# Wavelet transforms and filtering

This simple example illustrates how to compute wavelet transforms and perform filtering and mesh adaptation. We do this in one dimension, using bi-trees.

#include "grid/bitree.h"

A simple function to output fields on each level.

void write_level (scalar * list, const char * tag, FILE * fp)
{
for (int l = 0; l <= depth(); l++)
foreach_level (l, serial) {
fprintf (fp, "%s%d %g ", tag, l, x);
for (scalar s in list)
fprintf (fp, "%g ", s[]);
fputc ('\n', fp);
}
}

int main()
{

We consider a periodic 1D domain.

  init_grid (128);
periodic (right);
size (2);

scalar s[], w[], s2[];

We can optionally change the prolongation operator. We need to make sure that all fields use the same prolongation.

#if 0
for (scalar i in {s,w,s2})
#endif

We initialise the field with a function containing low-frequency and localised high-frequency components.

  foreach()
s[] = sin(2.*pi*x) + 0.4*sin(15*pi*x)*max(sin(2.*pi*x), 0);

The w field contains the wavelet transform of s.

  wavelet (s, w);

We check that the inverse wavelet transform recovers the initial signal (almost) exactly.

  inverse_wavelet (s2, w);
foreach()
assert (fabs (s2[] - s[]) < 1e-12);
write_level ({s,w}, "", stderr);

## The original signal and its wavelet transform.

maxlevel = 6
spacing = 1.5
set ytics -maxlevel,1,0
set xlabel 'x'
set ylabel 'Level'
set grid ytics ls 0
max(a,b) = a > b ? a : b
set samples 500
f(x)=sin(2.*pi*x)+0.4*sin(15.*pi*x)*max(sin(2.*pi*x), 0);
plot for [i=0:maxlevel] '< grep ^'.i.' log' u 2:($3/spacing - i) \ w lp t '' lt 7, \ for [i=0:maxlevel] f(x)/spacing - i t '' lt 1 plot for [i=0:maxlevel] '< grep ^'.i.' log' u 2:($4/spacing - i) \
w lp t '' lt 7,			   \
for [i=0:maxlevel] f(x)/spacing - i t '' lt 1

## Localised low-pass filtering

We want to filter the high-frequency components of the signal, but only locally i.e. for x > 1. To do this we cancel the high-frequency (i.e. high-level) wavelet coefficients and recover the corresponding filtered signal s2 with the inverse wavelet transform.

  for (int l = 5; l <= 6; l++) {
foreach_level (l)
w[] *= (x <= 1);
boundary_level ({w}, l);
}
inverse_wavelet (s2, w);
write_level ({s2,w}, "a", stdout);
plot for [i=0:maxlevel] '< grep ^a'.i.' out' u 2:($3/spacing - i) \ w lp t '' lt 7, \ for [i=0:maxlevel] f(x)/spacing - i t '' lt 1 ## High-pass filtering Here we filter out the low-frequency components everywhere.  wavelet (s, w); for (int l = 0; l < 5; l++) { foreach_level (l) w[] = 0.; boundary_level ({w}, l); } inverse_wavelet (s2, w); write_level ({s2,w}, "b", stdout); plot for [i=0:maxlevel] '< grep ^b'.i.' out' u 2:($3/spacing - i) \
w lp t '' lt 7,			   \
for [i=0:maxlevel] f(x)/spacing - i t '' lt 1

We can use the wavelet coefficients to decide where to coarsen the mesh.

  wavelet (s, w);
unrefine (fabs(w[]) < 0.1);
write_level ({s,w}, "c", stdout);
}
plot for [i=0:maxlevel-1] '< grep ^c'.i.' out' u 2:($3/spacing - i) \ w lp t '' lt 7, \ '< grep ^c6 out' u 2:($3/spacing - 6) w p t '' lt 7,		\
for [i=0:maxlevel] f(x)/spacing - i t '' lt 1

## Discussion

• Note that the restriction operators assume that the discrete quantity is the average of the function over the cell, not its point value. While this makes no difference for second-order (i.e. linear) prolongation operators, it matters for higher-order prolongation but is not taken into account in this example.
• Boundary conditions for the wavelet transform are important and non-trivial. This is not an issue here since we use a periodic domain.
• The link between the transform implemented here and formal wavelets is not entirely clear. It is probably a kind of second-generation wavelet.
• The filters applied in the example could be formulated in a more generic way using a filtering function \phi(x,l) and a code looking like:
  foreach_cell()
w[] *= phi(x,level);

combined with proper application of boundary conditions.

## References

 [sweldens1998] Wim Sweldens. The lifting scheme: A construction of second generation wavelets. SIAM journal on mathematical analysis, 29(2):511–546, 1998.