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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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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.
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
Interferometric synthetic aperture microscopy: physicsbased image reconstruction from optical coherence tomography data
 In International Conference on Image Processing
, 2007
"... sensors ..."
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...
Reconstruction in Diffraction Ultrasound Tomography Using Nonuniform FFT
 IEEE Trans. Medical Imaging
, 2002
"... We show an iterative reconstruction framework for diffraction ultrasound tomography. The use of broadband illumination allows significant reduction of the number of projections compared to straight ray tomography. The proposed algorithm makes use of forward nonuniform fast Fourier transform (NUFFT) ..."
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We show an iterative reconstruction framework for diffraction ultrasound tomography. The use of broadband illumination allows significant reduction of the number of projections compared to straight ray tomography. The proposed algorithm makes use of forward nonuniform fast Fourier transform (NUFFT) for iterative Fourier inversion. Incorporation of total variation regularization allows the reduction of noise and Gibbs phenomena while preserving the edges. The complexity of the NUFFTbased reconstruction is comparable to the frequencydomain interpolation (gridding) algorithm, whereas the reconstruction accuracy (in sense of the and the norm) is better. Index TermsAcoustic diffraction tomography, image reconstruction, nonuniform fast Fourier transform (NUFFT).
Sampling In ParallelBeam Tomography
 in: Inverse Problems and Imaging, A.G. Ramm (editor), Plenum
, 1998
"... We pesent Shannon sampling theory for functions defined on T \Theta IR, where T denotes the circle group, prove a new estimate for the aliasing error, and apply the result to parallelbeam diffraction tomography. The class of admissible sampling lattices is characterized and general sampling conditi ..."
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Cited by 3 (1 self)
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We pesent Shannon sampling theory for functions defined on T \Theta IR, where T denotes the circle group, prove a new estimate for the aliasing error, and apply the result to parallelbeam diffraction tomography. The class of admissible sampling lattices is characterized and general sampling conditions are derived which lead to the identification of new efficient sampling schemes. Corresponding results for xray tomography are obtained in the highfrequency limit. 1. INTRODUCTION Sampling theorems provide interpolation formulas for functions whose Fourier transform is compactly supported. If the Fourier transform does not have compact support, a socalled aliasing error occurs. In this paper we pesent a new estimate for the aliasing error for functions defined on T \Theta IR, where T denotes the circle group, and work out its application to computed tomography. In computed tomography (CT) an object is exposed to radiation which is measured after passing through the object. From the...
Highresolution ab initio threedimensional xray diffraction microscopy
 Journal of the Optical Society of America A
, 2006
"... Coherent Xray diffraction microscopy is a method of imaging nonperiodic isolated objects at resolutions only limited, in principle, by the largest scattering angles recorded. We demonstrate Xray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative anal ..."
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Coherent Xray diffraction microscopy is a method of imaging nonperiodic isolated objects at resolutions only limited, in principle, by the largest scattering angles recorded. We demonstrate Xray diffraction imaging with high resolution in all three dimensions, as determined by a quantitative analysis of the reconstructed volume images. These images are retrieved from the 3D diffraction data using no a priori knowledge about the shape or composition of the object, which has never before been demonstrated on a nonperiodic object. We also construct 2D images of thick objects with infinite depth of focus (without loss of transverse spatial resolution). These methods can be used to image biological and materials science samples at high resolution using Xray undulator radiation, and establishes the techniques to be used in atomicresolution ultrafast imaging at Xray freeelectron laser sources. OCIS codes: 340.7460, 110.1650, 110.6880, 100.5070, 100.6890, 070.2590, 180.6900 1.
Signal Processing Issues In Synthetic Aperture Radar And Computer Tomography
, 1998
"... This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2D) trapezoidal rule. In addition, the possibility of reconstruction from a concentricsquares raster was discussed. Numerous simple interpolators have bee ..."
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This paper also proposed another reconstruction method based on a direct approximation of the Fourier inversion formula using a twodimensional (2D) trapezoidal rule. In addition, the possibility of reconstruction from a concentricsquares raster was discussed. Numerous simple interpolators have been tried in DF reconstruction with the results compared with CBP [33]. In [34] and [35], the concept of angular bandlimiting was used to interpolate the polar data onto a Cartesian grid. In [36], a DF reconstruction using bilinear interpolation for diffraction tomography provided image quality that was comparable to that produced by the CBP algorithm. Very good reconstruction quality was obtained in [37] and [38] using a spline interpolator, or a hybrid type of spline interpolator. The notion of "gridding" was introduced in [39] as a method of obtaining optimal inversion of Fourier data. An optimal gridding function was proposed, and successful results were obtained when applied to the tomographic reconstruction problem. In [40], several different gridding functions were tried for DF reconstruction, and the performances were compared. In [41, 42], the linogram reconstruction method was proposed as a form of DF reconstruction. The data collection grid in the linogram method is the same as in the concentricsquares sampling scheme. The inversion of the Fourier data in [41, 42] was accomplished by first applying the chirpz transform in one direction and then computing FFTs in the other direction. In CT, many of these attempts at DF reconstruction have given a poorer result than the CBP algorithm, due to the error incurred in the process of the polartoCartesian interpolation. The attraction of DF reconstruction, however, is that it is thought to require less computation than ...
Accurate image reconstruction from fewview and limitedangle data in diffraction tomography
, 2007
"... We present a method for obtaining accurate image reconstruction from highly sparse data in diffraction tomography (DT). A practical need exists for reconstruction from fewview and limitedangle data, as this can greatly reduce required scan times in DT. Our method does this by minimizing the total ..."
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We present a method for obtaining accurate image reconstruction from highly sparse data in diffraction tomography (DT). A practical need exists for reconstruction from fewview and limitedangle data, as this can greatly reduce required scan times in DT. Our method does this by minimizing the total variation (TV) of the estimated image, subject to the constraint that the Fourier transform of the estimated image matches the measured Fourier data samples. Using simulation studies, we show that the TVminimization algorithm allows accurate reconstruction in a variety of fewview and limitedangle situations in DT. Accurate image reconstruction is obtained from far fewer data samples than are required by common algorithms such as the filteredbackpropagation algorithm. Overall our results indicate that the TVminimization algorithm can be successfully applied to DT image reconstruction under a variety of scan configurations and data conditions of practical