Wavelets - Caltech Multi-Res Modeling Group

MULTIRESOLUTION AND WAVELETS 39
original. The normalization conditions for the filters imply that the filter (hi) constitutes a lowpass filter
which smooths data and the filter (gi) a highpass filter which picks out the detail; this difference in filter
roles can also be seen in the examples in the previous section.
Reconstruction is performed in the opposite direction using the adjoint filtering operation:
xi =
In matrix form: x = H T y + G T z,whereH T is the transpose of H:
X
0
B
@
k
hi,2kyk + gi,2kzk: (2)
h0 ::: :::
h1 0 0
h2 h0 0
h3 h1 0
0 h2 h0
0 h3 h1
0 0 h2
:::
The reconstruction step filters and upsamples: the upsampling produces a sequence twice as long as the
sequences started with.
Note: Some of the filter requirements often show up in slightly different forms in the literature. For instance,
the normalization to p 2 is a convention stemming from the derivation of the filters, but it is also common
to normalize the sum of the filter elements to equal 1. Similarly, the wavelet filter definition can appear with
different indices: the filter elements can for instance be shifted by an even number of steps. The differences
due to these changes are minor.
Similarly, decomposition filtering is often defined using the following convention:
y 0 i =
X
k
h2i,kxk; z 0 i =
1
C
A
X
k
g2i,kxk: (3)
The only difference is that the filter is applied “backwards” in the scheme (3), conforming to the usual
convolution notation, and forwards in (1). Again, there is no real difference between the definitions. We
choose the “forward” one only because it agrees notationally with the standard definition of wavelets via
dilations and translations. If decomposition is performed as in (3), the reconstruction operation (2) should
be replaced by:
xi =
X
k
hky 0 i+k
2
Siggraph ’95 Course Notes: #26 Wavelets
+ gkz 0 i+k : (4)
2