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82
On Projection Algorithms for Solving Convex Feasibility Problems
, 1996
"... Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of the ..."
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Cited by 142 (24 self)
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Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated . Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 6502, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and phrases. Angle between two subspaces, averaged mapping, Cimmino's method, computerized tomography, convex feasibility problem, convex function, convex inequalities, convex programming, convex set, Fej'er monotone sequence, firmly nonexpansive mapping, H...
Volume Rendering by Adaptive Refinement
, 1989
"... Volume rendering is a technique for visualizing sampled scalar functions of three spatial dimensions by computing 2D projections of a colored semitransparent gel. This paper presents a volume rendering algorithm in which image quality is adaptively refined over time. An initial image is generated b ..."
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Cited by 55 (3 self)
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Volume rendering is a technique for visualizing sampled scalar functions of three spatial dimensions by computing 2D projections of a colored semitransparent gel. This paper presents a volume rendering algorithm in which image quality is adaptively refined over time. An initial image is generated by casting a small number of rays into the data, less than one ray per pixel, and interpolating between the resulting colors. Subsequent images are generated by alternately casting more rays and interpolating. The usefulness of these rays is maximized by distributing them according to measures of local image complexity. Examples from two applications are given: molecular graphics and medical imaging. Key words: Volume rendering, voxel, adaptive refinement, adaptive sampling, ray tracing. 1. Introduction In this paper, we address the problem of visualizing sampled scalar functions of three spatial dimensions, henceforth referred to as volume data. We focus on a relatively new visualization tec...
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 48 (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.
Reconstructing Polygons from Moments with Connections to Array Processing
 IEEE Transactions on Signal Processing
, 1995
"... In this paper, we establish a set of results slowing that the vertices of any simply connected planar polygonal region can be reconstructed from a finite number of its complex moments. These results find applications in a variety of apparently disparate arcas such as computerized tomography and inve ..."
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Cited by 32 (4 self)
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In this paper, we establish a set of results slowing that the vertices of any simply connected planar polygonal region can be reconstructed from a finite number of its complex moments. These results find applications in a variety of apparently disparate arcas such as computerized tomography and inverse potential theory, where in the former, it is of interest to estimate the shape of an object from a finite number of its projections, whereas in the latter, the objective is to extract the shape of a gravitating body from measurements of its exterior logarithmic potentials at a finite number of points. We show that the problem of polygonal vertex reconstruction from moments can in fact be posed as an array processing problem, and taking advantage of this relationship, we derive and illustrate several new algorithms for the reconstruction of the vertices of simply connected polygons from moments.
Mathematics of thermoacoustic tomography
 European Journal Applied Mathematics
"... The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1 ..."
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Cited by 29 (6 self)
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The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacoustic or Optoacoustic) Tomography. 1
On the optimality of the gridding reconstruction algorithm
 IEEE Trans.Med.Imag.,vol.19,no.4,pp.306–317,2000
"... Abstract—Gridding reconstruction is a method to reconstruct data onto a Cartesian grid from a set of nonuniformly sampled measurements. This method is appreciated for being robust and computationally fast. However, it lacks solid analysis and design tools to quantify or minimize the reconstruction e ..."
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Cited by 26 (0 self)
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Abstract—Gridding reconstruction is a method to reconstruct data onto a Cartesian grid from a set of nonuniformly sampled measurements. This method is appreciated for being robust and computationally fast. However, it lacks solid analysis and design tools to quantify or minimize the reconstruction error. Least squares reconstruction (LSR), on the other hand, is another method which is optimal in the sense that it minimizes the reconstruction error. This method is computationally intensive and, in many cases, sensitive to measurement noise. Hence, it is rarely used in practice. Despite their seemingly different approaches, the gridding and LSR methods are shown to be closely related. The similarity between these two methods is accentuated when they are properly expressed in a common matrix form. It is shown that the gridding algorithm can be considered an approximation to the least squares method. The optimal gridding parameters are defined as the ones which yield the minimum approximation error. These parameters are calculated by minimizing the norm of an approximation error matrix. This problem is studied and solved in the general form of approximation using linearly structured matrices. This method not only supports more general forms of the gridding algorithm, it can also be used to accelerate the reconstruction techniques from incomplete data. The application of this method to a case of twodimensional (2D) spiral magnetic resonance imaging shows a reduction of more than 4 dB in the average reconstruction error. Index Terms—Gridding reconstruction, image reconstruction, matrix approximation, nonuniform sampling. I.
A MomentBased Variational Approach to Tomographic Reconstruction
 IEEE Transactions on Image Processing
, 1996
"... In this paper, we describe a variational framework for the tomographic reconstruction of an image from the maximum likelihood (ML) estimates of its orthogonal moments. We show how these estimated moments and their (correlated) error statistics can be computed directly, and in a linear fashion from g ..."
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Cited by 21 (6 self)
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In this paper, we describe a variational framework for the tomographic reconstruction of an image from the maximum likelihood (ML) estimates of its orthogonal moments. We show how these estimated moments and their (correlated) error statistics can be computed directly, and in a linear fashion from given noisy and possibly sparse projection data. Moreover, thanks to the consistency properties of the Radon transform, this twostep approach (moment estimation followed by image reconstruction) can be viewed as a statistically optimal procedure. Furthermore, by focusing on the important role played by the mmnents of projection data, we immediately see the close connection between tomographic reconstruction of nonnegativevalued images and the problem of nonparametric estimation of probability densities given estimates of their moments. Taking advantage of this connection, our proposed variational algorithm is based on the minimization of a cost functional composed of a term measuring the divergence between a given prior estimate of the image and the current estimate of the image and a second quadratic term based on the error incurred in the estimation of the moments of the underlying image from the noisy projection data. We show that an iterative refinement of this algorithm leads to a practical algorithm for the solution of the highly complex equality constrained divergence minimization problem. We show tbat this iterative refinement results in superior reconstructions of images from very noisy data as compared with the classical filtered backprojection (FBP) algorithm.
Inversion Of LargeSupport IllPosed Linear . . .
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 1998
"... We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is illposed and we resolve it by incorporating the prior information that the reconstructed object ..."
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Cited by 20 (12 self)
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We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is illposed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weakstring in one dimension and the weakmembrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite
A randomized Kaczmarz algorithm with exponential convergence
"... The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We i ..."
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Cited by 20 (1 self)
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The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system, but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling. ∗ T.S. was supported by NSF DMS grant 0511461. R.V. was supported by the Alfred P.
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