Fourier-based reprojection methods have the potential to reduce the computation time in iterative tomographic image reconstruction. Interpolation errors are a limitation of Fourier-based reprojection methods. We apply a min-max interpolation method for the nonuniform fast Fourier transform (NUFFT) to minimize the interpolation errors. Numerical results show that the min-max NUFFT approach provides substantially lower approximation errors in tomographic reprojection and backprojection than conventional interpolation methods.

Consider a piecewise smooth function for which the (pseudo-)spectral coefficients are given. It is well known that while spectral partial sums yield exponentially convergent approximations for smooth functions, the results for piecewise smooth functions are poor, with spurious oscillations developing near the discontinuities and a much reduced overall convergence rate. This behavior, known as the Gibbs phenomenon, is considered as one of the major drawbacks in the application of spectral methods. Various types of reconstruction methods developed for the recovery of piecewise smooth functions have met with varying degrees of success. The Gegenbauer reconstruction method, originally proposed by Gottlieb et. al. has the particularly impressive ability to reconstruct piecewise analytic functions with exponential convergence up to the points of discontinuity. However, it has been sharply criticized for its high cost and susceptibility to round-off error. In this paper, a new approach to Gegenbauer reconstruction is considered, resulting in a reconstruction method that is less computationally intensive and costly, yet still enjoys superior convergence. The idea is to create a procedure that combines the well known exponential filtering method in smooth regions away from the discontinuities with the Gegenbauer reconstruction method in regions close to the discontinuities. This hybrid approach benefits from both the simplicity of exponential filtering and the high resolution properties of the Gegenbauer reconstruction method. Additionally, a new way of computing the Gegenbauer coefficients from Jacobian polynomial expansions is introduced that is both more cost effective and less prone to round-off errors. AMS(MOS) subject classification. 42A10, 65T40, 74S25.

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Abstract. The Gegenbauer reconstruction method effectively eliminates the Gibbs phenomenon and restores exponential accuracy to the approximations of piecewise smooth functions. Recent investigations show that its success depends upon choosing parameters in such a way that the regularization and the truncation error estimates are equally considered. This paper shows that the underlying analyticity of the function in smooth regions plays a critical role in the regularization error estimate. Hence we develop a technique that first analyzes the behavior of the function in its regions of smoothness and then applies this knowledge to refine the regularization error estimate. Such refinement yields better parameter choices for the Gegenbauer reconstruction method, and is confirmed both by better accuracy and more robustness in the approximation of piecewise smooth functions. Key words. Fourier pseudo-spectral approximation, Gegenbauer reconstruction, exponential convergence. AMS subject classifications. 42A25, 42A30, 65D10

Fourier spectral methods have proven to be powerful tools that are frequently employed in image reconstruction. However, since images can be typically viewed as piecewise smooth functions, the Gibbs phenomenon often hinders accurate reconstruction. Much work has been done to combat the Gibbs phenomenon, including the development of numerical edge detection methods as well as reconstruction techniques that effectively reduce the Gibbs oscillations while maintaining high resolution accuracy at the edges.

We present an open-source ITK implementation of a direct Fourier method for tomographic reconstruction, applicable to parallel-beam x-ray images. Direct Fourier reconstruction makes use of the central-slice theorem to build a polar 2D Fourier space from the 1D transformed projections of the scanned object, that is resampled into a Cartesian grid. Inverse 2D Fourier transform eventually yields the reconstructed image. Additionally, we provide a complex wrapper to the BSplineInterpolateImageFunction to overcome ITK’s current lack for image interpolators dealing with complex data types. A sample application is presented and extensively illustrated on the Shepp-Logan head phantom. We show that appropriate input zeropadding and 2D-DFT oversampling rates together with radial cubic b-spline interpolation improve 2D-DFT interpolation quality and are efficient remedies to reduce reconstruction artifacts.

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

This paper was published in SPIE Medical Imaging (2008), 6914, and is made available as an electronic reprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited. Bi-planar 2D-to-3D Registration in Fourier Domain for Stereoscopic

Non-Cartesian sampling is widely used for fast magnetic resonance imaging (MRI). Accurate and fast image reconstruction from non-Cartesian k-space data becomes a challenge and gains a lot of attention. Images provided by conventional direct reconstruction methods usually bear ringing, streaking, and other leakage artifacts caused by discontinuous structures. In this paper, we tackle these problems by analyzing the principal point spread function (PSF) of non-Cartesian reconstruction and propose a leakage reduction reconstruction scheme based on discontinuity subtraction. Data fidelity in k-space is enforced during each iteration. Multidimensional nonuniform fast Fourier transform (NUFFT) algorithms are utilized to simulate the k-space samples as well as to reconstruct images. The proposed method is compared to the direct reconstruction method on computer-simulated phantoms and physical scans. Non-Cartesian sampling trajectories including 2D spiral, 2D and 3D radial trajectories are studied. The proposed method is found useful on reducing artifacts due to high image discontinuities. It also improves the quality of images reconstructed from undersampled data. Copyright © 2006 J. Song and Q. H. Liu. 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.