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11
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 23 (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.
A framework for discrete integral transformations II – the 2D 31 Radon transform
"... This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopola ..."
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Cited by 20 (10 self)
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This paper is dedicated to the memory of Professor Moshe Israeli 19402007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudopolar Fourier transform that samples the Fourier transform on the pseudopolar grid, also known as the concentric squares grid. The pseudopolar grid consists of equally spaced samples along rays, where different rays are equally spaced and not equally angled. The pseudopolar Fourier transform Fourier transform is shown to be fast (the same complexity as the FFT), stable, invertible, requires only
Using NFFT 3  a software library for various nonequispaced fast Fourier transforms
, 2008
"... NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and ..."
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Cited by 12 (8 self)
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NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.
On the computation of the polar FFT
 Appl. Comput. Harmon. Anal
, 2007
"... We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT. ..."
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Cited by 9 (7 self)
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We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT.
Discrete Analytical Ridgelet Transform
 Signal Processing
, 2004
"... In this paper, we propose an implementation of the 3D ridgelet transform: The 3D Discrete Analytical Ridgelet Transform (3D DART). This transform uses the Fourier strategy for the computation of the associated 3D discrete Radon transform. The innovative step is the definition of a discrete 3D t ..."
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Cited by 5 (0 self)
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In this paper, we propose an implementation of the 3D ridgelet transform: The 3D Discrete Analytical Ridgelet Transform (3D DART). This transform uses the Fourier strategy for the computation of the associated 3D discrete Radon transform. The innovative step is the definition of a discrete 3D transform with the discrete analytical geometry theory by the construction of 3D discrete analytical lines in the Fourier domain. We propose two types of 3D discrete lines: 3D discrete radial lines going through the origin defined from their orthogonal projections and 3D planes covered with 2D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3D DART adapted to a specific application. Indeed, the 3D DART representation is not orthogonal, It is associated with a flexible redundancy factor. The 3D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3D DART and its extension to the LocalDART (with smooth windowing) to the denoising of 3D image and colour video. These experimental results show that the simple thresholding of the 3D DART coefficients is efficient.
Discrete diffraction transform
, 2004
"... In this paper we define a discrete analogue of the continuous diffracted projection. We define a discrete diffracted transform (DDT) as a collection of the discrete diffracted projections taken at specific set of angles along specific set of lines. We define ‘discrete diffracted projection ’ to be a ..."
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Cited by 4 (3 self)
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In this paper we define a discrete analogue of the continuous diffracted projection. We define a discrete diffracted transform (DDT) as a collection of the discrete diffracted projections taken at specific set of angles along specific set of lines. We define ‘discrete diffracted projection ’ to be a discrete transform that is similar in its properties to the continuous diffracted projection. We prove that when the DDT is applied on a set of samples of a continuous object, it approximates a set of continuous vertical diffracted projections of a horizontally sheared object and a set of continuous horizontal diffracted projections of a vertically sheared object. We prove that a similar statement, where diffracted projections are replaced by the Xray projections, holds in the case of the discrete 2D Radon transform (DRT). We prove that the discrete diffraction transform is rapidly computable and invertible. Some of the underlying ideas came from the definition of DRT. Unlike the DRT, though, this transform cannot be used for reconstruction of the object from the set of rotated projections. Key word: diffraction tomography, discrete diffraction transform, Radon transform.
Numerical stability of nonequispaced fast Fourier transforms
"... Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case ..."
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Cited by 3 (1 self)
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Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case study, we show that both approximation error and roundoff error have a strong influence on the numerical stability of NFFT. Numerical tests confirm the theoretical estimates of numerical stability.
A probability argument in favor of ignoring small singular values
 Operators and Matrices, 1:31 – 43
, 2007
"... O perators a nd M atrices ..."
NOISE PROPERTIES OF REGULARIZED IMAGE RECONSTRUCTION IN XRAY COMPUTED TOMOGRAPHY
, 2007
"... is the endofjourney harvest of my five and half years of hard work whereby I have been inspired and encouraged by many people. It is my greatest pleasure to express my deepest and sincerest gratitude for all of them. This thesis would not exist without my advisor, Professor Jeffrey A. Fessler. His ..."
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is the endofjourney harvest of my five and half years of hard work whereby I have been inspired and encouraged by many people. It is my greatest pleasure to express my deepest and sincerest gratitude for all of them. This thesis would not exist without my advisor, Professor Jeffrey A. Fessler. His enlightening guidance, colossal support and sincere friendship helped me from day one and throughout my graduate study. I would also like to express my gratitude to Professor Neal H. Clinthorne, Professor Mitchell M. Goodsitt, Professor Alfred O. Hero, and Professor David C. Munson for their expertise and valuable feedbacks on this work. I give special thanks to Professor Anthony W. England for his understanding and assistance in my transition of research fields. I further present my gratefulness to the colleagues at University
Advance Access publication on April 29, 2008 The discrete diffraction transform
, 2006
"... In this paper, we define a discrete analogue of the continuous diffracted projection. We define the discrete diffraction transform (DDT) as a collection of the discrete diffracted projections (DDPs) taken at specific set of angles along specific set of lines. The ‘DDP ’ is defined to be a discrete t ..."
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In this paper, we define a discrete analogue of the continuous diffracted projection. We define the discrete diffraction transform (DDT) as a collection of the discrete diffracted projections (DDPs) taken at specific set of angles along specific set of lines. The ‘DDP ’ is defined to be a discrete transform that is similar in its properties to the continuous diffracted projection. We prove that when the DDT is applied to a set of samples of a continuous object, it approximates a set of continuous vertical diffracted projections of a horizontally sheared object and a set of continuous horizontal diffracted projections of a vertically sheared object. A similar statement, where diffracted projections are replaced by the Xray projections, that holds for the 2D discrete Radon transform (DRT), is also proved. We prove that the DDT is rapidly computable and invertible.