We present an explicit formula for B-spline convolution kernels; these are defined as the convolution of several B-splines of variable widths hi and degrees rzl. We apply our results to derive spline-convolution-based 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 least-squares sense. We then consider the reverse problem and introduce a new spline-convolution version of the filtered back-projection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of high-degree splines; these offer better approximation performance than the conventional lower-degree 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.

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

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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 CT-scanners are immobile. However, a new era in x-ray production could usher in new, lightweight, flexible and portable x-ray devices. The proposed device, currently under development, is envisioned as a flexible band with tiny x-ray 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 parallel-beam 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.

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.

We present an approximate algorithm for image reconstruction in spiral cone-beam CT at small cone-angles. 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 z-axis by their cone angle and the rays within each view have different z-positions. 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...

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2008 i A novel systematic framework of medical image analysis, weighted Fourier series (WFS) analysis is introduced. WFS is a combination of Fourier series and heat kernel smoothing. WFS effectively reduces the Gibbs phenomenon, improves the signal to noise ratio, and increases normality of the estimated errors in the WFS-based generalized linear models. In estimating the parameters of WFS, the least squares estimation of WFS has been widely used but it is computationally inefficient. To address the computational inefficiency in the least squares estimation, much faster but less accurate iterative residual fitting (IRF) method has been proposed. The proposed adaptive iterative regression (AIR) technique inherits the computational efficiency of IRF and improves accuracy of IRF. AIR partitions the function space into a set of subspaces, and performs an extra orthogonalization procedure to reduce the bias of IRF estimation. A complimentary tool, the fast weighted

Currently, images acquired via Magnetic Resonance Imaging (MRI) and functional Magnetic Resonance Imaging (fMRI) technology are reconstructed using the discrete inverse Fourier transform. While computationally convenient, this approach is not able to filter out noise. This is a serious limitation because the amount of noise in MRI and fMRI can be substantial. In this paper, we propose an alternative approach to reconstruction, based on penalized likelihood methodology. In particular, we focus on non-linear shrinkage estimators and show that this approach achieves a great reduction in Integrated Mean Squared Error (IMSE) of the estimated image with respect to the currently used estimator. This approach is extremely fast and easy to implement computationally. In addition, it can be combined with various alternative approaches to MR image reconstruction and can be easily adapted to other, non-MRI contexts, in which the observed data and the quantities of interest are related via a linear transform.

Image reconstruction from nonuniformly sampled spatial frequency domain data is an important problem that arises in computed imaging. Current reconstruction techniques suffer from limitations in their model and implementation. In this paper, we present a new reconstruction method that is based on solving a system of linear equations using an efficient iterative approach. Image pixel intensities are related to the measured frequency domain data through a set of linear equations. Although the system matrix is too dense and large to solve by direct inversion in practice, a simple orthogonal transformation to the rows of this matrix is applied to convert the matrix into a sparse one up to a certain chosen level of energy preservation. The transformed system is subsequently solved using the conjugate gradient method. This method is applied to reconstruct images of a numerical phantom as well as magnetic resonance images from experimental spiral imaging data. The results support the theory and demonstrate that the computational load of this method is similar to that of standard gridding, illustrating its practical utility. Copyright © 2006 Yasser M. Kadah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.