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35
Accelerating the nonuniform Fast Fourier Transform
 SIAM REVIEW
, 2004
"... The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in recon ..."
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Cited by 35 (2 self)
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The nonequispaced Fourier transform arises in a variety of application areas, from medical imaging to radio astronomy to the numerical solution of partial differential equations. In a typical problem, one is given an irregular sampling of N data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain. When the sampling is uniform, the fast Fourier transform (FFT) allows this calculation to be computed in O(N log N) operations rather than O(N 2) operations. Unfortunately, when the sampling is nonuniform, the FFT does not apply. Over the last few years, a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform FFTs (NUFFTs). These rely on a mixture of interpolation and the judicious use of the FFT on an oversampled grid [A. Dutt and V. Rokhlin, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1383]. In this paper, we observe that one of the standard interpolation or “gridding ” schemes, based on Gaussians, can be accelerated by a significant factor without precomputation and storage of the interpolation weights. This is of particular value in two and threedimensional settings, saving either 10dN in storage in d dimensions or a factor of about 5–10 in CPUtime (independent of dimension).
On the optimality of the gridding reconstruction algorithm
 IEEE Trans.Med.Imag.,vol.19,no.4,pp.306–317,2000
"... Abstract—Gridding reconstruction is a method to reconstruct data onto a Cartesian grid from a set of nonuniformly sampled measurements. This method is appreciated for being robust and computationally fast. However, it lacks solid analysis and design tools to quantify or minimize the reconstruction e ..."
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Cited by 26 (0 self)
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Abstract—Gridding reconstruction is a method to reconstruct data onto a Cartesian grid from a set of nonuniformly sampled measurements. This method is appreciated for being robust and computationally fast. However, it lacks solid analysis and design tools to quantify or minimize the reconstruction error. Least squares reconstruction (LSR), on the other hand, is another method which is optimal in the sense that it minimizes the reconstruction error. This method is computationally intensive and, in many cases, sensitive to measurement noise. Hence, it is rarely used in practice. Despite their seemingly different approaches, the gridding and LSR methods are shown to be closely related. The similarity between these two methods is accentuated when they are properly expressed in a common matrix form. It is shown that the gridding algorithm can be considered an approximation to the least squares method. The optimal gridding parameters are defined as the ones which yield the minimum approximation error. These parameters are calculated by minimizing the norm of an approximation error matrix. This problem is studied and solved in the general form of approximation using linearly structured matrices. This method not only supports more general forms of the gridding algorithm, it can also be used to accelerate the reconstruction techniques from incomplete data. The application of this method to a case of twodimensional (2D) spiral magnetic resonance imaging shows a reduction of more than 4 dB in the average reconstruction error. Index Terms—Gridding reconstruction, image reconstruction, matrix approximation, nonuniform sampling. I.
Overview of methods for image reconstruction from projections in emission computed tomography
 PROC. IEEE
, 2003
"... Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in t ..."
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Cited by 17 (1 self)
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Emission computed tomography (ECT) is a technology for medical imaging whose importance is increasing rapidly. There is a growing appreciation for the value of the functional (as opposed to anatomical) information that is provided by ECT and there are significant advancements taking place, both in the instrumentation for data collection, and in the computer methods for generating images from the measured data. These computer methods are designed to solve the inverse problem known as “image reconstruction from projections.” This paper uses the various models of the data collection process as the framework for presenting an overview of the wide variety of methods that have been developed for image reconstruction in the major subfields of ECT, which are positron emission tomography (PET) and singlephoton emission computed tomography (SPECT). The overall sequence of the major sections in the paper, and the presentation within each major section, both proceed from the more realistic and general models to those that are idealized and application specific. For most of the topics, the description proceeds from the threedimensional case to the twodimensional case. The paper presents a broad overview of algorithms for PET and SPECT, giving references to the literature where these algorithms and their applications are described in more detail.
DirectFourier Reconstruction In Tomography And Synthetic Aperture Radar
 Intl. J. Imaging Sys. and Tech
, 1998
"... We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR ..."
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Cited by 9 (0 self)
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We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobianweighted 2D periodic sinckernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the leastsquares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentricsquares sampling scheme, DF interpolation can be performed accurately and efficiently...
