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Dilation equations

A basic property of wavelets is that the scaling function (father wavelet) satisfies a dilation equation (also called a two-scale difference equation) of the form

where the ck are (up to a constant multiple), the filter coefficients h of the wavelet. The function dilation solves these equations numerically and can be used to graph scaling functions and wavelets (once a solution for the dilation equation has been obtained the function waveletd computes the values of the corresponding wavelet).

        dilation -- solution to a dilation equation

f = dilation(c,levels)

Inputs:
c        Coefficients from the dilation equation (automatically
normalized).
levels   How deep to iterate, result will be calculated on points
2^(-levels) apart.

Output:
f       The solution.

Note! This program first solves the exact values of f on
integers. This means solving an eigenvalue problem, which sometimes
fails. For discontinuous solutions, you must supply the initial data
explicitely, see below.

Optional arguments:
[f,x] = dilation(c,levels,initf)
initf   Iteration is started with this vector as initial data; in
this case 'levels' gives how many new levels to calculate.
x       Points at which f is given.



Example:

        [h,g] = wavecoef('dau',8);
[f,x] = dilation(h,8);
plot(x,f, x,waveletd(f,x,g), '--');


The algorithm fails for some dilation equations (e.g., when the solution is not continuous). The Vaidyanathan wavelet is such an example.

See also wavdemo, which provides a graphical interface for these routines. Figure 1 shows an example.

Harri Ojanen
1998-05-02