Wavelets - Caltech Multi-Res Modeling Group

7.5 Multiresolution Analysis
INTRODUCTION 17
A theoretical framework for wavelet decomposition [130] can be summarized as follows. Given functions
in L 2 (this applies as well to vectors) assume a sequence of nested subspaces Vi such that:
V,2 V,1 V0 V1 V2
If a function f (t) 2 Vi then all translates by multiples of 2 ,i also belongs (f (t , 2 ,i k) 2 Vi). We also
want that f (2t) 2 Vi+1. If we call Wi the orthogonal complement of Vi with respect to Vi+1. We write it:
Vi+1 = Wi Vi
In words, Wi has the details missing from Vi to go to Vi+1. By iteration, any space can be reached by:
Vi = Wi Wi,1 Wi,2 Wi,3 (7)
Therefore every function in L 2 can be expressed as the sum of the spaces Wi. IfV0 admits an orthonormal
basis j(t , j) and its integer translates (2 0 = 1), then Vi has ij = cj (2 i , j) as bases. There will exist
a wavelet 0() which spans the space W0 with its translates, and its dilations ij() will span Wi. Because
of (7), therefore, every function in L 2 can be expressed as a sum of ij(),awavelet basis. We then see that
a function can be expressed as a sum of wavelets, each representing details of the function at finer and finer
scales.
A simple example of a function is a box of width 1. If we take as V0 the space of all functions constant
within each integer interval [ j; j + 1 ), it is clear that the integer translates of the box spans that space.
Exercise 3: Show that boxes of width 2 i span the spaces Vi. Should there be a scaling factor when
going from width 2 i to 2 i,1 . Show that the Haar wavelets are the basis for Wi corresponding to the box for
Vi. 2
7.6 Constructing Wavelets
7.6.1 Smoothing Functions
To develop new dyadic wavelets, we need to find smoothing functions () which obey the basic dilation
equation:
(t) =
X
k
ck (2t , k)
Siggraph ’95 Course Notes: #26 Wavelets