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Journal articles on the topic 'Wavelet approximation'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Polanowski, Stanisław. "Application of movable approximation and wavelet decomposition to smoothing-out procedure of ship engine indicator diagrams." Polish Maritime Research 14, no. 2 (April 1, 2007): 12–17. http://dx.doi.org/10.2478/v10012-007-0008-y.

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Application of movable approximation and wavelet decomposition to smoothing-out procedure of ship engine indicator diagrams In this paper - on the basis of indicator diagram processing taken as an example - were shown possibilities of the smoothing-out and decomposing of run disturbances with the use of the movable multiple approximation based on the least squares criterion. The notion was defined of movable approximating object and constraints used to form approximation features. It was demonstrated that the multiple approximation can be used to decompose disturbances out of an analyzed run. The obtained smoothing-out results were compared with those obtained from full-interval approximation of runs by means of splines as well as wavelet decomposition with using various wavelets, Wavelet Explorer and Mathematica software. Smoothing-out quality was assessed by comparing runs of first derivatives which play crucial role in the advanced processing of indicator diagrams.

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2

BANAKAR, AHMAD, MOHAMMAD FAZLE AZEEM, and VINOD KUMAR. "COMPARATIVE STUDY OF WAVELET BASED NEURAL NETWORK AND NEURO-FUZZY SYSTEMS." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 06 (November 2007): 879–906. http://dx.doi.org/10.1142/s0219691307002099.

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Based on the wavelet transform theory and its well emerging properties of universal approximation and multiresolution analysis, the new notion of the wavelet network is proposed as an alternative to feed forward neural networks and neuro-fuzzy for approximating arbitrary nonlinear functions. Earlier, two types of neuron models, namely, Wavelet Synapse (WS) neuron and Wavelet Activation (WA) functions neuron have been introduced. Derived from these two neuron models with different non-orthogonal wavelet functions, neural network and neuro-fuzzy systems are presented. Comparative study of wavelets with NN and NF are also presented in this paper.

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3

Romanchak, V. M. "Local transformations with a singular wavelet." Informatics 17, no. 1 (March 29, 2020): 39–46. http://dx.doi.org/10.37661/1816-0301-2020-17-1-39-46.

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The paper considers a local wavelet transform with a singular basis wavelet. The problem of nonparametric approximation of a function is solved by the use of the sequence of local wavelet transforms. Traditionally believed that the wavelet should have an average equal to zero. Earlier, the author considered singular wavelets when the average value is not equal to zero. As an example, the delta-shaped functions, participated in the estimates of Parzen – Rosenblatt and Nadara – Watson, were used as a wavelet. Previously, a sequence of wavelet transforms for the entire numerical axis and finite interval was constructed for singular wavelets. The paper proposes a sequence of local wavelet transforms, a local wavelet transform is defined, the theorems that formulate the properties of a local wavelet transform are proved. To confirm the effectiveness of the algorithm an example of approximating the function by use of the sum of discrete local wavelet transforms is given.

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4

Romanchak, V. M. "APPROXIMATELY SINGULAR WAVELET." «System analysis and applied information science», no. 2 (August 7, 2018): 23–28. http://dx.doi.org/10.21122/2309-4923-2018-2-23-28.

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The problem of approximation is relevant for most engineering applications. In this connection, the universal methods of approximation are of interest. The method of nonparametric approximation is developing in the paper – the method of singular wavelets. The method includes an effective numerical algorithm based on the summation of a recursive sequence of functions. The universal algorithm of approximation makes it possible to apply it to approximate one-dimensional and multidimensional functions, in decision support systems, in the processing of stochastic information, pattern recognition, and solution of boundary-value problems.The introduction explain the idea of the method of singular wavelets – to combine the theory of wavelets with the Nadaraya-Watson kernel regression estimator. Usually, Nadaraya-Watson kernel regression are considered as an example of non- parametric estimation. However, one parameter, the smoothing parameter, is still present in the traditional kernel regression algorithm. The choice of the optimal value of this parameter is a complex mathematical problem, and numerous studies have been devoted to this question. In the approximation by the method of singular wavelets, summation of Nadaraya-Watson kernel regression estimates with the smoothing parameter takes place, which solves the problem of the optimal choice of this parameter.In the main part of the paper theorems are formulated that determine the properties of the regularized wavelet transform. Sufficient conditions for uniform convergence of the wavelet series are obtained for the first time. To illustrate the effectiveness of the numerical approximation algorithm, we consider an example of the quasi-interpolation of the Runge function by wavelets with a uniform distribution of interpolation nodes.

