The discrete wavelet transform is now extremely well-known
and is described in numerous references.
In Dirac it plays the same role of the DCT in MPEG-2 in
decorrelating data in a roughly frequency-sensitive way, whilst
having the advantage of preserving fine details better. In one
dimension it consists of the iterated application of a
complementary pair of half-band filters followed by subsampling
by a factor 2:
Figure: Perfect reconstruction analysis and synthesis
filter pairs
These filters are termed the analysis filters. Corresponding
synthesis filters can undo the aliasing introduced by the
critical sampling and perfectly reconstruct the input. Clearly
not just any pair of half-band filters can do this, and there
is an extensive mathematical theory of wavelet filter banks.
The filters split the signal into a LH and HF part; the
wavelet transform then iteratively decomposes the LF component
to produce an octave-band decomposition of the signal.
Applied to two-dimensional images, wavelet filters are
normally applied in both vertical and horizontal directions to
each image component to produce four so-called subbands
termed Low-Low (LL), Low-High
(LH), High-Low (HL) and High-High (HH). In the case of two
dimensions, only the LL band is iteratively decomposed to
obtain the decomposition of the two-dimensional spectrum shown
below:
Figure: wavelet transform frequency
decomposition
The number of samples in each resulting subband is as
implied by the diagram: the critical sampling ensures that after
each decomposition the resulting bands all have one quarter of
the samples of the input signal.
Figure: 3-level wavelet transform of LENA
Wavelet filters
The choice of wavelet filters has an impact on compression
performance, filters having to have both compact impulse
response in order to reduce ringing artefacts and other
properties in order to represent smooth areas compactly. It also has an
impact on encoding and decoding speed in software.
There are numerous filters supported by Dirac to allow a trade-off
between complexity and performance. These are configurable in
the reference software.
These filters
are all defined using the 'lifting scheme' for speed.
The lifting stages are defined as follows. One fiter available in Dirac
is a lifting approximation of the Daubechies (9,7)
wavelet: for this we have (s denoting sum and d denoting difference),
The magic numbers are integer approximations of the Daubechies lifting ceofficients. This makes the transform fully
invertible, where a floating point implementation wouldn't quite be. The implementation ignores scaling
coefficients, since these can be taken into account in quantiser selection by weighting quantiser noise
appropriately. The problem with this filter is that it has four lifting stages, and so it takes longer
in software. At the other extreme, there is the (5,3) filter:
We can improve the (5,3) high pass filter, and get an approximating of the Daubechies
low-pass filter by having more taps in the first lifing stage. This is the Deslauriers-Dubuc
(9,7) filter:
The Deslauriers-Dubuc (13,7) extends this further to provide better frequency selectivity in
the low pass filter, still only using two lifting stages:
Padding and invertibility
Clearly, applying an N-level wavelet transform requires N levels of
subsamplings, and so for reversibility, it is necessary that 2N
divides all the dimensions of each component.
So if this condition is not
met, the input picture components are padded as they are read in, by edge values
for best compression performance.
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