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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.
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 21 (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
Fast and accurate Polar Fourier transform
 Appl. Comput. Harmon. Anal.
, 2006
"... In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is pr ..."
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Cited by 19 (1 self)
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In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given twodimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2DFFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudoPolar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudoPolar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudoPolar FFT plays the role of a halfway point—a nearlyPolar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesianbased unequallysampled FFT method to ours, both algorithms using a smallsupport interpolation and no precompensating, and show marked advantage to the use of the pseudoPolar initial grid.
Fast Xray and beamlet transforms for threedimensional data
 in Modern Signal Processing
, 2002
"... Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop t ..."
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Cited by 14 (8 self)
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Abstract. Threedimensional volumetric data are becoming increasingly available in a wide range of scientific and technical disciplines. With the right tools, we can expect such data to yield valuable insights about many important phenomena in our threedimensional world. In this paper, we develop tools for the analysis of 3D data which may contain structures built from lines, line segments, and filaments. These tools come in two main forms: (a) Monoscale: the Xray transform, offering the collection of line integrals along a wide range of lines running through the image — at all different orientations and positions; and (b) Multiscale: the (3D) beamlet transform, offering the collection of line integrals along line segments which, in addition to ranging through a wide collection of locations and positions, also occupy a wide range of scales. We describe different strategies for computing these transforms and several basic applications, for example in finding faint structures buried in noisy data. 1.
DirectFourier Reconstruction In Tomography And Synthetic Aperture Radar
 Intl. J. Imaging Sys. and Tech
, 1998
"... We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR ..."
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Cited by 9 (0 self)
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We investigate the use of directFourier (DF) image reconstruction in computerized tomography and synthetic aperture radar (SAR). One of our aims is to determine why the convolutionbackprojection (CBP) method is favored over DF methods in tomography, while DF methods are virtually always used in SAR. We show that the CBP algorithm is equivalent to DF reconstruction using a Jacobianweighted 2D periodic sinckernel interpolator. This interpolation is not optimal in any sense, which suggests that DF algorithms utilizing optimal interpolators may surpass CBP in image quality. We consider use of two types of DF interpolation: a windowed sinc kernel, and the leastsquares optimal Yen interpolator. Simulations show that reconstructions using the Yen interpolator do not possess the expected visual quality, because of regularization needed to preserve numerical stability. Next, we show that with a concentricsquares sampling scheme, DF interpolation can be performed accurately and efficiently...
Signal Processing Issues In Synthetic Aperture Radar And Computer Tomography
, 1998
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Lynx: A highresolution synthetic aperture radar
"... Lynx is a high resolution, synthetic aperture radar (SAR) that has been designed and built by Sandia National Laboratories in collaboration with General Atomics (GA). Although Lynx may be operated on a wide variety of manned and unmanned platforms, it is primarily intended to be fielded on unmanned ..."
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Lynx is a high resolution, synthetic aperture radar (SAR) that has been designed and built by Sandia National Laboratories in collaboration with General Atomics (GA). Although Lynx may be operated on a wide variety of manned and unmanned platforms, it is primarily intended to be fielded on unmanned aerial vehicles. In particular, it may be operated on the Predator, IGNAT, or Prowler II platforms manufactured by GA Aeronautical Systems, Inc. The Lynx production weight is less than 120 lb. and has a slant range of 30 km (in 4 mm/hr rain). It has operator selectable resolution and is capable of 0.1 m resolution in spotlight mode and 0.3 m resolution in stripmap mode. In ground moving target indicator mode, the minimum detectable velocity is 6 knots with a minimum target crosssection of 10 dBsm. In coherent change detection mode, Lynx makes registered, complex image comparisons either of 0.1 m resolution (minimum) spotlight images or of 0.3 m resolution (minimum) strip images. The Lynx user interface features a view manager that allows it to pan and zoom like a video camera. Lynx was developed under corporate funding from GA and will be manufactured by GA for both military and commercial applications. The Lynx system architecture will be presented and some of its unique features will be described. Imagery at the finest resolutions in both spotlight and strip modes have been obtained and will also be presented.
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
Digital Implementation of Ridgelet Packets A.G. Flesia, H.
"... Abstract. The Ridgelet Packets library provides a large family of orthonormal bases for functions f(x, y) in L 2 (dxdy) which includes orthonormal ridgelets as well as bases deriving from tilings reminiscent from the theory of wavelets and the study of oscillatory Fourier integrals. An intuitively a ..."
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Abstract. The Ridgelet Packets library provides a large family of orthonormal bases for functions f(x, y) in L 2 (dxdy) which includes orthonormal ridgelets as well as bases deriving from tilings reminiscent from the theory of wavelets and the study of oscillatory Fourier integrals. An intuitively appealing feature: many of these bases have elements whose envelope is strongly aligned along specified ‘ridges ’ while displaying oscillatory components across the main ‘ridge’. There are two approaches to constructing ridgelet packets; the most direct is a frequencydomain viewpoint. We take a recursive dyadic partition of the polar Fourier domain into a collection of rectangular tiles of various widths and lengths. Focusing attention on each tile in turn, we take a tensor basis, using windowed sinusoids in θ times windowed sinusoids in r. There is also a Radondomain approach to constructing ridgelet packets, which involves applying the Radon isometry and then, in the Radon plane, using wavelets in θ times wavelet packets in t, with