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| An '''orthogonal wavelet''' is a [[wavelet]] whose associated [[Discrete wavelet transform|wavelet transform]] is [[Orthogonality|orthogonal]].
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| That is, the inverse wavelet transform is the [[Adjoint of an operator|adjoint]] of the wavelet transform.
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| If this condition is weakened you may end up with [[biorthogonal wavelet]]s.
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| == Basics ==
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| The [[Wavelet#Scaling_function|scaling function]] is a [[refinable function]].
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| That is, it is a [[fractal]] functional equation, called the '''refinement equation''' ('''twin-scale relation''' or '''dilation equation'''):
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| :<math>\phi(x)=\sum_{k=0}^{N-1} a_k\phi(2x-k)</math>, | |
| where the sequence <math>(a_0,\dots, a_{N-1})</math> of [[real number]]s is called a scaling sequence or scaling mask.
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| The wavelet proper is obtained by a similar linear combination,
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| :<math>\psi(x)=\sum_{k=0}^{M-1} b_k\phi(2x-k)</math>,
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| where the sequence <math>(b_0,\dots, b_{M-1})</math> of real numbers is called a wavelet sequence or wavelet mask.
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| A necessary condition for the ''orthogonality'' of the wavelets is that the scaling sequence is orthogonal to any shifts of it by an even number of coefficients:
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| :<math>\sum_{n\in\Z} a_n a_{n+2m}=2\delta_{m,0}</math> | |
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| In this case there is the same number ''M=N'' of coefficients in the scaling as in the wavelet sequence, the wavelet sequence can be determined as <math>b_n=(-1)^n a_{N-1-n}</math>. In some cases the opposite sign is chosen.
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| == Vanishing moments, polynomial approximation and smoothness ==
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| A necessary condition for the existence of a solution to the refinement equation is that some power ''(1+Z)<sup>A</sup>'', ''A>0'', divides the polynomial <math>a(Z):=a_0+a_1Z+\dots+a_{N-1}Z^{N-1}</math> (see [[Z-transform]]). The maximally possible power ''A'' is called '''polynomial approximation order''' (or pol. app. power) or '''number of vanishing moments'''. It describes the ability to represent polynomials up to degree ''A-1'' with linear combinations of integer translates of the scaling function.
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| In the biorthogonal case, an approximation order ''A'' of <math>\phi</math> corresponds to ''A'' '''vanishing moments''' of the dual wavelet <math>\tilde\psi</math>, that is, the [[dot product|scalar products]] of <math>\tilde\psi</math> with any polynomial up to degree ''A-1'' are zero. In the opposite direction, the approximation order ''Ã'' of <math>\tilde\phi</math> is equivalent to ''Ã'' vanishing moments of <math>\psi</math>. In the orthogonal case, ''A'' and ''Ã'' coincide.
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| A sufficient condition for the existence of a scaling function is the following: if one decomposes <math>a(Z)=2^{1-A}(1+Z)^Ap(Z)</math>, and the estimate
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| :<math>1\le\sup_{t\in[0,2\pi]}|p(e^{it})|<2^{A-1-n}</math> for some <math>n\in\N</math>, | |
| holds, then the refinement equation has a ''n'' times continuously differentiable solution with compact support.
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| Examples:
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| *<math>a(Z)=2^{1-A}(1+Z)^A</math>, that is ''p(Z)=1'', has ''n=A-2''. The solutions are Schoenbergs [[B-spline]]s of order ''A-1'', where the ''(A-1)''-th derivative is piecewise constant, thus the ''(A-2)''-th derivative is [[Lipschitz continuity|Lipschitz-continuous]]. ''A=1'' corresponds to the index function of the unit interval.
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| *''A=2'' and ''p'' linear may be written as <math>a(Z)=\frac14(1+Z)^2\,((1+Z)+c(1-Z))</math>. Expansion of this degree 3 polynomial and insertion of the 4 coefficients into the orthogonality condition results in ''c²=3''. The positive root gives the scaling sequence of the D4-wavelet, see below.
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| ==References==
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| * [[Ingrid Daubechies]]: ''Ten Lectures on Wavelets'', SIAM 1992,
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| [[Category:Orthogonal wavelets|*]]
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