Spectral Transformations - Biomedical Signal Analysis: Contemporary Methods and Applications - page 55

approximation P −1 f .

2 (R)
f L

f

P -1 f

P 0 f

Q 0 f

t

(b)

(a)

(c)

Figure 2.11
Approximation of (a) P 0 f, (b) P −1 f, and (c) the detail signal Q 0 f, with
P 0 f+Q 0 f=P −1 f.

The scaling function coecients for the Haar wavelet at scale m are
given by

c m,n = <f,ϕ mn > =2 −m/2 2 m (n+1)
2 m n

f ( t ) dt (2.66)

This yields an approximation of f at scale m :

P m f =
n

c m,n ϕ mn ( t )=
n

c m,n 2 −m/2 ϕ (2 −m t − n ) (2.67)

In spite of their simplicity, the Haar wavelets exhibit some undesirable
properties which pose a di culty in many practical applications. Other
wavelet families, such as Daubechies wavelets and Coiflet basis [4, 278]
are more attractive in practice. Daubechies wavelets are quite often used
in image compression. The scaling function coecients h 0 ( n )andthe
wavelet function coecients h 1 ( n ) for the Daubechies-4 family are nearly
impossible to determine. They were obtained based on iterative methods
[38].

Multiscale Signal Decomposition and Reconstruction

In this section we will illustrate multiscale pyramid decomposition.
Based on a wavelet family, a signal can be decomposed into scaled and
translated copies of a basic function. As discussed in the preceeding
sections, a wavelet family consists of scaling functions, which are scalings
and translations of a father wavelet, and wavelet functions, which are