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Fast slant stack: A notion of Radon transform for data in a Cartesian grid which is rapidly computible, algebraically exact, geometrically faithful and invertible
 SIAM J. Sci. Comput
, 2001
"... Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition i ..."
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Cited by 53 (11 self)
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Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at nonCartesian locations defined using trigonometric interpolation on a zeropadded grid. The definition is geometrically faithful: the lines exhibit no ‘wraparound effects’. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n2 is the number of pixels. This relies on a discrete projectionslice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2D Fourier transform on a nonCartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT – the process of rapid exact evaluation of the 2D Fourier transform at these nonCartesian grid points – is possible using chirpZ transforms. This Radon transform is onetoone and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3D version of the transform.
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 25 (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.
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 ..."
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Cited by 11 (4 self)
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In this paper, we propose two new algorithms for high quality Fourier reconstructions of digital N &times; N images from their Radon transform. Both algorithms are based on fast Fourier transforms for nonequispaced data (NFFT) and require only O(N&sup2;log 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.
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"... Abstract. In this paper, we propose a new linogram algorithm for the high quality Fourier reconstruction of digital N \Theta N images from their Radon transform. The algorithm is based on univariate fast Fourier transforms for nonequispaced data in the time domain and in the frequency domain. The al ..."
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Abstract. In this paper, we propose a new linogram algorithm for the high quality Fourier reconstruction of digital N \Theta N images from their Radon transform. The algorithm is based on univariate fast Fourier transforms for nonequispaced data in the time domain and in the frequency domain. The algorithm requires only O(N 2 log N) arithmetic operations and preserves the good reconstruction quality of the filtered backprojection. 1991 Mathematics Subject Classification. 44A12, 65T50 Key words and phrases. Fast Fourier transform for nonequispaced data, Radon transform, computerized tomography, gridding, linogram, chirpz transform 1 Introduction We are interested in efficient and high quality reconstructions of digital N \Theta N medical images from their Radon transform. The standard reconstruction algorithm, the filtered backprojection, ensures a good quality of the images at the expense of O(N
and
"... Abstract. In this paper, we propose a new linogram algorithm for the high quality Fourier reconstruction of digital N N images from their Radon transform. The algorithm is based on univariate fast Fourier transforms for nonequispaced data in the time domain and in the frequency domain. The algorithm ..."
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Abstract. In this paper, we propose a new linogram algorithm for the high quality Fourier reconstruction of digital N N images from their Radon transform. The algorithm is based on univariate fast Fourier transforms for nonequispaced data in the time domain and in the frequency domain. The algorithm requires only O(N 2 logN) arithmetic operations and preserves the good reconstruction quality of the ltered backprojection. 1991 Mathematics Subject Classication. 44A12, 65T50 Key words and phrases. Fast Fourier transform for nonequispaced data, Radon transform, computerized tomography, gridding, linogram, chirp{z transform 1
Acknowledgments
, 2013
"... I would like to first and foremost thank my parents and brother. Without their unwavering love, support, and commonsense, I would not be where and who I am today. I am immeasurably thankful for all they have done and continue to do for me. Next, I would like to thank my advisor, Dr. Adel Faridani. H ..."
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I would like to first and foremost thank my parents and brother. Without their unwavering love, support, and commonsense, I would not be where and who I am today. I am immeasurably thankful for all they have done and continue to do for me. Next, I would like to thank my advisor, Dr. Adel Faridani. His guidance, help, and knowledge made this entire research paper possible. I would also like to thank Charlie Robson for always giving just the right amount of support and positive encouragement, Nancy Scherich for putting up with me even in my “early morning ” mode, and Chelsea Hall for always lending an ear when asked. 2 1