This paper presents an analysis on and experimental comparison of several typical fast algorithms for discrete wavelet transform (DWT) and their implementation in image compression, particularly the Mallat algorithm, ...
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This paper presents an analysis on and experimental comparison of several typical fast algorithms for discrete wavelet transform (DWT) and their implementation in image compression, particularly the Mallat algorithm, FFT-based algorithm, Short- length based algorithm and Lifting algorithm. The principles, structures and computational complexity of these algorithms are explored in details respectively. The results of the experiments for comparison are consistent to those simulated by MATLAB. It is found that there are limitations in the implementation of DWT. Some algorithms are workable only for special wavelet transform, lacking in generality. Above all, the speed of wavelet transform, as the governing element to the speed of image processing, is in fact the retarding factor for real-time image processing.
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions generally exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid...
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When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions generally exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing such solutions using standard techniques grows with the magnitudes of the eigenvalues. As a consequence, the running times of standard solvers for ordinary differential equations also grow with the size of these eigenvalues. The solutions of scalar equations with slowly-varying coefficients, however, can be represented via slowly-varying phase functions at a cost which is bounded independent of the magnitudes of the eigenvalues of the corresponding coefficient matrix. Here we couple an existing solver for scalar equations which exploits this observation with a well-known technique for transforming a system of linear ordinary differential equations into scalar form. The result is a method for solving a large class of systems of linear ordinary differential equations in time independent of the magnitudes of the eigenvalues of their coefficient matrices. We discuss the results of numerical experiments demonstrating the properties of our algorithm.
Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by formin...
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Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansions which are then evaluated to compute the near or on-surface value of the potential. Recent work towards coupling of a fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon this approach with a modified algorithm for which we give a comprehensive analysis of error and cost in the case of the Laplace equation in two dimensions. For the same levels of (user-specified) accuracy, the new algorithm empirically has cost-per-accuracy comparable to prior approaches. We provide experimental results to demonstrate scalability and numerical accuracy. (C) 2018 Elsevier Inc. All rights reserved.
The need to filter functions defined on the sphere arises in a number of applications, such as climate modeling, electromagnetic and acoustic scattering, and several other areas. Recently, it has been observed that th...
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The need to filter functions defined on the sphere arises in a number of applications, such as climate modeling, electromagnetic and acoustic scattering, and several other areas. Recently, it has been observed that the problem of uniform resolution filtering on the sphere can be performed efficiently via the fast multipole method (FMM) in one dimension. In this paper, we introduce a generalization of the FMM that leads to an accelerated version of the filtering process. Instead of multipole expansions, the scheme uses special-purpose bases constructed via the singular value decomposition of appropriately chosen submatrices of the filtering matrix. The algorithm is applicable to a fairly wide class of projection operators;its performance is illustrated with several numerical examples. (C) 1998 Academic Press.
The fast Newton transversal filter (FNTF) algorithm starts from the recursive least-squares algorithm for adapting a finite impulse response filter. The FNTF algorithm approximates the Kalman gain by replacing the sam...
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The fast Newton transversal filter (FNTF) algorithm starts from the recursive least-squares algorithm for adapting a finite impulse response filter. The FNTF algorithm approximates the Kalman gain by replacing the sample covariance matrix inverse by a banded matrix of total bandwidth 2M + 1 (AR(M) assumption for the input signal). In this algorithm, the approximate Kalman gain can still be computed using an exact recursion that involves the prediction parts of two fast transversal filter (FTF) algorithms of order M. We introduce the subsampled updating (SU) approach in which the FNTF filter weights and the Kalman gain are provided at a subsampled rate, say every L samples. Because of its low computational complexity, the prediction part of the FNTF algorithm is kept as such, A Schur type procedure is used to compute various filter outputs at the intermediate time instants, while some products of vectors with Toeplitz matrices are carried out with the FFT. This leads to the fast subsampled-updating FNTF (FSU FNTF) algorithm, an algorithm that is mathematically equivalent to the FNTF algorithm in the sense that exactly the same filter output is produced. However, it shows a significantly smaller computational complexity for large filter lengths at the expanse of some relatively small delay, The FSU FNTF algorithm (like the FNTF algorithm) has good convergence and tracking properties. This renders the FSU FNTF algorithm very interesting for applications such as acoustic echo cancellation.