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Tahami, M., A. Askari Hemmat, and S. A. Yousefi. "Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet." International Journal of Wavelets, Multiresolution and Information Processing 14, no. 01 (January 2016): 1650004. http://dx.doi.org/10.1142/s0219691316500041.

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In one-dimensional problems, the Legendre wavelets are good candidates for approximation. In this paper, we present a numerical method for solving two-dimensional first kind Fredholm integral equation. The method is based upon two-dimensional linear Legendre wavelet basis approximation. By applying tensor product of one-dimensional linear Legendre wavelet we construct a two-dimensional wavelet. Finally, we give some numerical examples.

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6

Khanna, Nikhil, Varinder Kumar, and S. K. Kaushik. "Wavelet packet approximation." Integral Transforms and Special Functions 27, no. 9 (June 6, 2016): 698–714. http://dx.doi.org/10.1080/10652469.2016.1189912.

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7

Guo, Shun Sheng, and A. S. Cavaretta. "LINEAR WAVELET APPROXIMATION." Analysis 13, no. 4 (December 1993): 351–62. http://dx.doi.org/10.1524/anly.1993.13.4.351.

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8

DeVore, R. A., S. V. Konyagin, and V. N. Temlyakov. "Hyperbolic Wavelet Approximation." Constructive Approximation 14, no. 1 (January 1, 1997): 1–26. http://dx.doi.org/10.1007/s003659900060.

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9

Pourakbari, Fatemeh, and Ali Tavakoli. "Modification of Multiple Knot B-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem." Advances in Applied Mathematics and Mechanics 4, no. 06 (December 2012): 799–820. http://dx.doi.org/10.4208/aamm.12-12s10.

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AbstractA construction of multiple knotB-spline wavelets has been given in [C. K. Chui and E. Quak, Wavelet on a bounded interval, In: D. Braess and L. L. Schumaker, editors. Numerical methods of approximation theory. Basel: Birkhauser Verlag; (1992), pp. 57-76]. In this work, we first modify these wavelets to solve the elliptic (partially) Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods. We generalize this construction to two dimensional case by Tensor product space. In addition, the solution of the system discretized by Galerkin method with modified multiple knotB-spline wavelets is discussed. We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant. Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel, it is solved by multiple knotB-spline wavelet method that yields a very well approximation. Finally, some numerical examples are given to support our theoretical results.

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10

Cattani, Carlo, and Aleksey Kudreyko. "Application of Periodized Harmonic Wavelets towards Solution of Eigenvalue Problems for Integral Equations." Mathematical Problems in Engineering 2010 (2010): 1–8. http://dx.doi.org/10.1155/2010/570136.

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This article deals with the application of the periodized harmonic wavelets for solution of integral equations and eigenvalue problems. The solution is searched as a series of products of wavelet coefficients and wavelets. The absolute error for a general case of the wavelet approximation was analytically estimated.

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11

ZHANG, LI, WEIDA ZHOU, and LICHENG JIAO. "SUPPORT VECTOR MACHINES BASED ON THE ORTHOGONAL PROJECTION KERNEL OF FATHER WAVELET." International Journal of Computational Intelligence and Applications 05, no. 03 (September 2005): 283–303. http://dx.doi.org/10.1142/s1469026805001489.

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Recently the study on the theory of wavelets shows that the wavelets have not only the multi-resolution property both in frequency and time domain, but also the good approximation ability. SVMs based on the statistical learning theory are a kind of general and effective learning machines, and have described for us the nice application blueprint in machine learning domain. There exists a bottleneck problem, or the pre-selection of kernel parameter for SVMs. In this paper, the orthogonal projection kernels of father wavelet (OPFW kernels) are introduced into SVMs. In doing so SVMs based on the OPFW kernels can have good performance in both approximation and generalisation. Simultaneously the parameter pre-selection of wavelet kernels can be implemented by discrete wavelet transform. Experiments on regression estimation illustrate the approximation and generalisation ability of our method.

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12

SHUKLA, NIRAJ K. "NON-MSF A-WAVELETS FROM A-WAVELET SETS." International Journal of Wavelets, Multiresolution and Information Processing 11, no. 01 (January 2013): 1350002. http://dx.doi.org/10.1142/s0219691313500021.

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Generalizing the result of Bownik and Speegle [Approximation Theory X: Wavelets, Splines and Applications, Vanderbilt University Press, pp. 63–85, 2002], we provide plenty of non-MSF A-wavelets with the help of a given A-wavelet set. Further, by showing that the dimension function of the non-MSF A-wavelet constructed through an A-wavelet set W coincides with the dimension function of W, we conclude that the non-MSF A-wavelet and the A-wavelet set through which it is constructed possess the same nature as far as the multiresolution analysis is concerned. Some examples of non-MSF d-wavelets and non-MSF A-wavelets are also provided. As an illustration we exhibit a pathwise connected class of non-MSF non-MRA wavelets sharing the same wavelet dimension function.

