Vol. 1, No. 1, 2006

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ISSN: 2157-5452 (e-only)
ISSN: 1559-3940 (print)
The fast sinc transform and image reconstruction from nonuniform samples in $k$-space

Leslie Greengard, June-Yub Lee and Souheil Inati

Vol. 1 (2006), No. 1, 121–131
Abstract

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(NlogN) 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.

Keywords
sinc interpolation, fast transform, nonuniform fast Fourier transform, density compensation weights, iterative methods, Fourier analysis, image reconstruction, magnetic resonance imaging (MRI)
Milestones
Received: 30 November 2005
Accepted: 20 May 2006
Published: 9 May 2007
Authors
Leslie Greengard
Courant Institute
New York University
New York, NY 10012
United States
June-Yub Lee
Department of Mathematics
Ewha Women’s University
Seoul 120-750
Korea
Souheil Inati
Center for Neural Science and Department of Psychology
New York University
New York, NY 10003
United States