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fast algorithms and Efficient GPU Implementations for the Radon Transform and the Back-Projection Operator Represented as Convolution Operators

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SIAM JOURNAL ON IMAGING SCIENCES 2016年第2期9卷 637-664页

作者： Andersson, Fredrik Carlsson, Marcus Nikitin, Viktor V. Lund Univ Ctr Math Sci S-22100 Lund Sweden

The Radon transform and its adjoint, the back-projection operator, can both be expressed as convolutions in log-polar coordinates. Hence, fast algorithms for the application of these operators can be constructed by using the FFT, if data is resampled at log-polar coordinates. Radon data is typically measured on an equally spaced grid in polar coordinates, and reconstructions are represented (as images) in Cartesian coordinates. Therefore, in addition to FFT, several steps of interpolation have to be conducted in order to apply the Radon transform and the back-projection operator by means of convolutions. However, in comparison to the interpolation conducted in Fourier-based gridding methods, the interpolation performed in the Radon and image domains will typically deal with functions that are substantially less oscillatory. Reasonable reconstruction results can thus be expected using interpolation schemes of moderate order. The approach also provides better control over the artifacts that can appear due to measurement errors. Both the interpolation and the FFT operations can be efficiently implemented on graphical processor units (GPUs). For the interpolation, it is possible to make use of the fact that linear interpolation is hard-wired on GPUs, meaning that it has the same computational cost as direct memory access. Cubic order interpolation schemes can be constructed by combining linear interpolation steps, and this provides important computation speedup. We provide details about how the Radon transform and the back-projection can be implemented efficiently as convolution operators on GPUs. For large data sizes, these algorithms are several times faster than those of other software packages based on GPU implementations of the Radon transform and the back-projection operator. Moreover, the gain in computational speed is substantially higher when comparing against other CPU-based algorithms.

关键词： Radon transform fast algorithms FFT GPU

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fast algorithms for direction-of-arrival finding using large ESPRIT arrays

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SIGNAL PROCESSING 1996年第2期48卷 111-121页

作者： Zha, HY Department of Computer Science and Engineering The Pennsylvania State University University Park PA 16802 USA

To explore the redundancy in large ESPRIT arrays, we propose fast algorithms for direction-of-arrival (DOA) finding without computing (partial) eigendecompositions. For batch processing, the algorithms have computational complexity O(M(2)d) for the covariance matrix case and O(MNd) for the data matrix case, where M is the number of sensors, d the number of signals and N the number of snapshots. For a real-time implementation of the proposed algorithms, the computational complexity is O(Md)+O(d(3)) per data vector update. We also present numerical simulation results to illustrate the efficiency of the proposed algorithms.

关键词： array signal processing direction-of-arrival ESPRIT eigenvalue decomposition singular value decomposition QR decomposition fast algorithms updating

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fast algorithms for area morphology

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DIGITAL SIGNAL PROCESSING 2001年第3期11卷 187-203页

作者： Acton, ST Univ Virginia Dept Elect Engn Charlottesville VA 22904 USA

Efficient algorithms are developed for area morphology. As opposed to traditional morphological operations that alter grayscale images via a concatenation of order statistic filters, the area morphological operators manipulate connected components within the image level sets. Essentially, the area morphology filters are capable of removing objects based on the object area solely. These operators can then be effectively used in important multiscale and scale space tasks such as object-based coding and hierarchical image searches. Unfortunately, the traditional implementation of these filters based on level set theory precludes real-time implementation. This paper reviews previous fast algorithms and introduces a pyramidal approach. The full pyramidal algorithm is over 1000 times faster than the standard algorithm for typical image sizes. The paper provides supporting simulation results in terms of computational complexity and solution quality. (C) 2001 Academic Press.

关键词： nonlinear filters image morphology level set theory fast algorithms

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fast algorithms for spherical harmonic expansions

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2006年第6期27卷 1903-1928页

作者： Rokhlin, V Tygert, M Yale Univ Dept Comp Sci New Haven CT 06511 USA Yale Univ Dept Math New Haven CT 06511 USA Yale Univ Dept Appl Math New Haven CT 06511 USA

An algorithm is introduced for the rapid evaluation at appropriately chosen nodes on the two-dimensional sphere S-2 in R-3 of functions specified by their spherical harmonic expansions ( known as the inverse spherical harmonic transform), and for the evaluation of the coefficients in spherical harmonic expansions of functions specified by their values at appropriately chosen points on S-2 ( known as the forward spherical harmonic transform). The procedure is numerically stable and requires an amount of CPU time proportional to N-2( logN) log( 1/epsilon), where N-2 is the number of nodes in the discretization of S-2, and e is the precision of computations. The performance of the algorithm is illustrated via several numerical examples.

关键词： spherical harmonics fast algorithms expansions

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fast algorithms FOR GENERALIZED PREDICTIVE CONTROL

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SYSTEMS & CONTROL LETTERS 1994年第5期23卷 339-348页

作者： CHISCI, L GARULLI, A ZAPPA, G UNIV FLORENCE DIPARTIMENTO SISTEMI & INFORMATVIA SANTA MARTA 3I-50139 FLORENCEITALY

fast algorithms for generalized predictive control (GPC) are derived by adopting an approach whereby dynamic programming and a polynomial formulation are jointly exploited. They consist of a set of coupled linear polynomial recursions by which the dynamic output feedback GPC law is recursively computed with only O(Nn) computations for an n-th order plant and N-steps prediction horizon.

