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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.
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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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.
Algorithms in Tomography
 The State of the Art in Numerical Analysis
, 1997
"... this paper is as follows. In section 2 we survey the mathematical models used in tomography. In section 3 we give a fairly detailed survey on 2D reconstruction algorithm which still are the work horse of tomography. In section 4 we describe recent developments in 3D reconstruction. In section 5 we m ..."
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this paper is as follows. In section 2 we survey the mathematical models used in tomography. In section 3 we give a fairly detailed survey on 2D reconstruction algorithm which still are the work horse of tomography. In section 4 we describe recent developments in 3D reconstruction. In section 5 we make a few remarks on the beginning development of algorithms for nonstraight line tomography. 2 Mathematical Models in Tomography
General geometry CT reconstruction
 in Proc. of the International Conference on Image Processing and Computer Vision (IPCV’06), 2006
"... Abstract — We present an efficient and accurate reconstruction technique for computed tomography (CT) images from a geometrically unconstrained set of ray sums. Current CTscanners are immobile. However, a new era in xray production could usher in new, lightweight, flexible and portable xray devic ..."
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Abstract — We present an efficient and accurate reconstruction technique for computed tomography (CT) images from a geometrically unconstrained set of ray sums. Current CTscanners are immobile. However, a new era in xray production could usher in new, lightweight, flexible and portable xray devices. The proposed device, currently under development, is envisioned as a flexible band with tiny xray emitters and detectors attached. The device is to be wrapped around an appendage and an image obtained. Such a device would be desirable as it can increase the speed of medical diagnosis. To evaluate the feasibility of reconstructing a CT image from such a device, a simulation testbed was built to generate simulated CT ray sums of a test image. This data was then used in our reconstruction method, which involves slotting each ray sum onto a grid of data that represents what would have been acquired by a parallelbeam CT scanner. Once in that form, Filtered Back Projection can be used to perform the recostruction. Observation of sample reconstructions, as well as quantitative results, suggest that this simple method is efficient and effective.
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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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.
High Performance Computers
, 1991
"... ACM, (2008). This is the author’s version of the work. It is posted here by permission of ACM for your personal use. Not for ..."
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ACM, (2008). This is the author’s version of the work. It is posted here by permission of ACM for your personal use. Not for
New, efficient Fourierreconstruction method for approximate image reconstruction in spiral conebeam CT at small cone angles
"... We present an approximate algorithm for image reconstruction in spiral conebeam CT at small coneangles. The first step of the algorithm is a rebinning from fanbeam to parallel beam projections, that is performed independently for each detector row. The result of this rebinning procedure is a set o ..."
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We present an approximate algorithm for image reconstruction in spiral conebeam CT at small coneangles. The first step of the algorithm is a rebinning from fanbeam to parallel beam projections, that is performed independently for each detector row. The result of this rebinning procedure is a set of parallel views, where all rays are tilted against the zaxis by their cone angle and the rays within each view have different zpositions. After that, as with most of the approximate algorithms, the basic idea is to let each oblique ray contribute to the image with a weight that depends on its distance to the image plane. Because the distance of a ray to the image plane changes along the ray, so does its weight. This is the one important difference compared to standard 2D reconstruction procedures. However, for small cone angles, the variation of the weight along a ray is rather smooth, so that we can synthesize it as a Fourier series using only few Fourier coefficients. In standard 2D ima...
CT reconstruction from parallel and fanbeam projections by 2D discrete Radon transform, submitted
"... Abstract—The discrete Radon transform (DRT) was defined by Abervuch et al. as an analog of the continuous Radon transform for discrete data. Both the DRT and its inverse are computable in ( 2 log) operations for images of size. In this paper, we demonstrate the applicability of the inverse DRT for t ..."
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Abstract—The discrete Radon transform (DRT) was defined by Abervuch et al. as an analog of the continuous Radon transform for discrete data. Both the DRT and its inverse are computable in ( 2 log) operations for images of size. In this paper, we demonstrate the applicability of the inverse DRT for the reconstruction of a 2D object from its continuous projections. The DRT and its inverse are shown to model accurately the continuum as the number of samples increases. Numerical results for the reconstruction from parallel projections are presented. We also show that the inverse DRT can be used for reconstruction from fanbeam projections with equispaced detectors. Index Terms—Computed tomography (CT), fanbeam projections, parallel projections, tomography, 2D discrete Radon transform (DRT).