Book ChapterDOI
Using the refinement equations for the construction of Pre-Wavelets II: powers and two
Rong-Qing Jia,Charles A. Micchelli +1 more
- pp 209-246
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TLDR
The notion of extensibility of a finite set of Laurent polynomials is shown to be central in the construction of wavelet decompositions by decomposition of spaces in a multiresolution analysis.Abstract:
We study basic questions of wavelet decompositions associated with multiresolution analysis. A rather complete analysis of multiresolution associated with the solution of a refinement equation is presented. The notion of extensibility of a finite set of Laurent polynomials is shown to be central in the construction of wavelets by decomposition of spaces. Two examples of extensibility, first over the torus and then in complex space minus the coordinate axes are discussed. In each case we are led to a decomposition of the fine space in a multiresolution analysis as a sum of the adjacent coarse space plus an additional space spanned by the multiinteger translates of a finite number of pre-wavelets. Several examples are provided throughout to illustrate the general theory.read more
Citations
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The lifting scheme: A construction of second generation wavelets
TL;DR: The lifting scheme is presented, a simple construction of second generation wavelets; these are wavelets that are not necessarily translates and dilates of one fixed function, and can be adapted to intervals, domains, surfaces, weights, and irregular samples.
Journal ArticleDOI
The lifting scheme: a construction of second generation wavelets
TL;DR: The lifting wavelet as discussed by the authors is a simple construction of second generation wavelets that can be adapted to intervals, domains, surfaces, weights, and irregular samples, and it leads to a faster, in-place calculation of the wavelet transform.
Journal ArticleDOI
Multiresolution analysis for surfaces of arbitrary topological type
TL;DR: Whereas previous two-dimensional methods were restricted to functions difined on R2, the subdivision wavelets developed here may be applied to functions defined on compact surfaces of arbitrary topological type.
Journal ArticleDOI
Framelets: MRA-based constructions of wavelet frames☆☆☆
TL;DR: Wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames, are discussed and it is shown how they can be used for systematic constructions of spline, pseudo-spline tight frames, and symmetric bi-frames with short supports and high approximation orders.
Journal ArticleDOI
Affine Systems in L2(Rd): The Analysis of the Analysis Operator.
Amos Ron,Zuowei Shen +1 more
TL;DR: In this paper, the affine product and quasi-affine system were introduced to characterize the structure of affine systems, and sufficient conditions for constructing tight affine frames from multiresolution were given.
References
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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
Multiresolution approximations and wavelet orthonormal bases of L^2(R)
TL;DR: 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.
Book
An Introduction to Homological Algebra
TL;DR: An Introduction to Homological Algebra as discussed by the authors discusses the origins of algebraic topology and presents the study of homological algebra as a two-stage affair: first, one must learn the language of Ext and Tor and what it describes.