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Author

Gregory Beylkin

Other affiliations: New York University, University of Tennessee, Yale University  ...read more
Bio: Gregory Beylkin is an academic researcher from University of Colorado Boulder. The author has contributed to research in topics: Multiresolution analysis & Wavelet. The author has an hindex of 42, co-authored 140 publications receiving 9991 citations. Previous affiliations of Gregory Beylkin include New York University & University of Tennessee.


Papers
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Journal ArticleDOI
TL;DR: The algorithms presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators, and indicate that many previously intractable problems become manageable with the techniques presented here.
Abstract: A class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N × N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.

1,841 citations

Journal ArticleDOI
TL;DR: In this paper, the linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids.
Abstract: This paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers. The linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids. This integral equation is connected with the causal generalized Radon transform (GRT), and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the GRT. The first term of this asymptotic expansion is interpreted as a migration algorithm. As a result, this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium. Also, a partial reconstruction operator is explicitly derived for a limited aperture. When specialized to a constant background velocity and specific source–receiver geometries our results are directly related to some known migration algorithms.

671 citations

Journal ArticleDOI
TL;DR: Beylkin and Rokhlin this paper presented exact and explicit representations of the differential operators in orthonormal bases of compactly supported wavelets as well as the representations of Hilbert transform and fractional derivatives.
Abstract: This paper describes exact and explicit representations of the differential operators, ${{d^n } / {dx^n }}$, $n = 1,2, \cdots $, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators.Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring $O(N\log N)$ operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length $N = 2^n $ is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141–183]) may be reduced from $O(N)$ to $O(\log ^2 N)$ significant entries.

614 citations

Journal ArticleDOI
TL;DR: In this paper, the authors formalize the classical diffraction stack by relating it to linearized seismic inversion and the generalized Radon transform, which can handle both complex velocity models and arbitrary configurations of sources and receivers.
Abstract: A new approach to seismic migration formalizes the classical diffraction (or common-tangent) stack by relating it to linearized seismic inversion and the generalized Radon transform. This approach recasts migration as the problem of reconstructing the earth’s acoustic scattering potential from its integrals over isochron surfaces. The theory rests on a solution of the wave equation with the geometrical-optics Green function and an approximate inversion formula for the generalized Radon transform. The method can handle both complex velocity models and (nearly) arbitrary configurations of sources and receivers. In this general case, the method can be implemented as a weighted diffraction stack, with the weights determined by tracing rays from image points to the experiment’s sources and receivers. When tested on a finite-difference simulation of a deviated-well vertical seismic profile (a hybrid experiment which is difficult to treat with conventional wave-equation methods), the algorithm accurately reconstructed faulted-earth models. Analytical reconstruction formulas are derived from the general formula for zero-offset and fixed-offset surface experiments in which the background velocity is constant. The zero-offset inversion formula resembles standard Kirchhoff migration. Our analysis provides a direct connection between the experimental setup (source and receiver positions, source wavelet, background velocity) and the spatial resolution of the reconstruction. Synthetic examples illustrate that the lateral resolution in seismic images is described well by the theory and is improved greatly by combining surface data and borehole data. The best resolution is obtained from a zero-offset experiment that surrounds the region to be imaged.

430 citations

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TL;DR: It is shown that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves.
Abstract: This paper describes the discrete Radon transform (DRT) and the exact inversion algorithm for it. Similar to the discrete Fourier transform (DFT), the DRT is defined for periodic vector-sequences and studied as a transform in its own right. Casting the forward transform as a matrix-vector multiplication, the key observation is that the matrix-although very large-has a block-circulant structure. This observation allows construction of fast direct and inverse transforms. Moreover, we show that the DRT can be used to compute various generalizations of the classical Radon transform (RT) and, in particular, the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves. In fact, we describe not a single transform, but a class of transforms, representatives of which correspond in one way or another to discrete versions of the RT and its generalizations. An interesting observation is that the exact inversion algorithm cannot be obtained directly from Radon's inversion formula. Given the fact that the RT has no nontrivial one-dimensional analog, exact invertibility makes the DRT a useful tool geared specifically for multidimensional digital signal processing. Exact invertibility of the DRT, flexibility in its definition, and fast computational algorithm affect present applications and open possibilities for new ones. Some of these applications are discussed in the paper.

426 citations


Cited by
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28 Jul 2005
TL;DR: PfPMP1)与感染红细胞、树突状组胞以及胎盘的单个或多个受体作用,在黏附及免疫逃避中起关键的作�ly.
Abstract: 抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1(PfPMP1)与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用,在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员,通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

18,940 citations

Book
01 Jan 1998
TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

17,693 citations

Journal ArticleDOI
TL;DR: This survey provides an overview of higher-order tensor decompositions, their applications, and available software.
Abstract: This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

9,227 citations

Journal ArticleDOI
TL;DR: It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.
Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p-penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.

4,339 citations

Journal ArticleDOI
TL;DR: A scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed and it is shown that the wavelet transform is particularly well adapted to progressive transmission.
Abstract: A scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed. This method involves two steps. First, a wavelet transform used in order to obtain a set of biorthogonal subclasses of images: the original image is decomposed at different scales using a pyramidal algorithm architecture. The decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image. Second, according to Shannon's rate distortion theory, the wavelet coefficients are vector quantized using a multiresolution codebook. To encode the wavelet coefficients, a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed. In order to allow the receiver to recognize a picture as quickly as possible at minimum cost, a progressive transmission scheme is presented. It is shown that the wavelet transform is particularly well adapted to progressive transmission. >

3,925 citations