# Wavelets and Dilation Equations: A Brief Introduction

@article{Strang1989WaveletsAD,
title={Wavelets and Dilation Equations: A Brief Introduction},
author={Gilbert Strang},
journal={SIAM Rev.},
year={1989},
volume={31},
pages={614-627}
}
• G. Strang
• Published 1 December 1989
• Computer Science
• SIAM Rev.
Wavelets are new families of basis functions that yield the representation $f(x) = \sum {b_{jk} W(2^j x - k)}$. Their construction begins with the solution $\phi (x)$ to a dilation equation with coefficients $c_k$. Then W comes from $\phi$, and the basis comes by translation and dilation of W. It is shown in Part 1 how conditions on the $c_k$ lead to approximation properties and orthogonality properties of the wavelets. Part 2 describes the recursive algorithms (also based on the $c_k… 1,091 Citations The Construction and Application of Wavelets in Numerical Analysis This thesis investigates the use of wavelets in numerical analysis problems by constructing and studying wavelets adapted to a weighted inner product, and showing how one can use those wavelets for the rapid solution of ordinary di erential equations. Wavelet transforms versus Fourier transforms This note is a very basic introduction to wavelets, starting with an orthogonal basis of piecewise constant functions, constructed by dilation and translation, and leading to dilation equations and their unusual solutions. Methods of solving dilation equations A wavelet basis is an orthonormal basis for L2(ℜ), the space of squareintegrable functions on the real line, of the form {gnkn,k∈z, where gnk (t) = 2n/2 g (2nt−k) and g is a single fixed function, Multiresolution and wavelets • Mathematics • 1994 Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspace S of L 2 (ℝ d ), let S k be the 2 k -dilate of S ( k ∈ℤ). A necessary and Asymptotic error expansion of wavelet approximations of smooth functions II • Mathematics • 1994 Summary.We generalize earlier results concerning an asymptotic error expansion of wavelet approximations. The properties of the monowavelets, which are the building blocks for the error expansion, Introduction to Wavelets and Wavelet Transforms: A Primer • Computer Science • 1997 This work describes the development of the Basic Multiresolution Wavelet System and some of its components, as well as some of the techniques used to design and implement these systems. Compactly Supported Wavelets and Their Generalizations: An Algebraic Approach In a classical sense, a wavelet basis is an orthonormal basis formed by translates of dyadic dilates of a single function. Usually, such a wavelet basis is associated with a multiresolution analysis, Harmonic wavelet analysis • D. Newland • Mathematics Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences • 1993 A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure ω(x) = Dilation equations with exponential decay coefficients Dilation equations with finitely many nonzero coefficients have been discussed by many people, since those equations are related to compactly supported wavelets, which constitute an important family Fractional Splines and Wavelets • Mathematics SIAM Rev. • 2000 The symmetric version of the B-splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative, and may be used to build new families of wavelet bases with a continuously varying order parameter. ## References SHOWING 1-10 OF 20 REFERENCES Continuous and Discrete Wavelet Transforms • Mathematics SIAM Rev. • 1989 This paper is an expository survey of results on integral representations and discrete sum expansions of functions in$L^2 ({\bf R})\$ in terms of coherent states, focusing on Weyl–Heisenberg coherent states and affine coherent states.
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It is proved that a multiresolution approximation is characterized by a 2π periodic function which is further described and provides a new approach for understanding and computing wavelet orthonormal bases.
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• Computer Science
IEEE Trans. Pattern Anal. Mach. Intell.
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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/ 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.
DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE
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An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant
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• Computer Science
IEEE Trans. Acoust. Speech Signal Process.
• 1989
The author describes the mathematical properties of such decompositions and introduces the wavelet transform, which relates to the decomposition of an image into a wavelet orthonormal basis.
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The theory of edge detection explains several basic psychophysical findings, and the operation of forming oriented zero-crossing segments from the output of centre-surround ∇2G filters acting on the image forms the basis for a physiological model of simple cells.
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This work focuses on orthonormal bases of wavelets, in particular bases ofwavelets with finite support, and defines wavelets and the wavelet transform.
Orthonormal bases of compactly supported wavelets
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.
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• 1989
The definitive, authoritative text on DSP, written by prominent, DSP pioneers, it provides thorough treatment of the fundamental theorems and properties of discrete- time linear systems, filtering, sampling, and discrete-time Fourier Analysis.