Wavelets - Caltech Multi-Res Modeling Group

INTRODUCTION 7
If we look at the operations as applying a filter to the signal, we can see easily that the successive filters
in the difference pyramid are (1/2, 1/2) and (1/2, -1/2), their scales and translates. We will see that they
are characteristics of the Haar transform. Notice also that this scheme computes the pyramid in O(n)
operations.
2 Frequency
The standard Fourier transform is especially useful for stationary signals, that is for signals whose properties
do not change much (stationarity can be defined more precisely for stochastic processes, but a vague concept
is sufficient here) with time (or through space for images). For signals such as images with sharp edges
and other discontinuities, however, one problem with Fourier transform and Fourier synthesis is that in
order to accommodate a discontinuity high frequency terms appear and they are not localized, but are added
everywhere. In the following examples we will use for simplicity and clarity piece-wise constant signals and
piece-wise constant basis functions to show the characteristics of several transforms and encoding schemes.
Two sample 1-D signals will be used, one with a single step, the other with a (small) range of scales in
constant spans. The signals are shown in Figure I.4 and Figure I.5.
10
8
6
4
2
0
12
10
8
6
4
2
0
0 5 10 15 20 25 30
Figure I.4: Piece-wise constant 1-D signal (signal 1)
0 5 10 15 20 25 30
Figure I.5: Piece-wise constant 1-D signal (signal 2)
Siggraph ’95 Course Notes: #26 Wavelets