# Wavelet Frames

To cover the time frequency plane with
wavelet Heisenberg boxes, a regular grid is not used; time steps
which are inverse proportional to the frequecy step are used instead,
the latter being itself proportional to the scale.

The wavelet is assumed to satisfy the
reconstruction condition

which garantees the invertibility of the wavelet
transform. Daubechies gives necessary conditions for the previous
tiling to yield a frame:

Sufficient conditions also exist.

The following differences with the windowed
Fourier frames should be emphasized:

- there are continuously differentable wavelets
that generate frames (look at the construction of wavelet bases for more)
- in the general case, the dual frame of a
wavelet frame is not a wavelet frame. However, in the cases of
bases, a dual wavelet basis can be built by other means (look at
the wavelet bases for more,
especially the biorthogonal ones)

##

## Translation invariance

In both cases (Fourier or wavelets), the frame
representation has the drawback of **not
being translation invariant** with respect
to time or frequency. Now, most interesting signal patterns are not
naturally synchronized with frame intervals. In particular, the
structure of a signal may be degraded at the lower
resolutions.

This motivates the study of the dyadic wavelet transform,
which is discrete in scale but not in time (in practice, this means
that signals are oversampled when switching to coarser
resolutions).

Another time invariant represenation is the
representation by dyadic wavelet maxima. It
is less redundant, but is not complete.

## Why wavelet bases are studied nonetheless

In practice, the dyadic wavelet transform is
implemented by perfect reconstruction filter banks. These fast filter banks correspond to wavelet bases which
are built from **multiresolution
approximations**.

Multiresolution Approximations and Wavelet
Bases