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13

EHLER, MARTIN. "COMPACTLY SUPPORTED MULTIVARIATE, PAIRS OF DUAL WAVELET FRAMES OBTAINED BY CONVOLUTION." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 02 (March 2008): 183–208. http://dx.doi.org/10.1142/s0219691308002306.

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In this paper, we present a construction of compactly supported multivariate pairs of dual wavelet frames. The approach is based on the convolution of two refinable distributions. We obtain smooth wavelets with any preassigned number of vanishing moments. Their underlying refinable function is fundamental. In the examples, we obtain symmetric wavelets with small support from optimal refinable functions, i.e. the refinable function has minimal mask size with respect to smoothness and approximation order of its generated multiresolution analysis. The wavelet system has maximal approximation order with respect to the underlying refinable function.

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14

Sahito, Faisal, Pan Zhiwen, Junaid Ahmed, and Raheel Ahmed Memon. "Wavelet-Integrated Deep Networks for Single Image Super-Resolution." Electronics 8, no. 5 (May 17, 2019): 553. http://dx.doi.org/10.3390/electronics8050553.

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We propose a scale-invariant deep neural network model based on wavelets for single image super-resolution (SISR). The wavelet approximation images and their corresponding wavelet sub-bands across all predefined scale factors are combined to form a big training data set. Then, mappings are determined between the wavelet sub-band images and their corresponding approximation images. Finally, the gradient clipping process is used to boost the training speed of the algorithm. Furthermore, stationary wavelet transform (SWT) is used instead of a discrete wavelet transform (DWT), due to its up-scaling property. In this way, we can preserve more information about the images. In the proposed model, the high-resolution image is recovered with detailed features, due to redundancy (across the scale) property of wavelets. Experimental results show that the proposed model outperforms state-of-the algorithms in terms of peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM).

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15

Cattani, Carlo. "Fractional Calculus and Shannon Wavelet." Mathematical Problems in Engineering 2012 (2012): 1–26. http://dx.doi.org/10.1155/2012/502812.

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An explicit analytical formula for the any order fractional derivative of Shannon wavelet is given as wavelet series based on connection coefficients. So that for anyL2(ℝ)function, reconstructed by Shannon wavelets, we can easily define its fractional derivative. The approximation error is explicitly computed, and the wavelet series is compared with Grünwald fractional derivative by focusing on the many advantages of the wavelet method, in terms of rate of convergence.

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16

Anastassiou, G. A., and X. M. Yu. "Probabilistic discrete wavelet approximation." Numerical Functional Analysis and Optimization 13, no. 1-2 (January 1992): 117–21. http://dx.doi.org/10.1080/01630569208816464.

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17

Anastassiou, George, and X. M. Yu. "Bivariate constrained wavelet approximation." Journal of Computational and Applied Mathematics 53, no. 1 (July 1994): 1–9. http://dx.doi.org/10.1016/0377-0427(92)00131-r.

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18

Burma Saparova, Roza Mamytova, Nurjamal Kurbanbaeva, and Anvarjon Akhatjonovich Ahmedov. "A Haar Wavelet Series Solution of Heat Equation with Involution." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 86, no. 2 (August 22, 2021): 50–55. http://dx.doi.org/10.37934/arfmts.86.2.5055.

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It is well known that the wavelets have widely applied to solve mathematical problems connected with the differential and integral equations. The application of the wavelets possess several important properties, such as orthogonality, compact support, exact representation of polynomials at certain degree and the ability to represent functions on different levels of resolution. In this paper, new methods based on wavelet expansion are considered to solve problems arising in approximation of the solution of heat equation with involution. We have developed new numerical techniques to solve heat equation with involution and obtained new approximative representation for solution of heat equations.

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19

Luo, Hongjun, Dietmar Kolb, Heinz-Jürgen Flad, Wolfgang Hackbusch, and Thomas Koprucki. "Wavelet approximation of correlated wave functions. II. Hyperbolic wavelets and adaptive approximation schemes." Journal of Chemical Physics 117, no. 8 (August 22, 2002): 3625–38. http://dx.doi.org/10.1063/1.1494800.

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20

Cheng, Bing, and Xiaokun Zhu. "A Multiresolution Approximation Theory of Fractal Transform." Fractals 05, supp01 (April 1997): 173–86. http://dx.doi.org/10.1142/s0218348x97000747.

