Wavelets - Caltech Multi-Res Modeling Group

7.3 Dyadic Wavelet Transforms
INTRODUCTION 15
We have sort of “stumbled” upon the Haar wavelet, from two different directions (from the windowed Walsh
transform and from the difference pyramid). We need better methods than that to construct new wavelets
and express their basic properties. This is exactly what most of the recent work on wavelets is about [50].
We reproduce the following development from Mallat & Zhong [133]. Consider a wavelet function (t).
All we ask is that its average R (t) dt = 0. Let us write i(t) its dilation by a factor of 2 i :
i = 1 t
(
2i 2
The wavelet transform of f (t) at scale 2 i is given by:
WFi(t) = f i(t) =
Z 1
The dyadic wavelet transform is the sequence of functions
,1
i )
f ( ) i(t , ) d
WF[f ()] = [WFi(t)] i 2 Z
We want to see how well WF represents f (t) and how to reconstruct it from its transform. Looking at the
Fourier transform (we use F (f ) or F[f (t)] as notation for the Fourier transform of f (t)):
F[WFi(t)] = F (f ) Ψ(2 i f ) (1)
If we impose that there exists two strictly positive constants A and B such that:
8f; A
1X
jΨ(2
i=,1
if )j2 B (2)
we guarantee that everywhere on the frequency axis the sum of the dilations of () have a finite norm.
If this is true, then F (f ), and therefore f (t) can be recovered from its dyadic wavelet transform. The
reconstructing wavelet (t) is any function such that its Fourier transform X(f ) satisfies:
1X
i=,1
Ψ(2 i f ) X(2 i f )=1 (3)
An infinity of () satisfies (3) if (2) is valid. We can then reconstruct f (t) using:
f (t) =
1X
WFi(t) i(t) (4)
i=,1
Siggraph ’95 Course Notes: #26 Wavelets