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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
Multilevel Image Reconstruction with Natural Pixels
 SIAM J. Sci. Comp
, 1995
"... The sampled Radon transform of a 2D function can be represented as a continuous linear map A : L 2(\Omega\Gamma ! R N , where (Au) j = hu; / j i and / j is the characteristic function of a strip through \Omega approximating the set of line integrals in the sample. The image reconstruction problem ..."
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Cited by 9 (0 self)
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The sampled Radon transform of a 2D function can be represented as a continuous linear map A : L 2(\Omega\Gamma ! R N , where (Au) j = hu; / j i and / j is the characteristic function of a strip through \Omega approximating the set of line integrals in the sample. The image reconstruction problem is: given a vector b 2 R N , find an image (or density function) u(x; y) such that Au = b. In general there are infinitely many solutions; we seek the solution with minimal 2norm, which leads to a matrix equation Bw = b, where B is a square dense matrix with several convenient properties. We analyze the use of GaussSeidel iteration applied to the problem, observing that while the iteration formally converges, there exists a near null space into which the error vectors migrate, after which the iteration stalls. The null space and near null space of B are characterized in order to develop a multilevel scheme. Based on the principles of the Multilevel Projection Method (PML), this scheme l...
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...
AntiAliased 3D ConeBeam Reconstruction Of LowContrast Objects With Algebraic Methods
 IEEE Trans. Med. Imag
, 1999
"... This paper examines the use of the Algebraic Reconstruction Technique (ART) and related techniques to reconstruct 3D objects from a relatively sparse set of conebeam projections. Although ART has been widely used for conebeam reconstruction of highcontrast objects, e.g. in computed angiography ..."
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Cited by 9 (3 self)
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This paper examines the use of the Algebraic Reconstruction Technique (ART) and related techniques to reconstruct 3D objects from a relatively sparse set of conebeam projections. Although ART has been widely used for conebeam reconstruction of highcontrast objects, e.g. in computed angiography, the work presented here explores the more challenging lowcontrast case which represents a little investigated scenario for ART. Preliminary experiments indicate that for cone angles greater than 20, traditional ART produces reconstructions with strong aliasing artifacts. These artifacts are in addition to the usual offmidplane inaccuracies of conebeam tomography with planar orbits. We find that the source of these artifacts is the nonuniform reconstruction grid sampling and correction by the conebeam rays during the ART projection/backprojection procedure. A new method to compute the weights of the reconstruction matrix is devised which replaces the usual constantsize interpol...
On the Use of Graphics Hardware to Accelerate Algebraic Reconstruction Methods
 In Proceedings of SPIE Medical Imaging Conference 1999, number 365962
, 1999
"... The Algebraic Reconstruction Technique (ART) reconstructs a 2D or 3D object from its projections. It has, in certain scenarios, many advantages over the more popular Filtered Backprojection approaches and has also recently been shown to perform well for 3D conebeam reconstruction. However, so far, ..."
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Cited by 8 (2 self)
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The Algebraic Reconstruction Technique (ART) reconstructs a 2D or 3D object from its projections. It has, in certain scenarios, many advantages over the more popular Filtered Backprojection approaches and has also recently been shown to perform well for 3D conebeam reconstruction. However, so far, ART's slow speed has prohibited its routine use in clinical applications. Currently, a software implementation requires several hours for a 3D reconstruction, even on modest reconstruction grid sizes. Although one solution to combat these problems would be the timeconsuming design of expensive custom accelerator boards, we would rather like to resort to existing and widely available hardware for our purposes. In this sense, we find that ART's main operations, i.e., volume projections and image backprojections, can be performed very rapidly on standard 2D texture mapping hardware, resident in many graphics workstations and PC graphics boards. In this paper, we discuss the use of this hardwar...
Discretization of the Radon Transform and of its Inverse by Spline Convolutions
, 2002
"... We present an explicit formula for Bspline convolution kernels; these are defined as the convolution of several Bsplines of variable widths hi and degrees rzl. We apply our results to derive splineconvolutionbased algorithms for two closely related problems: the computation of the Radon transfor ..."
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Cited by 8 (3 self)
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We present an explicit formula for Bspline convolution kernels; these are defined as the convolution of several Bsplines of variable widths hi and degrees rzl. We apply our results to derive splineconvolutionbased algorithms for two closely related problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the leastsquares sense. We then consider the reverse problem and introduce a new splineconvolution version of the filtered backprojection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of highdegree splines; these offer better approximation performance than the conventional lowerdegree formulations (e.g., piecewise constant or piecewise linear models). We present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity. In particular, we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate.