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In this paper, we show that the fractal transform (FT) constitutes a multiresolution approximation to the square-integrable space L2(Td) for d≥1, where T is the interval (-∞,∞). This provides a theoretical basis for the successful applications of the fractal transform algorithms in signal/image encoding. There are many similarities between fractal-based and wavelet-based approximations. However, they are undamentally different from each other in many aspects. Fractal-based multiresolution approximation to signals/images is by a way of self-increasing model complexity, and wavelet-based multiresolution approximation to signals/images is by a way of decomposing data complexity into single (time domain) components.

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21

Khanna, Nikhil, S. K. Kaushik, and A. M. Jarrah. "Wavelet packets: Uniform approximation and numerical integration." International Journal of Wavelets, Multiresolution and Information Processing 18, no. 02 (November 27, 2019): 2050004. http://dx.doi.org/10.1142/s0219691320500046.

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In this paper, it is proved that under some conditions, wavelet packet basis of [Formula: see text] can be used as a tool for the uniform approximation of an [Formula: see text]-times ([Formula: see text]) continuously differentiable and square integrable function [Formula: see text]. Sufficient conditions which establish that the approximations of wavelet packet sequences of square integrable function [Formula: see text] at lower levels are uniformly reliable and they uniformly approach zero as [Formula: see text] are given. Finally, a method based on wavelet packet expansion to find the definite integral of a function in [Formula: see text] is given and its error analysis has been discussed.

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22

Liu, Yanan, and Keqin Din. "A Numerical Method Based on Daubechies Wavelet Basis and B-Spline Patches for Elasticity Problems." Mathematical Problems in Engineering 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/2549213.

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The Daubechies (DB) wavelets are used for solving 2D plane elasticity problems. In order to improve the accuracy and stability in computation, the DB wavelet scaling functions in0,+∞)comprising boundary scaling functions are chosen as basis functions for approximation. The B-spline patches used in isogeometry analysis method are constructed to describe the problem domain. Through the isoparametric analysis approach, the function approximation and relevant computation based on DB wavelet functions are implemented on B-spline patches. This work makes an attempt to break the limitation that problems only can be discretized on uniform grids in the traditional wavelet numerical method. Numerical examples of 2D elasticity problems illustrate that this kind of analysis method is effective and stable.

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Romanchak, V. M., and M. A. Hundzina. "Isolation of a periodic component by singular wavelet decomposition." «System analysis and applied information science», no. 3 (September 25, 2020): 4–8. http://dx.doi.org/10.21122/2309-4923-2020-3-4-8.

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In this paper, we propose to use a discrete wavelet transform with a singular wavelet to isolate the periodic component from the signal. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet is zero). For singular wavelets, the validity condition is not met. As a singular wavelet, you can use the Delta-shaped functions, which are involved in the estimates of Parzen-Rosenblatt, Nadaraya-Watson. Using singular value of a wavelet is determined by the discrete wavelet transform. This transformation was studied earlier for the continuous case. Theoretical estimates of the convergence rate of the sum of wavelet transformations were obtained; various variants were proposed and a theoretical justification was given for the use of the singular wavelet method; sufficient conditions for uniform convergence of the sum of wavelet transformations were formulated. It is shown that the wavelet transform can be used to solve the problem of nonparametric approximation of the function. Singular wavelet decomposition is a new method and there are currently no examples of its application to solving applied problems. This paper analyzes the possibilities of the singular wavelet method. It is assumed that in some cases a slow and fast component can be distinguished from the signal, and this hypothesis is confirmed by the numerical solution of the real problem. A similar analysis is performed using a parametric regression equation, which allows you to select the periodic component of the signal. Comparison of the calculation results confirms that nonparametric approximation based on singular wavelets and the application of parametric regression can lead to similar results.

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24

LEWALLE, JACQUES. "FIELD RECONSTRUCTION FROM SINGLE SCALE CONTINUOUS WAVELET COEFFICIENTS." International Journal of Wavelets, Multiresolution and Information Processing 07, no. 01 (January 2009): 131–42. http://dx.doi.org/10.1142/s0219691309002738.

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The redundancy of continuous wavelet transforms implies that the wavelet coefficients are not independent of each other. This interdependence allows the reconstruction or approximation of the wavelet transform, and of the original field, from a subset of the wavelet coefficients. Contrasting with lines of modulus maxima, known to provide useful partition functions and some data compaction, the reconstruction from single-scale coefficients is derived for the Hermitian family of wavelets. The formula is exact in the continuum for d-dimensional fields, and its limitations under discretization are illustrated.