关键词： LINEAR QUADRATIC CONTROL PREDICTIVE CONTROL COMPUTATIONAL METHODS fast algorithms

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fast algorithms FOR BAYESIAN UNCERTAINTY QUANTIFICATION IN LARGE-SCALE LINEAR INVERSE PROBLEMS BASED ON LOW-RANK PARTIAL HESSIAN APPROXIMATIONS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2011年第1期33卷 407-432页

作者： Flath, H. P. Wilcox, L. C. Akcelik, V. Hill, J. Waanders, B. van Bloemen Ghattas, O. Univ Texas Austin Inst Computat Engn & Sci Austin TX 78712 USA Stanford Linear Accelerator Ctr Menlo Pk CA 94025 USA Univ Texas Austin Computat Sci Engn & Math Grad Program Austin TX 78712 USA Sandia Natl Labs Uncertainty & Optimizat Dept Albuquerque NM 87185 USA Univ Texas Austin Dept Mech Engn Austin TX 78712 USA Univ Texas Austin Jackson Sch Geosci Inst Computat Engn & Sci Austin TX 78712 USA

We consider the problem of estimating the uncertainty in large-scale linear statistical inverse problems with high-dimensional parameter spaces within the framework of Bayesian inference. When the noise and prior probability densities are Gaussian, the solution to the inverse problem is also Gaussian and is thus characterized by the mean and covariance matrix of the posterior probability density. Unfortunately, explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters and is thus prohibitive when the forward problem is expensive and the parameter dimension is large. However, for many ill-posed inverse problems, the Hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero. We present a fast method for computation of an approximation to the posterior covariance that exploits the low-rank structure of the preconditioned (by the prior covariance) Hessian of the data misfit. Analysis of an infinite-dimensional model convection-diffusion problem, and numerical experiments on large-scale three-dimensional convection-diffusion inverse problems with up to 1.5 million parameters, demonstrate that the number of forward PDE solves required for an accurate low-rank approximation is independent of the problem dimension. This permits scalable estimation of the uncertainty in large-scale ill-posed linear inverse problems at a small multiple (independent of the problem dimension) of the cost of solving the forward problem.

关键词： large-scale statistical inverse problem Bayesian inference uncertainty quantification fast algorithms low-rank approximation Lanczos Hessian convection-diffusion contaminant transport

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fast algorithms for generalized discrete Hartley transform of composite sequence lengths

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-ANALOG AND DIGITAL SIGNAL PROCESSING 2000年第9期47卷 893-901页

作者： Bi, GA Chen, YQ Zeng, YH Nanyang Technol Univ Sch Elect & Elect Engn Singapore 639798 Singapore

This paper presents fast algorithms for type-II, type-III, and type-IV generalized discrete Hartley transform. In particular, new odd-factor algorithms are derived to support transforms whose sequence length contains multiple odd factors. By jointly using the odd-factor and radix-2 algorithms, fast computation for arbitrarily composite sequence length can be achieved. Compared to other reported algorithms, the proposed ones have a regular computational structure, achieve a substantial reduction of computational complexity, and support a wider range of choices on the sequence length.

关键词： generalized discrete Hartley transform fast algorithms

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fast algorithms for implementation of Montgomery's modular multiplication technique

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CIRCUITS SYSTEMS AND SIGNAL PROCESSING 2004年第6期23卷 463-478页

作者： Mohan, PVA ECIL Bangalore 560052 Karnataka India

In this paper, algorithms for fast implementations of Montgomery's modular multiplication algorithm are proposed. These algorithms use nonredundant multibit recoding. Two techniques, one based on multiplication followed by estimation of scaled residue and the other involving integrated multiplication with scaled residue estimation, have been considered in detail. Techniques for simplifying the computation to estimate the multiple of the modulus to be added have been described. The area and computation time requirements of the proposed techniques are estimated for a general radix in order to highlight the time-space trade-offs.

关键词： fast algorithms modular multiplication Montgomery's algorithm multibit recoding Rivest Shamir and Adleman (RSA) implementation modular exponentiation high-radix implementations very large-scale integration (VLSI) architectures

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fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces

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MATHEMATICS OF COMPUTATION 2006年第254期75卷 903-920页

作者： Nuyens, D Cools, R Katholieke Univ Leuven Dept Comp Sci B-3001 Heverlee Belgium

We reformulate the original component-by-component algorithm for rank-1 lattices in a matrix-vector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost O(sn log(n)), in contrast with the original algorithm which has construction cost O(sn(2)). Herein s is the number of dimensions and n the number of points ( taken prime). In contrast to other approaches to speed up construction, our fast algorithm computes exactly the same quantity as the original algorithm. The presented algorithm can also be used to construct randomly shifted lattice rules in weighted Sobolev spaces.

关键词： numerical integration quasi-Monte Carlo rank-1 lattice rules component-by-component construction fast algorithms

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fast algorithms for computing isogenies between elliptic curves

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MATHEMATICS OF COMPUTATION 2008年第263期77卷 1755-1778页

作者： Bostan, A. Morain, F. Salvy, B. Schost, E. INRIA Rocquencourt Algorithms Project F-78153 Le Chesnay France Univ Paris Sud INRIA Ecole Polytech CNRSProjet Tanc Pole Commun Rech Informat Platea Paris France Univ Western Ontario Dept Comp Sci London ON N6A 5B7 Canada

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree l (l different from the characteristic) in time quasi-linear with respect to l. This is based in particular on fast algorithms for power series expansion of the Weierstrass P-function and related functions.

关键词： fast algorithms elliptic curves finite fields isogenies Schoof-Elkies-Atkin algorithm Newton iteration

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