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
Fractional Gaussian noise, functional MRI and Alzheimer's disease.
Voichita Maxim,Levent Sendur,Jalal M. Fadili,John Suckling,Rebecca L. Gould,Robert Howard,Edward T. Bullmore +6 more
TL;DR: Parsimonious mapping of fMRI noise properties in terms of fGn parameters efficiently estimated in the wavelet domain is feasible and can enhance insight into the pathophysiology of Alzheimer's disease.
Journal ArticleDOI
Detection of microcalcifications in digital mammograms using wavelets
T.C. Wang,N.B. Karayiannis +1 more
TL;DR: Preliminary experiments indicate that further studies are needed to investigate the potential of wavelet-based subband image decomposition as a tool for detecting microcalcifications in digital mammograms.
Journal ArticleDOI
Multilevel preconditioning
Wolfgang Dahmen,Angela Kunoth +1 more
TL;DR: In this article, a multilevel technique for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems is presented. But the results are restricted to the setting of refinable shift-invariant spaces, in particular those induced by wavelets.
Journal ArticleDOI
WaveCluster: a wavelet-based clustering approach for spatial data in very large databases
TL;DR: WaveCluster is proposed, a novel clustering approach based on wavelet transforms, which satisfies all the above requirements and can effectively identify arbitrarily shaped clusters at different degrees of detail.
Journal ArticleDOI
Wavelet Methods in Computational Fluid Dynamics
Kai Schneider,Oleg V. Vasilyev +1 more
TL;DR: In this paper, state-of-the-art adaptive, multiresolution wavelet methodologies for modeling and simulation of turbulent flows with various examples are reviewed, and different numerical methods for solving the Navier-Stokes equations in adaptive wavelet bases are 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.