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25

Anastassiou, G. A., and X. M. Yu. "Monotone and probabilistic wavelet approximation." Stochastic Analysis and Applications 10, no. 3 (January 1992): 251–64. http://dx.doi.org/10.1080/07362999208809268.

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26

Skopina, Maria. "Wavelet Approximation of Periodic Functions." Journal of Approximation Theory 104, no. 2 (June 2000): 302–29. http://dx.doi.org/10.1006/jath.1999.3434.

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27

OTHMANI, MOHAMED, WAJDI BELLIL, CHOKRI BEN AMAR, and ADEL M. ALIMI. "A NEW STRUCTURE AND TRAINING PROCEDURE FOR MULTI-MOTHER WAVELET NETWORKS." International Journal of Wavelets, Multiresolution and Information Processing 08, no. 01 (January 2010): 149–75. http://dx.doi.org/10.1142/s0219691310003353.

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This paper deals with the features of a new wavelet network structure founded on several mother wavelets families. This new structure is similar to the classic wavelets network but it admits some differences eventually. The wavelet network basically uses the dilations and translations versions of only one mother wavelet to construct the network, but the new one uses several mother wavelets and the objective is to maximize the probability of selection of the best wavelets. Two methods are presented to assist the training procedure of this new structure. On one hand, we have an optimal selection technique that is based on an improved version of the Orthogonal Least Squares method; on the other, the Generalized Cross-Validation method to determine the number of wavelets to be selected for every mother wavelet. Some simulation results are reported to demonstrate the performance and the effectiveness of the new structure and the training procedure for function approximation in one and two dimensions.

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Tamimi, E. "Wavelet series approximation using wavelet function with compactly support." Journal of Fundamental and Applied Sciences 8, no. 2 (June 14, 2016): 132. http://dx.doi.org/10.4314/jfas.8vi2s.9.

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29

Islam, M. R., S. F. Ahemmed, and S. M. A. Rahman. "Comparison of wavelet approximation order in different smoothness spaces." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–7. http://dx.doi.org/10.1155/ijmms/2006/63670.

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In linear approximation by wavelet, we approximate a given function by a finite term from the wavelet series. The approximation order is improved if the order of smoothness of the given function is improved, discussed by Cohen (2003), DeVore (1998), and Siddiqi (2004). But in the case of nonlinear approximation, the approximation order is improved quicker than that in linear case. In this study we proved this assumption only for the Haar wavelet. Haar function is an example of wavelet and this fundamental example gives major feature of the general wavelet. A nonlinear space comes from arbitrary selection of wavelet coefficients, which represent the target function almost equally. In this case our computational work will be reduced tremendously in the sense that approximation error decays more quickly than that in linear case.

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Lal, Shyam, and Susheel Kumar. "Quasi-Positive Delta Sequences and Their Applications in Wavelet Approximation." International Journal of Mathematics and Mathematical Sciences 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/9121249.

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A sufficient literature is available for the wavelet error of approximation of certain functions in theL2-norm. There is no work in context of multiresolution approximation of a function in the sense of sup-error. In this paper, for the first time, wavelet estimator for the approximation of a functionfbelonging toLipα[a,b]class under supremum norm has been obtained. Working in this direction, four new theorems on the wavelet approximation of a functionfbelonging toLipα,0<α≤1class using the projectionPmfof its wavelet expansions have been estimated. The calculated estimator is best possible in wavelet analysis.

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Romanchak, V. M. "Wavelet transformation on a finite interval." Informatics 17, no. 4 (January 3, 2021): 22–35. http://dx.doi.org/10.37661/10.37661/1816-0301-2020-17-4-22-35.

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Integral transformations on a finite interval with a singular basis wavelet are considered. Using a sequence of such transformations, the problem of nonparametric approximation of a function is solved. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet must be zero). The paper develops the previously proposed method of singular wavelets when the tolerance condition is not met. In this case Delta-shaped functions that participate in Parzen – Rosenblatt and Nadaray – Watson estimations can be used as a basic wavelet. The set of wavelet transformations for a function defined on a numeric axis, defined locally, and on a finite interval were previously investigated. However, the study of the convergence of the decomposition on a finite interval was carried out only in one particular case. It was due to technical difficulties when trying to solve this problem directly. In the paper the idea of evaluating the periodic continuation of a function defined initially on a finite interval is implemented. It allowed to formulate sufficient convergence conditions for the expansion of the function in a series. An example of approximation of a function defined on a finite interval using the sum of discrete wavelet transformations is given.

