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76
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 287 (27 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Wavelet Processes and Adaptive Estimation of the Evolutionary Wavelet Spectrum
, 1998
"... This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power va ..."
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Cited by 60 (28 self)
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This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable timelocalized autocovariance. We illustrate our theory with a pedagogical example based on discrete nondecimated Haar wavelets and also a real medical time series example.
Parallel scientific computing in C++ and MPI: a seamless approach to parallel algorithms and their implementation
, 2003
"... Scientific computing is by its very nature a practical subject it requires tools and a lot of practice. To solve realistic problems we need not only fast algorithms but also a combination of good tools and fast computers. This is the subject of the current book, which emphasizes equally all three: ..."
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Cited by 26 (1 self)
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Scientific computing is by its very nature a practical subject it requires tools and a lot of practice. To solve realistic problems we need not only fast algorithms but also a combination of good tools and fast computers. This is the subject of the current book, which emphasizes equally all three: algorithms, tools, and computers. Often times such concepts and tools are taught serially across different courses and different textbooks, and hence the interconnection between them is not immediately apparent. We believe that such a close integration is important from the outset. The book starts with a heavy dosage of C++ and basic mathematical and computational concepts, and it ends emphasizing advanced parallel algorithms that are used in modern simulations. We have tried to make this book fun to read, to somewhat demystify the subject, and thus the style is sometimes informal and personal. It may seem that this happens at the expense of rigor, and indeed we have tried to limit notation and theorem proofing. Instead, we emphasize concepts and useful tricksofthetrade with many code segments, remarks, reminders, and warnings throughout the book. The material of this book has been taught at different times to students in engineering,
Adaptive ENOWavelet Transforms for Discontinuous Functions
, 2001
"... this report is patent pending. ..."
Ridge functions and orthonormal ridgelets
 J. Approx. Theory
"... Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coeff ..."
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Cited by 16 (2 self)
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Orthonormal ridgelets are a specialized set of angularlyintegrated ridge functions which make up an orthonormal basis for L 2 (R 2). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coefficients of a ridge function in terms of the 1D wavelet coefficients of the ridge profile r(t), and we study the properties of the linear approximation operator which ‘kills ’ coefficients at high angular scale or high ridge scale. We also show that partial orthonormal ridgelet expansions can give efficient nonlinear approximations to pure ridge functions. In effect, the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1D wavelet coefficients of the 1D ridge profile r(t). This shows that simple thresholding in the ridgelet basis is, for certain purposes, equally as good as ideal nonlinear ridge approximation. Key Words and Phrases. Wavelets. Ridge function. Ridgelet. Radon transform. Best mterm approximation. Thresholding of wavelet coefficients.
ENOwavelet Transforms for Piecewise Smooth Functions
 SIAM J. Numer. Anal., Vol
"... Abstract. We have designed an adaptive essentially nonoscillatory (ENO)wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The ..."
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Cited by 14 (2 self)
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Abstract. We have designed an adaptive essentially nonoscillatory (ENO)wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENOwavelet transforms we adaptively change the function and use the same uniform stencils. The ENOwavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENOwavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate. Key words. ENO, wavelet, image compression, image denoising, signal processing
Parallel Performance Of Fast Wavelet Transforms
 INTERNATIONAL JOURNAL OF HIGH SPEED COMPUTING
, 2000
"... We present a parallel 2D wavelet transform algorithm with modest communication requirements. Data are transmitted between nearest neighbors only and the amount is independent of the problem size as well as the number of processors. An analysis of the theoretical performance shows that the algorithm ..."
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Cited by 12 (1 self)
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We present a parallel 2D wavelet transform algorithm with modest communication requirements. Data are transmitted between nearest neighbors only and the amount is independent of the problem size as well as the number of processors. An analysis of the theoretical performance shows that the algorithm is scalable approaching perfect speedup as the problem size is increased. This performance is realized in practice on the IBM SP2 as well as on the Fujitsu VPP300 where it will form part of the Scientific Software Library.
Assessing Nonstationary Time Series Using Wavelets
, 1998
"... The discrete wavelet transform has be used extensively in the field of Statistics, mostly in the area of "denoising signals" or nonparametric regression. This thesis provides a new application for the discrete wavelet transform, assessing nonstationary events in time series  especially l ..."
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Cited by 10 (4 self)
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The discrete wavelet transform has be used extensively in the field of Statistics, mostly in the area of "denoising signals" or nonparametric regression. This thesis provides a new application for the discrete wavelet transform, assessing nonstationary events in time series  especially long memory processes. Long memory processes are those which exhibit substantial correlations between events separated by a long period of time. Departures from stationarity in these heavily autocorrelated time series, such as an abrupt change in the variance at an unknown location or "bursts" of increased variability, can be detected and accurately located using discrete wavelet transforms  both orthogonal and overcomplete. A cumulative sum of squares method, utilizing a KolomogorovSmirnovtype
A new wavelet transform preconditioner for iterative solution of elastohydrodynamic lubrication problems
 Int. J. Comput. Maths
, 2000
"... \Lambda y ..."
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A Scalable Parallel 2D Wavelet Transform Algorithm
, 1997
"... We present a new parallel 2D wavelet transform algorithm with minimal communication requirements. Data are transmitted between nearest neighbors only and the amount is independent of the problem size as well as the number of processors. An analysis of the theoretical performance shows that our algor ..."
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Cited by 6 (0 self)
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We present a new parallel 2D wavelet transform algorithm with minimal communication requirements. Data are transmitted between nearest neighbors only and the amount is independent of the problem size as well as the number of processors. An analysis of the theoretical performance shows that our algorithm is highly scalable approaching perfect speedup as the problem size is increased. This performance is realized in practice on the IBM SP2 as well as on the Fujitsu VPP300 where it will form part of the Scientific Software Library.