A new version of the fast multipole method (FMM) for potential fields is presented. We introduce a new representation of potentials, in which most translation operators are diagonal. As a result, for double precision ...
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A new version of the fast multipole method (FMM) for potential fields is presented. We introduce a new representation of potentials, in which most translation operators are diagonal. As a result, for double precision calculations in two dimensions we obtain an improvement of a factor of two to four in speed, compared to previously published algorithms;the improvement is expected to be much greater in three dimensions. The performance of the method is illustrated with several numerical examples.
In the paper a class of fast adaptive Fourier-based transforms were used for spectroscopic data compression. These transforms are based on adaptive modification of the Cooley-Tukey's signal flow graph. The adaptiv...
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In the paper a class of fast adaptive Fourier-based transforms were used for spectroscopic data compression. These transforms are based on adaptive modification of the Cooley-Tukey's signal flow graph. The adaptive versions of the cosine, cosine-Haar and cosine-Walsh transform of various degrees were taken as a base for the experiments. The transform kernels are modified according to reference vectors representing a given class of processed data. The results obtained using these transforms for gamma-gamma ray coincidence spectra compression are presented and compared with the results obtained by use of classical transforms. Both classical and adaptive transforms can be used for off-line as well as for on-line compression. (C) 2004 Elsevier B.V. All rights reserved.
The analysis of (approximately) periodic signals is an important element in numerous applications. One generalization of standard periodic signals often occurring in practice is harmonic chirp signals where the instan...
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The analysis of (approximately) periodic signals is an important element in numerous applications. One generalization of standard periodic signals often occurring in practice is harmonic chirp signals where the instantaneous frequency increases/decreases linearly as a function of time. A statistically efficient estimator for extracting the parameters of the harmonic chirp model in additive white Gaussian noise is the maximum-likelihood (ML) estimator, which recently has been demonstrated to be robust to noise and accurate-evenwhen the model order is unknown. The main drawback of the ML estimator is that only very computationally demanding algorithms for computing an estimate are known. In this paper, we give an algorithm for computing an estimate to the ML estimator for a number of candidate model orders with a much lower computational complexity than previously reported in the literature. The lower computational complexity is achieved by exploiting recursive matrix structures, including a block Toeplitzplus-Hankel structure, the fast Fourier transform, and using a two-step approach composed of a grid and refinement step to reduce the number of required function evaluations. The proposed algorithms are assessed via Monte Carlo and timing studies. The timing studies show that the proposed algorithm is orders of magnitude faster than a recently proposed algorithm for practical sizes of the number of harmonics and the length of the signal.
The higher-order singular value decomposition (HOSVD) is a generalization of the singular value decomposition (SVD) to higher-order tensors (i.e., arrays with more than two indices) and plays an important role in vari...
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The higher-order singular value decomposition (HOSVD) is a generalization of the singular value decomposition (SVD) to higher-order tensors (i.e., arrays with more than two indices) and plays an important role in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a third-order tensor involves the computation of the SVD of three matrices, which are referred to as "modes" or "matrix unfoldings." In this paper, we present fast algorithms for computing the full and the rank-truncated HOSVD of third-order structured (symmetric, Toeplitz, and Hankel) tensors. These algorithms are derived by considering two specific ways to unfold a structured tensor, leading to structured matrix unfoldings whose SVD can be efficiently computed.
We propose a fast reconstruction method for a subband-decomposed, progressive signal coding system. We show that unlike the conventional approach which requires a fixed computational complexity, the computational comp...
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We propose a fast reconstruction method for a subband-decomposed, progressive signal coding system. We show that unlike the conventional approach which requires a fixed computational complexity, the computational complexity of the proposed approach is proportional to the number of refined coefficients at each level of progression. Therefore, unrefined coefficients do not add to the computational complexity of the proposed scheme. It is shown, through specific examples, that the proposed approach can lead to significant reductions in reconstruction complexity. Furthermore, the proposed approach provides the capability for an online updating of the reconstructed image based on receiving the refinement of each coefficient.
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