Wavelets - Caltech Multi-Res Modeling Group

1 Introduction
II: Multiresolution and Wavelets
Leena-Maija REISSELL
University of British Columbia
This section discusses the properties of the basic discrete wavelet transform. The concentration is on
orthonormal multiresolution wavelets, but we also briefly review common extensions, such as biorthogonal
wavelets. The fundamental ideas in the development of orthonormal multiresolution wavelet bases
generalize to many other wavelet constructions.
The orthonormal wavelet decomposition of discrete data is obtained by a pyramid filtering algorithm which
also allows exact reconstruction of the original data from the new coefficients. Finding this wavelet
decomposition is easy, and we start by giving a quick recipe for doing this. However, it is surprisingly
difficult to find suitable, preferably finite, filters for the algorithm. One objective in this chapter is to find
and characterize such filters. The other is to understand what the wavelet decomposition says about the
data, and to briefly justify its use in common applications.
In order to study the properties of the wavelet decomposition and construct suitable filters, we change our
viewpoint from pyramid filtering to spaces of functions. A discrete data sequence represents a function in a
given basis. Similarly, the wavelet decomposition of data is the representation of the function in a wavelet
basis, which is formed by the discrete dilations and translations of a suitable basic wavelet. This is analx
ogous to the control point representation of a function using underlying cardinal B-spline functions.
For simplicity, we will restrict the discussion to the 1-d case. There will be some justification of selected
results, but no formal proofs. More details can be found in the texts [50] and [26] and the review paper
[113]. A brief overview is also given in [177].
1.1 A recipe for finding wavelet coefficients
The wavelet decomposition of data is derived from 2-channel subband filtering with two filter sequences
(hk), thesmoothing or scaling filter, and(gk), thedetail, orwavelet, filter. These filters should have the
following special properties: