Multiresolution approximations and wavelet orthonormal bases of L^2(R)
TLDR
In this paper, the authors study the properties of multiresolution approximation and prove that it is characterized by a 2π periodic function, which is further described in terms of wavelet orthonormal bases.Abstract:
A multiresolution approximation is a sequence of embedded vector spaces V j jmember Z for approximating L 2 (R) functions. We study the properties of a multiresolution approximation and prove that it is characterized by a 2π periodic function which is further described. From any multiresolution approximation, we can derive a function ψ(x) called a wavelet such that √ 2 j ψ(2 j x −k) (k ,j)member Z 2 is an orthonormal basis of L 2 (R). This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space H s .read more
Citations
More filters
Journal ArticleDOI
Statistical Process Monitoring of Nonlinear Profiles Using Wavelets
TL;DR: The complex data records collected by many modern industrial processes do not readily allow the use of traditional statistical process control techniques as discussed by the authors, and therefore, traditional statistical data structures do not work well in industrial processes.
Journal ArticleDOI
Wavenet ability assessment in comparison to ANN for predicting the maximum surface settlement caused by tunneling
TL;DR: An alternative method of maximum ground surface settlement prediction, which is based on integration between wavelet theory and Artificial Neural Network (ANN), or wavelet network (wavenet), is presented and demonstrates its ability to enhance the function approximation capability and consequently exhibits excellent learning ability.
Journal ArticleDOI
Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering
G. Hariharan,K. Kannan +1 more
TL;DR: It is shown that the wavelet method is efficient and powerful in solving wide class of linear and nonlinear reaction–diffusion equations and future scope and directions involved in developing wavelet algorithm for solving reaction– Diffusion equations are addressed.
Journal ArticleDOI
Wavelet thresholding for some classes of non–Gaussian noise
TL;DR: This paper proposes a framework in which the notion of sparseness can be naturally expressed by a Bayesian model for the wavelet coefficients of the underlying signal and establishes close connections between wavelet thresholding techniques and Maximum A Posteriori estimation for two classes of noise distributions including heavy–tailed noises.
Journal ArticleDOI
Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems
Helen C. Yee,Björn Sjögreen +1 more
TL;DR: In this paper, a class of low dissipative high order (fourth-order or higher) filter schemes for multiscale Navier-Stokes, and ideal and non-ideal magnetohydrodynamics (MHD) systems is described.
References
More filters
Journal ArticleDOI
A theory for multiresolution signal decomposition: the wavelet representation
TL;DR: In this paper, it is shown that the difference of information between the approximation of a signal at the resolutions 2/sup j+1/ and 2 /sup j/ (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L/sup 2/(R/sup n/), the vector space of measurable, square-integrable n-dimensional functions.
Journal ArticleDOI
Orthonormal bases of compactly supported wavelets
TL;DR: This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.
Journal ArticleDOI
Decomposition of Hardy functions into square integrable wavelets of constant shape
A. Grossmann,J. Morlet +1 more
TL;DR: In this article, the authors studied square integrable coefficients of an irreducible representation of the non-unimodular $ax + b$-group and obtained explicit expressions in the case of a particular analyzing family that plays a role analogous to coherent states (Gabor wavelets) in the usual $L_2 $ -theory.
Journal ArticleDOI
Exact reconstruction techniques for tree-structured subband coders
TL;DR: It is shown that it is possible to design tree-structured analysis/reconstruction systems which meet the sampling rate condition and which result in exact reconstruction of the input signal.
Journal ArticleDOI
Analysis of sound patterns through wavelet transforms
TL;DR: The main features of so-called wavelet transforms are illustrated through simple mathematical examples and the first applications of the method to the recognition and visualisation of characteristic features of speech and of musical sounds are presented.