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
Compactly supported box-spline wavelets
TL;DR: In this article, a general procedure for constructing multivariate non-tensor product wavelets that generate an orthogonal decomposition of L 2(R)s,s s ≥ 1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis of any box spline whose direction set constitutes a unimodular matrix.
Journal ArticleDOI
Multiresolution analysis in extraction of reference lines from documents with gray level background
TL;DR: Experiments indicate that this new method based on wavelets can be applied to process documents with promising results, and is more efficient to process form documents with gray level background.
Proceedings ArticleDOI
GRADE: Gibbs reaction and diffusion equations
Song-Chun Zhu,David Mumford +1 more
TL;DR: This paper proposes a general statistical framework for designing a new class of PDEs the Gibbs Reaction And Diffusion Equations-GRADE and demonstrates experiments where GRADE are used for texture pattern formation, denoising, image enhancement, and clutter removal.
Journal ArticleDOI
The development of the quaternion wavelet transform
P. Fletcher,Stephen J. Sangwine +1 more
TL;DR: There is no consensus about what quaternion wavelet transforms should look like and what their properties should be, so this article reviews what has been written and concludes with an analysis of what it is that should define a QWT as being truly quaternionic.
Journal ArticleDOI
Using wavelets for solving PDEs: an adaptive collocation method
TL;DR: In this paper, the authors introduce the chemical engineering community to the basic features of wavelet-based adaptive collocation methods for solving PDEs and give an illustration of its powerful capabilities.
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.