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28
Nonuniform Fast Fourier Transforms Using MinMax Interpolation
 IEEE Trans. Signal Process
, 2003
"... The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several pap ..."
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Cited by 83 (13 self)
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The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformlyspaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the minmax sense of minimizing the worstcase approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the minmax approach provides substantially lower approximation errors than conventional interpolation methods. The minmax criterion is also useful for optimizing the parameters of interpolation kernels such as the KaiserBessel function.
Fast slant stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible
 SIAM J. Sci. Comput
, 2001
"... Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition i ..."
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Cited by 48 (11 self)
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Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition is geometrically faithful: the lines exhibit no ‘wraparound effects’. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n2 is the number of pixels. This relies on a discrete projectionslice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2D Fourier transform on a nonCartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT – the process of rapid exact evaluation of the 2D Fourier transform at these nonCartesian grid points – is possible using chirpZ transforms. This Radon transform is onetoone and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3D version of the transform.
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.
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 23 (4 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.
Resampling of data between arbitrary grids using convolution interpolation
 IEEE Trans. Med. Imag
, 1999
"... Abstract—For certain medical applications resampling of data is required. In magnetic resonance tomography (MRT) or computer tomography (CT), e.g., data may be sampled on nonrectilinear grids in the Fourier domain. For the image reconstruction a convolutioninterpolation algorithm, often called grid ..."
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Cited by 19 (0 self)
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Abstract—For certain medical applications resampling of data is required. In magnetic resonance tomography (MRT) or computer tomography (CT), e.g., data may be sampled on nonrectilinear grids in the Fourier domain. For the image reconstruction a convolutioninterpolation algorithm, often called gridding, can be applied for resampling of the data onto a rectilinear grid. Resampling of data from a rectilinear onto a nonrectilinear grid are needed, e.g., if projections of a given rectilinear data set are to be obtained. In this paper we introduce the application of the convolution interpolation for resampling of data from one arbitrary grid onto another. The basic algorithm can be split into two steps. First, the data are resampled from the arbitrary input grid onto a rectilinear grid and second, the rectilinear data is resampled onto the arbitrary output grid. Furthermore, we like to introduce a new technique to derive the sampling density function needed for the first step of our algorithm. For fast, samplingpatternindependent determination of the sampling density function the Voronoi diagram of the sample distribution is calculated. The volume of the Voronoi cell around each sample is used as a measure for the sampling density. It is shown that the introduced resampling technique allows fast resampling of data between arbitrary grids. Furthermore, it is shown that the suggested approach to derive the sampling density function is suitable even for arbitrary sampling patterns. Examples are given in which the proposed technique has been applied for the reconstruction of data acquired along spiral, radial, and arbitrary trajectories and for the fast calculation of projections of a given rectilinearly sampled image. Index Terms—Arbitrary grids, convolution interpolation, density function, Voronoi diagram. I.
Toeplitzbased iterative image reconstruction for MRI with correction for magnetic field inhomogeneity
 IEEE Trans. Signal Process
, 2005
"... Abstract — In some types of magnetic resonance (MR) imaging, particularly functional brain scans, the conventional Fourier model for the measurements is inaccurate. Magnetic field inhomogeneities, caused by imperfect main fields and by magnetic susceptibility variations, induce distortions in images ..."
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Cited by 18 (3 self)
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Abstract — In some types of magnetic resonance (MR) imaging, particularly functional brain scans, the conventional Fourier model for the measurements is inaccurate. Magnetic field inhomogeneities, caused by imperfect main fields and by magnetic susceptibility variations, induce distortions in images that are reconstructed by conventional Fourier methods. These artifacts hamper the use of functional MR imaging (fMRI) in brain regions near air/tissue interfaces. Recently, iterative methods that combine the conjugate gradient (CG) algorithm with nonuniform FFT (NUFFT) operations have been shown to provide considerably improved image quality relative to the conjugatephase method. However, for nonCartesian kspace trajectories, each CGNUFFT iteration requires numerous kspace interpolations, operations that are computationally expensive and poorly suited to fast hardware implementations. This paper proposes a faster iterative approach to fieldcorrected MR image reconstruction based on the CG algorithm and certain Toeplitz matrices. This CGToeplitz approach requires kspace interpolations only for the initial iteration; thereafter only FFTs are required. Simulation results show that the proposed CGToeplitz approach produces equivalent image quality as the CGNUFFT method with significantly reduced computation time. Index Terms — fMRI imaging, spiral trajectory, magnetic susceptibility, nonCartesian 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.
