Gauss–Manin connection: Difference between revisions

Latest revision as of 14:45, 20 April 2013

A biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal. Designing biorthogonal wavelets allows more degrees of freedom than orthogonal wavelets. One additional degree of freedom is the possibility to construct symmetric wavelet functions.

In the biorthogonal case, there are two scaling functions $ϕ, \tilde{ϕ}$ , which may generate different multiresolution analyses, and accordingly two different wavelet functions $ψ, \tilde{ψ}$ . So the numbers M and N of coefficients in the scaling sequences $a, \tilde{a}$ may differ. The scaling sequences must satisfy the following biorthogonality condition

\sum_{n \in ℤ} a_{n} {\tilde{a}}_{n + 2 m} = 2 \cdot δ_{m, 0}

.

Then the wavelet sequences can be determined as
$b_{n} = (- 1)^{n} {\tilde{a}}_{M - 1 - n} (n = 0, \dots, N - 1)$
${\tilde{b}}_{n} = (- 1)^{n} a_{M - 1 - n} (n = 0, \dots, N - 1)$ .

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+A '''biorthogonal wavelet''' is a [[wavelet]] where the associated [[Discrete wavelet transform|wavelet transform]] is [[invertible]] but not necessarily [[Orthogonality|orthogonal]]. Designing biorthogonal wavelets allows more degrees of freedom than [[orthogonal wavelet]]s. One additional degree of freedom is the possibility to construct symmetric wavelet functions.
+In the biorthogonal case, there are two [[Wavelet#Scaling_function|scaling functions]] <math>\phi,\tilde\phi</math>, which may generate different multiresolution analyses, and accordingly two different wavelet functions <math>\psi,\tilde\psi</math>. So the numbers ''M'' and ''N'' of coefficients in the scaling sequences <math>a,\tilde a</math> may differ. The scaling sequences must satisfy the following biorthogonality condition
+:<math>\sum_{n\in\Z} a_n \tilde a_{n+2m}=2\cdot\delta_{m,0}</math>.
+Then the wavelet sequences can be determined as <br>
+<math>b_n=(-1)^n \tilde a_{M-1-n} \quad \quad (n=0,\dots,N-1) </math><br>
+<math>\tilde b_n=(-1)^n a_{M-1-n} \quad \quad (n=0,\dots,N-1) </math>.<br>
+==References==
+* {{cite book| author = Stéphane G. Mallat| title = A Wavelet Tour of Signal Processing| year = 1999| publisher = Academic Press| isbn = 978-0-12-466606-1 }}
+[[Category:Biorthogonal wavelets| ]]

Gauss–Manin connection: Difference between revisions

Latest revision as of 14:45, 20 April 2013

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