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M. Griebel

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Univariate Wavelets

We start with some basic notation:

• $\phi^{l,0},...,\phi^{l,2^l}$ are the scaling functions of refinement level $l\ge L_0$. $L_0$ is a certain coarsest level of refinement. The larger the support of the scaling functions, the larger is $L_0$. For example, $L_0=1$ for Interpolets of order 2 and $L_0=3$ for Interpolets of order 4. The number of scaling functions for level $l$ is $2^l+1$, as we use dyadic refinement only.
• $\psi_{l,t}$ are the wavelets of level $l$. The translation index $t$ runs from $0,...,2^{l-1}-1$ for $l>L_0$ and from $0,...,2^{L_0}$ if $l=L_0$.

Up to now the following univariate wavelets are supported:

• Interpolets of order 2, 4 and 6. For example, Interpolets of order 2 are the piecewise linear hat functions.
• orthogonal Daubechies wavelets of order 1 .. 4 and 6. Daubechies wavelets of order 1 are the Haar wavelets.

Filter masks for general Interpolets/Daubechies wavelets can be generated using the MATLAB functions b10write{I}coeffFile. These scripts generate two files AWFD/MATLAB/Data/coeffL.dat AWFD/MATLAB/Data/coeffR.dat which must be concated to either AWFD/MATLAB/Data/Icoeff.dat (Interpolets) or AWFD/MATLAB/Data/coeff.dat (Daubechies).

Since most of the algorithms work for general biorthogonal wavelets it is straightforward to include further wavelet families. To do so, one just has to feed the coefficients for the low and high pass filters and all other required operators in struct Wavelets. By now, we provide member functions which read the coefficients for the desired wavelets from the files MATLAB/Data/Icoeff.dat or MATLAB/Data/coeff.dat.

Example: The following program reads the filter coefficients for 4th order Interpolets and dumps them on the screen.

#include "Wavelet.hpp"
int debugRefine ;
int main() {
    Wavelets WC("Interpolet",4) ;
    WC.Print(1) ;
}