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Romanchak, V. M. "Wavelet transformation on a finite interval." Informatics 17, no. 4 (January 3, 2021): 22–35. http://dx.doi.org/10.37661/10.37661/1816-0301-2020-17-4-22-35.

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Integral transformations on a finite interval with a singular basis wavelet are considered. Using a sequence of such transformations, the problem of nonparametric approximation of a function is solved. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet must be zero). The paper develops the previously proposed method of singular wavelets when the tolerance condition is not met. In this case Delta-shaped functions that participate in Parzen – Rosenblatt and Nadaray – Watson estimations can be used as a basic wavelet. The set of wavelet transformations for a function defined on a numeric axis, defined locally, and on a finite interval were previously investigated. However, the study of the convergence of the decomposition on a finite interval was carried out only in one particular case. It was due to technical difficulties when trying to solve this problem directly. In the paper the idea of evaluating the periodic continuation of a function defined initially on a finite interval is implemented. It allowed to formulate sufficient convergence conditions for the expansion of the function in a series. An example of approximation of a function defined on a finite interval using the sum of discrete wavelet transformations is given.

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33

Farkov, Yuri, Elena Lebedeva, and Maria Skopina. "Wavelet frames on Vilenkin groups and their approximation properties." International Journal of Wavelets, Multiresolution and Information Processing 13, no. 05 (September 2015): 1550036. http://dx.doi.org/10.1142/s0219691315500368.

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An explicit description of all Walsh polynomials generating tight wavelet frames is given. An algorithm for finding the corresponding wavelet functions is suggested, and a general form for all wavelet frames generated by an appropriate Walsh polynomial is described. Approximation properties of tight wavelet frames are also studied. In contrast to the real setting, it appeared that a wavelet tight frame decomposition has an arbitrary large approximation order whenever all wavelet functions are compactly supported.

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Zhang, Zhihua. "Approximation of Bivariate Functions via Smooth Extensions." Scientific World Journal 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/102062.

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For a smooth bivariate function defined on a general domain with arbitrary shape, it is difficult to do Fourier approximation or wavelet approximation. In order to solve these problems, in this paper, we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodic function in the whole space or to a smooth, compactly supported function in the whole space. These smooth extensions have simple and clear representations which are determined by this bivariate function and some polynomials. After that, we expand the smooth, periodic function into a Fourier series or a periodic wavelet series or we expand the smooth, compactly supported function into a wavelet series. Since our extensions are smooth, the obtained Fourier coefficients or wavelet coefficients decay very fast. Since our extension tools are polynomials, the moment theorem shows that a lot of wavelet coefficients vanish. From this, with the help of well-known approximation theorems, using our extension methods, the Fourier approximation and the wavelet approximation of the bivariate function on the general domain with small error are obtained.

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Capobianco, Enrico. "Computationally Efficient Atomic Representations for Nonstationary Stochastic Processes." International Journal of Wavelets, Multiresolution and Information Processing 01, no. 03 (September 2003): 325–51. http://dx.doi.org/10.1142/s0219691303000177.

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Function approximation methods based on frames or other overcomplete dictionaries of approximating functions offer advantages over the orthogonal schemes due to the fact that the associated redundancy may lead to better de-noising and reconstruction power. Wavelet packets represent special wavelet frames; they combine overcompleteness with high time-frequency localization power through an optimal frequency-then-time segmentation. Compared to cosine packets, which enable optimal adaptation through time-then-frequency segmentation, wavelet packets show a different time-frequency resolution trade-off that might be useful for analyzing some kinds of non-stationary phenomena. We study the properties of covariance non-stationary stochastic processes whose realizations are observed at very high frequencies; the data are supplied by time series of a stock market return index. For these complex processes the effectiveness of wavelet and cosine packets is explored by implementing entropic optimization, greedy approximation techniques and dimension reduction methods.

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36

Johnstone, Iain M., and Bernard W. Silverman. "Boundary coiflets for wavelet shrinkage in function estimation." Journal of Applied Probability 41, A (2004): 81–98. http://dx.doi.org/10.1017/s0021900200112227.

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There are standard modifications of certain compactly supported wavelets that yield orthonormal bases on a bounded interval. We extend one such construction to those wavelets, such as ‘coiflets', that may have fewer vanishing moments than had to be assumed previously. Our motivation lies in function estimation in statistics. We use these boundary-modified coiflets to show that the discrete wavelet transform of finite data from sampled regression models asymptotically provides a close approximation to the wavelet transform of the continuous Gaussian white noise model. In particular, estimation errors in the discrete setting of computational practice need not be essentially larger than those expected in the continuous setting of statistical theory.