Fast And Accurate ThreeDimensional Reconstruction From ConeBeam Projection Data Using Algebraic Methods
, 1998
"... Conebeam computed tomography (CT) is an emerging imaging technology, as it provides all projections needed for threedimensional (3D) reconstruction in a single spin of the Xray sourcedetector pair. This facilitates fast, lowdose data acquisition as required for imaging fast moving objects, such ..."
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Cited by 8 (1 self)
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Conebeam computed tomography (CT) is an emerging imaging technology, as it provides all projections needed for threedimensional (3D) reconstruction in a single spin of the Xray sourcedetector pair. This facilitates fast, lowdose data acquisition as required for imaging fast moving objects, such as the heart, and intraoperative CT applications. Current conebeam reconstruction algorithms mainly employ the FilteredBackprojection (FBP) approach. In this dissertation, a different class of reconstruction algorithms is studied: the algebraic reconstruction methods. Algebraic reconstruction starts from an initial guess for the reconstructed object and then performs a sequence of iterative grid projections and correction backprojections until the reconstruction has converged. Algebraic methods have many advantages over FBP, such as better noise tolerance and better handling of sparse and nonuniformly distributed projection datasets. So far, the main repellant for using algebraic methods...
AntiAliased ThreeDimensional ConeBeam Reconstruction of LowContrast Objects with Algebraic Methods
 IEEE Trans. Med. Imag
, 1999
"... This paper examines the use of the algebraic reconstruction technique (ART) and related techniques to reconstruct 3D objects from a relatively sparse set of conebeam projections. Although ART has been widely used for conebeam reconstruction of highcontrast objects, e.g., in computed angiography, ..."
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Cited by 8 (1 self)
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This paper examines the use of the algebraic reconstruction technique (ART) and related techniques to reconstruct 3D objects from a relatively sparse set of conebeam projections. Although ART has been widely used for conebeam reconstruction of highcontrast objects, e.g., in computed angiography, the work presented here explores the more challenging lowcontrast case which represents a littleinvestigated scenario for ART. Preliminary experiments indicate that for cone angles greater than 20 ffiffiffi , traditional ART produces reconstructions with strong aliasing artifacts. These artifacts are in addition to the usual offmidplane inaccuracies of conebeam tomography with planar orbits. We find that the source of these artifacts is the nonuniform reconstruction grid sampling and correction by the conebeam rays during the ART projectionbackprojection procedure. A new method to compute the weights of the reconstruction matrix is devised, which replaces the usual constantsize in...
The Weighted Distance Scheme: A Globally Optimizing Projection Ordering Method for ART
, 1997
"... The order in which the projections are applied in the Algebraic Reconstruction Technique (ART) has a great effect on speed of convergence, accuracy and the amount of noiselike artifacts in the reconstructed image. In this paper, a new projection ordering scheme for ART is presented: the Weighted ..."
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Cited by 7 (7 self)
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The order in which the projections are applied in the Algebraic Reconstruction Technique (ART) has a great effect on speed of convergence, accuracy and the amount of noiselike artifacts in the reconstructed image. In this paper, a new projection ordering scheme for ART is presented: the Weighted Distance Scheme (WDS). It heuristically optimizes the angular distance of a newly selected projection with respect to an extended sequence of previously applied projections. This sequence of influential projections may incorporate the complete set of all previously applied projections or any limited time interval subset thereof. The selection algorithm results in uniform sampling of the projection access space, minimizing correlation in the projection sequence. This produces more accurate images with less noiselike artifacts than previously suggested projection ordering schemes.
The approximate inverse in action II: convergence and stability
 Mathematics of Computation
, 2001
"... Abstract. The approximate inverse is a scheme for constructing stable inversion formulas for operator equations. Originally, it is defined on L2spaces. In the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of ..."
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Cited by 6 (4 self)
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Abstract. The approximate inverse is a scheme for constructing stable inversion formulas for operator equations. Originally, it is defined on L2spaces. In the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of the approximate inverse which actually underlies numerical computations. Indeed, we show convergence if the discretization parameter tends to zero. Further, we prove stability, that is, we show the regularization property. Finally we apply the results to the filtered backprojection algorithm in 2Dtomography to obtain convergence rates. 1. Setting the stage The approximate inverse is a numerical scheme for solving operator equations of the first kind in Hilbert spaces. In this paper we further develop its analytic convergence theory which we apply to classical Xray tomography. The concept of approximate inverse goes back to the article [9] by Louis and