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Dilation equations

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

$\begin{displaymath}\varphi(x) = \sum_k c_k \varphi(2x-k), \end{displaymath}$

where the c_k 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.
        
        See also WAVDEMO, WAVELETD.

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.

**Figure 1:** `wavdemo`

Harri Ojanen
1998-05-02