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Johnstone, Iain M., and Bernard W. Silverman. "Boundary coiflets for wavelet shrinkage in function estimation." Journal of Applied Probability 41, A (2004): 81–98. http://dx.doi.org/10.1239/jap/1082552192.

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Abstract:

There are standard modifications of certain compactly supported wavelets that yield orthonormal bases on a bounded interval. We extend one such construction to those wavelets, such as ‘coiflets', that may have fewer vanishing moments than had to be assumed previously. Our motivation lies in function estimation in statistics. We use these boundary-modified coiflets to show that the discrete wavelet transform of finite data from sampled regression models asymptotically provides a close approximation to the wavelet transform of the continuous Gaussian white noise model. In particular, estimation errors in the discrete setting of computational practice need not be essentially larger than those expected in the continuous setting of statistical theory.

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Chung, Hai‐Man, and Don C. Lawton. "Amplitude responses of thin beds: Sinusoidal approximation versus Ricker approximation." GEOPHYSICS 60, no. 1 (January 1995): 223–30. http://dx.doi.org/10.1190/1.1443750.

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The amplitude response of a thin bed with arbitrary upper and lower normal incidence reflection coefficients is studied. Two analytical expressions for the normal incidence amplitude response as a function of the thickness are derived and are both valid for weak reflectivities and for thicknesses below [Formula: see text], where [Formula: see text] is the dominant wavelength. The first expression is based on the substitution of a cosine wave for the source wavelet, and the second is based directly on the analytical expression for a Ricker wavelet. The results calculated from these two expressions are compared to numerical modeling results for a Ricker wavelet for several models. We found that the differences between the two expressions are small, and both are good approximations. Above the [Formula: see text] thickness, the percentage differences increase rapidly for both expressions, implying that the thin‐bed assumptions in both derivations break down rapidly beyond the [Formula: see text] thickness. Below the [Formula: see text] thickness, except in the case where the two reflection coefficients are equal in magnitude but opposite in sign, the amplitude dependence on the thickness is nonlinear.

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39

Yang, Zhibo, Xuefeng Chen, Yumin He, Zhengjia He, and Jie Zhang. "The Analysis of Curved Beam Using B-Spline Wavelet on Interval Finite Element Method." Shock and Vibration 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/738162.

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A B-spline wavelet on interval (BSWI) finite element is developed for curved beams, and the static and free vibration behaviors of curved beam (arch) are investigated in this paper. Instead of the traditional polynomial interpolation, scaling functions at a certain scale have been adopted to form the shape functions and construct wavelet-based elements. Different from the process of the direct wavelet addition in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space by aid of the corresponding transformation matrix. Furthermore, compared with the commonly used Daubechies wavelet, BSWI has explicit expressions and excellent approximation properties, which guarantee satisfactory results. Numerical examples are performed to demonstrate the accuracy and efficiency with respect to previously published formulations for curved beams.

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40

BADIEZADEGAN, SHIRIN, and HAMID SOLTANIAN-ZADEH. "DESIGN AND EVALUATION OF MATCHED WAVELETS WITH MAXIMUM CODING GAIN AND MINIMUM APPROXIMATION ERROR CRITERIA FOR R PEAK DETECTION IN ECG." International Journal of Wavelets, Multiresolution and Information Processing 06, no. 06 (November 2008): 799–825. http://dx.doi.org/10.1142/s0219691308002690.

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Recently, several wavelet-based algorithms have been proposed for feature extraction in non-stationary signals such as ECG. These methods, however, have mainly used general purpose (unmatched) wavelet bases such as Daubechies and Quadratic Spline. In this paper, five new matched wavelet bases, with minimum approximation error and maximum coding gain criteria, are designed and applied to ECG signal analysis. To study the effect of using different wavelet bases for this application, two different wavelet-based R peak detection algorithms are implemented: (1) a conventional wavelet-based method; and (2) a modified wavelet-based R peak detection algorithm. Both algorithms are evaluated using the MIT-BIH Arrhythmia database. Experimental results show lower computational complexity (up to 76%) of the proposed R peak detection method compared to the conventional method. They also show considerable decrease in the number of failed detections (up to 55%) for both the conventional and the proposed algorithms when using matched wavelets instead of Quadratic Spline wavelet which, according to the literature, has generated the best detection results among all conventional wavelet bases studied previously for ECG signal analysis.

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41

Hong, Dawei, Shushuang Man, Jean-Camille Birget, and Desmond S. Lun. "A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm." Journal of Applied Probability 51, no. 01 (March 2014): 1–18. http://dx.doi.org/10.1017/s0021900200010044.

