Multivariate wavelet frames

by Skopina, M. A., (Marii︠a︡ Aleksandrovna)

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Title Statements

Full title :Multivariate wavelet frames / by Maria Skopina, Aleksandr Krivoshein, Vladimir Protasov.

Summary

This book presents a systematic study of multivariate wavelet frames with matrix dilation, in particular, orthogonal and bi-orthogonal bases, which are a special case of frames. Further, it provides algorithmic methods for the construction of dual and tight wavelet frames with a desirable approximation order, namely compactly supported wavelet frames, which are commonly required by engineers. It particularly focuses on methods of constructing them. Wavelet bases and frames are actively used in numerous applications such as audio and graphic signal processing, compression and transmission of information. They are especially useful in image recovery from incomplete observed data due to the redundancy of frame systems. The construction of multivariate wavelet frames, especially bases, with desirable properties remains a challenging problem as although a general scheme of construction is well known, its practical implementation in the multidimensional setting is difficult. Another important feature of wavelet is symmetry. Different kinds of wavelet symmetry are required in various applications, since they preserve linear phase properties and also allow symmetric boundary conditions in wavelet algorithms, which normally deliver better performance. The authors discuss how to provide H-symmetry, where H is an arbitrary symmetry group, for wavelet bases and frames. The book also studies so-called frame-like wavelet systems, which preserve many important properties of frames and can often be used in their place, as well as their approximation properties. The matrix method of computing the regularity of refinable function from the univariate case is extended to multivariate refinement equations with arbitrary dilation matrices. This makes it possible to find the exact values of the Hölder exponent of refinable functions and to make a very refine analysis of their moduli of continuity.

Edition Statements

Publication Date:2017

Publication Statements

Publisher(s):Springer
Place of Publication:Singapore
Language:eng
Publication Date:2017

The Physical Object

Description:1 online resource

Edition Identifiers

ISBN:9789811032059, 981103205X
oclc:ocn971492253
LC Classification:QA403.3
Dewey Decimal:515/.2433

Preface; Contents; About the Authors; Basic Notation; 1 Bases and Frames in Hilbert Spaces; 1.1 Riesz Bases; 1.2 Frames ; 2 MRA-Based Wavelet Bases and Frames; 2.1 Dilation Matrix; 2.2 Shift-Invariant Systems and Bracket Product; 2.3 Multiresolution Analysis; 2.4 MRA-Based Wavelets and Matrix Extension Principle; 2.5 Matrix Extension Problem; 2.6 Refinable Functions; 2.7 Conditions of Biorthogonality; References; 3 Construction of Wavelet Frames Generated by MEP; 3.1 Sufficient Frame Conditions; 3.2 Approximation Order and Vanishing Moments; 3.3 Polyphase Characterization of Vanishing Moments

3.4 Construction of Compactly Supported Frames3.5 Examples; References; 4 Frame-Like Wavelet Expansions; 4.1 Dual Frame-Like Systems; 4.2 Scaling Expansions; 4.3 Wavelet Expansions; 4.4 Examples; References; 5 Symmetric Wavelets; 5.1 Symmetric Refinable Masks; 5.2 Interpolatory Symmetric Wavelets; 5.3 Symmetric Frame-Like Wavelets and Dual Wavelet Frames; 5.4 Examples; References; 6 Smoothness of Wavelets; 6.1 Tiles, Self-similar Tilings, and Supports of Refinable Functions; 6.2 The Matrix Method of Computing the Hölder Regularity; 6.3 Special Cases and Examples

6.4 Construction of the Cont…