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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 55 (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.
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 24 (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
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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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...
SNARK05: A PROGRAMMING SYSTEM FOR THE RECONSTRUCTION OF 2D IMAGES FROM 1D PROJECTIONS
, 2008
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Signal Processing Issues In Synthetic Aperture Radar And Computer Tomography
, 1998
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In memory of Moshe Israeli 1940–2007
"... Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for 2D discrete images which is algebraically exact, invertible, and rapidly computable. We define a no ..."
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Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for 2D discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for discrete 2D images, which is based on summation along lines of absolute slope less than 1. Values at nongrid locations are defined using trigonometric interpolation on a zeropadded grid. Our definition is shown to be geometrically faithful: the summation avoids wraparound effects. Our proposal uses a special collection of lines in R 2 for which the transform is rapidly computable and invertible. We describe a fast algorithm using O(N log N) operations, where N = n 2 is the number of pixels in the image. The fast algorithm exploits a discrete projectionslice theorem, which associates the discrete Radon transform with the pseudopolar Fourier transform [2]. Our definition for discrete images converges to a natural continuous counterpart with increasing refinement.
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"... 4Irregular sampling for multidimensional polar processing of integral transforms A. Averbuch, R. Coifman, M. Israeli, I. Sedelnikov, andY.Shkolnisky We survey a family of theories that enable to process polar data via integral transforms. We show the relation between irregular sampling and discrete ..."
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4Irregular sampling for multidimensional polar processing of integral transforms A. Averbuch, R. Coifman, M. Israeli, I. Sedelnikov, andY.Shkolnisky We survey a family of theories that enable to process polar data via integral transforms. We show the relation between irregular sampling and discrete integral transforms, demonstrate the application of irregular (polar) sampling to image processing problems, and derive approximation algorithms that are based on unequally spaced samples. It is based on sampling the Fourier domain. We describe 2D and 3D irregular sampling geometries of the frequency domain, derive efficient numerical algorithms that implement them, prove their correctness, and provide theory and algorithms that invert them. We also show that these sampling geometries are closely related to discrete integral transforms. The proposed underlying methodology bridges via sampling between the continuous nature of the physical
INTERPOLATION AND THE CHIRP TRANSFORM: DSP MEETS OPTICS
"... This paper considers the problem of interpolating a signal from one uniformlyspaced grid to another, where the grid spacings may be related by an arbitrary, irrational factor. Noting that interpolation is the digital equivalent of magnification, we begin by reviewing optical systems for magnificati ..."
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This paper considers the problem of interpolating a signal from one uniformlyspaced grid to another, where the grid spacings may be related by an arbitrary, irrational factor. Noting that interpolation is the digital equivalent of magnification, we begin by reviewing optical systems for magnification and “computation ” of the chirp Fourier transform. This route suggests several analog schemes for magnification, which can be discretized to produce algorithms for interpolation. We then derive one of these algorithms from first principles, using a digitalsignalprocessing perspective. The result is an important, but forgotten, algorithm for interpolation first suggested as an application of the chirpz transform by Rabiner, Schafer, and Rader. Unlike the earlier derivation, our approach is direct – we do not make use of Bluestein’s trick of completing the square. In addition, our approach identifies parameters under user control that can be optimized for best performance. 1.
Evaluation of Novel WholeBody High Resolution Rodent SPECT (Linoview) Based on Direct Acquisition of Linogram Projections
"... Studies of the biodistribution of radiolabeled compounds in rodents frequently are performed during the process of development of new pharmaceutical drugs. This article presents the evaluation of a new wholebody animal SPECT system, called the Linoview SPECT system. Methods: Linoview SPECT is base ..."
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Studies of the biodistribution of radiolabeled compounds in rodents frequently are performed during the process of development of new pharmaceutical drugs. This article presents the evaluation of a new wholebody animal SPECT system, called the Linoview SPECT system. Methods: Linoview SPECT is based on the linear orbit acquisition technique associated with slitaperture collimators mounted on 4 pixelated CsI(Na) detectors composed of an array of small, individual crystal elements. Sliding iridium rods allow variation of the collimator aperture. Hotrod and coldrod phantoms filled with 99mTc were imaged. Mice were imaged, and kidney radioactivity was measured after injection of 99mTcdimercaptosuccinic acid and 111Indiethylenetriaminepentaacetic acidDPhe1octreotide (111Inpentetreotide; OctreoScan111). Results: Phantom studies showed that hot rods