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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 ..."
Abstract

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.
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.
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
 Add to MetaCart
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.
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 commonly reconstructed with gridding methods and option ..."
Abstract
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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 commonly 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 addressed it differently so far. In the present work, a unified approach is proposed. An extension of the principle behind gridding methods is shown to permit its application to field inhomogeneity compensation. Based on this result, several new correction algorithms are derived from a straightforward embedding of the data into a higher dimensional space. They are evaluated in simulations and phantom experiments with spiral kspace sampling. Compared with existing algorithms, one of them promises to provide a favorable compromise between fidelity and complexity. Moreover, it allows a simple choice of key parameters involved in approximating an exponential function and a balance between reconstruction and correction accuracy. Index Terms — Magnetic resonance imaging, image reconstruction, gridding, field inhomogeneity, offresonance correction, conjugate phase reconstruction, iterative reconstruction, spiral imaging I.
Improved Time Bounds for NearOptimal Sparse Fourier Representations
"... We study the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N. The Fast Fourier Transform (FFT) can find the optimal Nterm representation in time O(N log N) time, but our goal is to get sublinear time algorithms when m ≪ N. ..."
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We study the problem of finding a Fourier representation R of m terms for a given discrete signal A of length N. The Fast Fourier Transform (FFT) can find the optimal Nterm representation in time O(N log N) time, but our goal is to get sublinear time algorithms when m ≪ N.
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 optionall ..."
Abstract
 Add to MetaCart
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 algorithms, which are compatible with a direct conjugate phase and an iterative algebraic reconstruction, 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—Conjugate phase reconstruction, field inhomogeneity, gridding, image reconstruction, iterative reconstruction, magnetic resonance imaging, offresonance correction, spiral imaging. I.