like to express my gratitude to Professor Vakur Ertürk who was my advisor during my graduate studies at Bilkent University. His inspiration, motivation, and mentorship helped me to get this far. I would like to thank to my colleagues Felipe Valdes and Francesco Andriulli at Michielssen research group. I would like to extend special thanks to Hakan Bağcı who was a postdoc in our group, now a Professor at KAUST for his help, knowledge, and friendship in the first years of my studies. I would like to thank to all my colleagues in the Radiation Laboratory (Radlab), University of Michigan. I would like to mention some of them here (in alphabetical order) Michael Benson, Mariko Burgin, Fikadu Dagefu,
THE FAST SINC TRANSFORM AND IMAGE RECONSTRUCTION FROM NONUNIFORM SAMPLES IN kSPACE
"... A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an a ..."
Abstract
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A number of problems in image reconstruction and image processing can be addressed, in principle, using the sinc kernel. Since the sinc kernel decays slowly, however, it is generally avoided in favor of some more local but less precise choice. In this paper, we describe the fast sinc transform, an algorithm which computes the convolution of arbitrarily spaced data with the sinc kernel in O.N log N / operations, where N denotes the number of data points. We briefly discuss its application to the construction of optimal density compensation weights for Fourier reconstruction and to the iterative approximation of the pseudoinverse of the signal equation in MRI. 1.