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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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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.
Iterative tomographic image reconstruction using Fourierbased forward and back projectors
 IEEE Trans. Med. Imag
, 2004
"... Fourierbased reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourierbased reprojection methods. We apply a minmax interpolation method for the nonuniform fast Fourier transform (NUFFT) t ..."
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Cited by 26 (5 self)
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Fourierbased reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourierbased reprojection methods. We apply a minmax interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the minmax NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.
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 5 (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.
DETECTION OF EDGES FROM NONUNIFORM FOURIER DATA
"... Abstract. Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing ..."
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Abstract. Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the locations of jump discontinuities, or edges, in a onedimensional periodic piecewisesmooth function from nonuniform Fourier coefficients. The technique employs the use of Fourier frames. Numerical examples are provided. 1.
On stable reconstructions from nonuniform Fourier measurements
 SIAM J. Imaging Sci
, 2014
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Deconvolutioninterpolation gridding (DING): accurate reconstruction for arbitrary kspace trajectories
 Magnetic Resonance in Medicine
, 2006
"... A simple iterative algorithm, termed deconvolutioninterpolation gridding (DING), is presented to address the problem of reconstructing images from arbitrarilysampled kspace. The new algorithm solves a sparse system of linear equations that is equivalent to a deconvolution of the kspace with a s ..."
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A simple iterative algorithm, termed deconvolutioninterpolation gridding (DING), is presented to address the problem of reconstructing images from arbitrarilysampled kspace. The new algorithm solves a sparse system of linear equations that is equivalent to a deconvolution of the kspace with a small window. The deconvolution operation results in increased reconstruction accuracy without grid subsampling, at some cost to computational load. By avoiding grid oversampling, the new solution saves memory, which is critical for 3D trajectories. The DING algorithm does not require the calculation of a sampling density compensation function, which is often problematic. DING’s sparse linear system is inverted efficiently using the conjugate gradient (CG) method. The reconstruction of the gridding system matrix is simple and fast, and no regularization is needed. This feature renders DING suitable for situations where the kspace trajectory is changed often or is not known a priori, such as when patient motion occurs during the scan. DING was compared with conventional gridding and an iterative reconstruction method in computer simulations and in vivo spiral MRI experiments. The results demonstrate a stable performance and reduced root mean square (RMS) error for DING in different kspace trajectories. Magn Reson Med 56:
On stable reconstructions from univariate nonuniform Fourier measurements. arXiv:1310.7820
, 2013
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Iterative NextNeighbor Regridding (INNG): improved reconstruction from nonuniformly sampled kspace data using rescaled matrices
 Magn Reson Med
, 2004
"... The reconstruction of MR images from nonrectilinearly sampled data is complicated by the fact that the inverse 2D Fourier transform (FT) cannot be performed directly on the acquired kspace data set. kSpace gridding is commonly used because it is an efficient reconstruction method. However, convent ..."
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The reconstruction of MR images from nonrectilinearly sampled data is complicated by the fact that the inverse 2D Fourier transform (FT) cannot be performed directly on the acquired kspace data set. kSpace gridding is commonly used because it is an efficient reconstruction method. However, conventional gridding requires optimized density compensation functions (DCFs) to avoid profile distortions. Oftentimes, the calculation of optimized DCFs presents an additional challenge in obtaining an accurately gridded reconstruction. Another type of gridding algorithm, the block uniform resampling (BURS) algorithm, often requires singular value decomposition (SVD) regularization to avoid amplification of data imperfections, and under some conditions it is difficult to adjust the regularization parameters. In this work, new reconstruction algorithms for nonuniformly sampled kspace data are presented. In the newly proposed algorithms, highquality reconstructed images are obtained from an iterative reconstruction that is performed using matrices scaled to sizes greater than that of the target image matrix. A second version partitions the sampled kspace region into several blocks to avoid limitations that could result from performing multiple 2DFFTs on large data matrices. The newly proposed algorithms are a simple alternative approach to previously proposed optimized gridding algorithms. Magn Reson