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We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1]of Hurst indexH∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM forH∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

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Damnjanovic, Djordje, Dejan Ciric, and Zoran Peric. "Wavelet-based audio features of DC motor sound." Facta universitatis - series: Electronics and Energetics 34, no. 1 (2021): 71–88. http://dx.doi.org/10.2298/fuee2101071d.

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The usage of wavelets is widespread in many fields nowadays, especially in signal processing. Their nature provides some advantages in comparison to the Fourier transform, and therefore many applications rely on wavelets rather than on other methods. The decomposition of wavelets into detail and approximation coefficients is one of the methods to extract representative audio features. They can be used in signal analysis and further classification. This paper investigates the usage of various wavelet families in the wavelet decomposition to extract audio features of direct current (DC) motor sounds recorded in the production environment. The purpose of feature representation and analysis is the detection of DC motor failures in motor production. The effects of applying different wavelet families and parameters in the decomposition process are studied using sounds of more than 60 motors. Time and frequency analysis is also done for the tested DC motor sounds.

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Yang, L. H., T. D. Bui, and C. Y. Suen. "Image Recognition Based on Nonlinear Wavelet Approximation." International Journal of Wavelets, Multiresolution and Information Processing 01, no. 02 (June 2003): 151–61. http://dx.doi.org/10.1142/s0219691303000104.

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This paper presents a novel approach to recognize images based on nonlinear wavelet approximation. The mathematical theory on nonlinear wavelet approximation is introduced, which shows that nonlinear approximation contains much more information of the original image than linear approximation. Based on this theory, a scheme to obtain the basic information of images with less data is provided. Experiments on face recognition produce effective matching rates.

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44

Hong, Dawei, Shushuang Man, Jean-Camille Birget, and Desmond S. Lun. "A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm." Journal of Applied Probability 51, no. 1 (March 2014): 1–18. http://dx.doi.org/10.1239/jap/1395771410.

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We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

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FENG, L., C. Y. SUEN, Y. Y. TANG, and L. H. YANG. "EDGE EXTRACTION OF IMAGES BY RECONSTRUCTION USING WAVELET DECOMPOSITION DETAILS AT DIFFERENT RESOLUTION LEVELS." International Journal of Pattern Recognition and Artificial Intelligence 14, no. 06 (September 2000): 779–93. http://dx.doi.org/10.1142/s0218001400000519.

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This paper describes a novel method for edge feature detection of document images based on wavelet decomposition and reconstruction. By applying the wavelet decomposition technique, a document image becomes a wavelet representation, i.e. the image is decomposed into a set of wavelet approximation coefficients and wavelet detail coefficients. Discarding wavelet approximation, the edge extraction is implemented by means of the wavelet reconstruction technique. In consideration of the mutual frequency, overlapping will occur between wavelet approximation and wavelet details, a multiresolution-edge extraction with respect to an iterative reconstruction procedure is developed to ameliorate the quality of the reconstructed edges in this case. A novel combination of this multiresolution-edge results in clear final edges of the document images. This multi-resolution reconstruction procedure follows a coarser-to-finer searching strategy. The edge feature extraction is accompanied by an energy distribution estimation from which the levels of wavelet decomposition are adaptively controlled. Compared with the scheme of wavelet transform, our method does not incur any redundant operation. Therefore, the computational time and the memory requirement are less than those in wavelet transform.

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Li, Hongmin, Yigang He, and Yichuang Sun. "A PSO Method for Wavelet Approximation." Information Technology Journal 13, no. 4 (February 1, 2014): 661–68. http://dx.doi.org/10.3923/itj.2014.661.668.

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Anastassiou, G. A., and X. M. Yu. "Convex and coconvex-probabilistic wavelet approximation." Stochastic Analysis and Applications 10, no. 5 (January 1992): 507–21. http://dx.doi.org/10.1080/07362999208809287.

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48

Unser, M. "Approximation power of biorthogonal wavelet expansions." IEEE Transactions on Signal Processing 44, no. 3 (March 1996): 519–27. http://dx.doi.org/10.1109/78.489025.

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Muraki, Shigeru. "Volume Data Approximation Using Wavelet Transforms." Journal of the Institute of Television Engineers of Japan 46, no. 12 (1992): 1635–42. http://dx.doi.org/10.3169/itej1978.46.1635.

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50

Romero, José, and Salvador Cerdá. "A first approximation to wavelet transform." Journal of the Acoustical Society of America 98, no. 5 (November 1995): 2968. http://dx.doi.org/10.1121/1.413970.

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