New Fourier reconstruction algorithms for computerized tomography
"... In this paper, we propose two new algorithms for high quality Fourier reconstructions of digital N × N images from their Radon transform. Both algorithms are based on fast Fourier transforms for nonequispaced data (NFFT) and require only O(N²log N) arithmetic operations. While the rst alg ..."
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Cited by 7 (3 self)
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In this paper, we propose two new algorithms for high quality Fourier reconstructions of digital N × N images from their Radon transform. Both algorithms are based on fast Fourier transforms for nonequispaced data (NFFT) and require only O(N²log N) arithmetic operations. While the rst algorithm includes a bivariate NFFT on the polar grid, the second algorithm consists of several univariate NFFTs on the socalled linogram.
FOURIER VOLUME RENDERING OF IRREGULAR DATA SETS
, 2002
"... Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We p ..."
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Cited by 6 (0 self)
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Examining irregularly sampled data sets usually requires gridding that data set. However, examination of a data set at one particular resolution may not be adequate since either fine details will be lost, or coarse details will be obscured. In either case, the original data set has been lost. We present an algorithm to create a regularly sampled data set from an irregular one. This new data set is not only an approximation to the original, but allows the original points to be accurately recovered, while still remaining relatively small. This result is accompanied by an efficient ‘zooming ’ operation that allows the user to increase the resolution while gaining new details, all without regridding the data. The technique is presented in Ndimensions, but is particularly well suited to Fourier Volume Rendering, which is the fastest known method of direct volume rendering. Together, these techniques allow accurate and efficient, multiresolution exploration of volume data.
An Efficient Fourier Method for 3D Radon Inversion in Exact ConeBeam CT Reconstruction
"... The 3D Radon transform of an object is an important intermediate result in many analytically exact conebeam reconstruction algorithms. In this paper, we present a new, highly efficient method for 3D Radon inversion, i.e. reconstruction of the image from the 3D Radon transform, called Direct Fourier ..."
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Cited by 5 (0 self)
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The 3D Radon transform of an object is an important intermediate result in many analytically exact conebeam reconstruction algorithms. In this paper, we present a new, highly efficient method for 3D Radon inversion, i.e. reconstruction of the image from the 3D Radon transform, called Direct Fourier Inversion (DFI). The method is based directly on the 3D Fourier Slice Theorem. From the 3D Radon data, which is assumed to be sampled on a polar grid, the 3D object spectrum is calculated by performing FFTs along radial lines in the Radon space. Then, an interpolation is performed from the polar to a cartesian grid using a 3D Gridding step in the frequency domain. Finally, this spectrum is transformed back to the spatial domain via 3D inverse FFT. The algorithm is extremely efficient with computational complexity in the order of N 3 log 2 (N). We have done reconstructions of simulated 3D Radon data assuming sampling conditions and image quality requirements similar to those in medical CT. ...
A New Solution to the Gridding Problem
 In Proceedings of SPIE Medical Imaging
, 2002
"... Image reconstruction from nonuniformly sampled frequency domain data is an important problem that arises in computed imaging. The current reconstruction techniques suffer from fundamental limitations in their model and implementation that result in blurred reconstruction and/or artifacts. Here, we p ..."
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Cited by 2 (0 self)
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Image reconstruction from nonuniformly sampled frequency domain data is an important problem that arises in computed imaging. The current reconstruction techniques suffer from fundamental limitations in their model and implementation that result in blurred reconstruction and/or artifacts. Here, we present a new approach for solving this problem that relies on a more realistic model and involves an explicit measure for the reconstruction accuracy that is optimized iteratively. The image is assumed piecewise constant to impose practical display constraints using pixels. We express the mapping of these unknown pixel values to the available frequency domain values as a linear system. Even though the system matrix is shown to be dense and too large to solve for practical purposes, we observe that applying a simple orthogonal transformation to the rows of this matrix converts the matrix into a sparse format. The transformed system is subsequently solved using the conjugate gradient method. The proposed method is applied to reconstruct images of a numerical phantom as well as actual magnetic resonance images using spiral sampling. The results support the theory and show that the computational load of this method is similar to that of other techniques. This suggests its potential for practical use.