A fast and accurate multilevel inversion of the radon transform
 SIAM J. Appl. Math
, 1999
"... Abstract. A number of imaging technologies reconstruct an image function from its Radon projection using the convolution backprojection method. The convolution is an O(N 2 log N) algorithm, where the image consists of N ×N pixels, while the backprojection is an O(N 3) algorithm, thus constituting th ..."
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Cited by 12 (2 self)
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Abstract. A number of imaging technologies reconstruct an image function from its Radon projection using the convolution backprojection method. The convolution is an O(N 2 log N) algorithm, where the image consists of N ×N pixels, while the backprojection is an O(N 3) algorithm, thus constituting the major computational burden of the convolution backprojection method. An O(N 2 log N) multilevel backprojection method is presented here. When implemented with a Fourierdomain postprocessing technique, also presented here, the resulting image quality is similar or superior to the image quality of the classical backprojection technique. Key words. Radon transform, inversion of the Radon transform, computed tomography, convolution backprojection, multilevel, Fourierdomain postprocessing AMS subject classifications. 92C55, 44A12, 65R10, 68U10 PII. S003613999732425X 1. Background. Reconstruction of a function of two or three variables from its Radon transform has proven vital in computed tomography (CT), nuclear magnetic resonance imaging, astronomy, geophysics, and a number of other fields [13]. One of the best known reconstruction algorithms is the convolution backprojection method (CB), which is widely used in commercial CT devices [13] (with rebinning for divergentbeam projections [18]). Recently, it has been applied to spotlightmode synthetic aperture radar image reconstruction [14, 23] in which the conventional method is the direct Fourier method (DF), i.e., Fourierdomain interpolation followed by twodimensional (2D) FFT [21]. Originally, CB was preferred to DF since the former provided better images [18, 20]. However, since the backprojection part of CB raises the computational complexity of the method to O(N 3), while DF’s complexity is O(N 2 log N), there has been
ConeBeam Reconstruction Using Filtered Backprojection
, 2001
"... To my parents The art of medical computed tomography is constantly evolving and the last years have seen new ground breaking systems with multirow detectors. These tomographs are able to increase both scanning speed and image quality compared to the singlerow systems more commonly found in hospita ..."
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Cited by 10 (1 self)
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To my parents The art of medical computed tomography is constantly evolving and the last years have seen new ground breaking systems with multirow detectors. These tomographs are able to increase both scanning speed and image quality compared to the singlerow systems more commonly found in hospitals today. This thesis deals with threedimensional image reconstruction algorithms to be used in future generations of tomographs with even more detector rows than found in current multirow systems. The first practical algorithm for threedimensional reconstruction from conebeam projections acquired from a circular source trajectory is the FDK method. We present a novel version of this algorithm that produces images of higher quality. We also formulate a version of the FDK method that performs the backprojection in O(N 3 log N) steps instead of the O(N 4) steps traditionally required. An efficient way to acquire volumetric patient data is to use a helical source
How GPUs can improve the quality of magnetic resonance imaging
 In The First Workshop on General Purpose Processing on Graphics Processing Units
, 2007
"... Abstract — In magnetic resonance imaging (MRI), nonCartesian scan trajectories are advantageous in a wide variety of emerging applications. Advanced reconstruction algorithms that operate directly on nonCartesian scan data using optimality criteria such as leastsquares (LS) can produce significan ..."
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Cited by 9 (3 self)
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Abstract — In magnetic resonance imaging (MRI), nonCartesian scan trajectories are advantageous in a wide variety of emerging applications. Advanced reconstruction algorithms that operate directly on nonCartesian scan data using optimality criteria such as leastsquares (LS) can produce significantly better images than conventional algorithms that apply a fast Fourier transform (FFT) after interpolating the scan data onto a Cartesian grid. However, advanced LS reconstructions require significantly more computation than conventional reconstructions based on the FFT. For example, one LS algorithm requires nearly six hours to reconstruct a single threedimensional image on a modern CPU. Our work demonstrates that this advanced reconstruction can be performed quickly and efficiently on a modern GPU, with the reconstruction of a 64 3 3D image requiring just three minutes, an acceptable latency for key applications. This paper describes how the reconstruction algorithm leverages the resources of the GeForce 8800 GTX (G80) to achieve over 150 GFLOPS in performance. We find that the combination of tiling the data and storing the data in the G80’s constant memory dramatically reduces the algorithm’s required bandwidth to offchip memory. The G80’s special functional units provide substantial acceleration for the trigonometric computations in the algorithm’s inner loops. Finally, experimentdriven code transformations increase the reconstruction’s performance by as much as 60 % to 80%. I.