An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal.That is, the inverse wavelet transform is the adjoint of the wavelet transform.If this condition is weakened one may end up with biorthogonal wavelets.
The scaling function is a refinable function.That is, it is a fractal functional equation, called the refinement equation (twin-scale relation or dilation equation):
N-1 | |
\phi(x)=\sum | |
k=0 |
ak\phi(2x-k)
(a0,...,aN-1)
M-1 | |
\psi(x)=\sum | |
k=0 |
bk\phi(2x-k)
(b0,...,bM-1)
A necessary condition for the orthogonality of the wavelets is that the scaling sequence is orthogonal to any shifts of it by an even number of coefficients:
\sumn\in\Zanan+2m=2\deltam,0
where
\deltam,n
In this case there is the same number M=N of coefficients in the scaling as in the wavelet sequence, the wavelet sequence can be determined as
n | |
b | |
n=(-1) |
aN-1-n
A necessary condition for the existence of a solution to the refinement equation is that there exists a positive integer A such that (see Z-transform):
(1+Z)A|a(Z):=a0+a1Z+...+aN-1ZN-1.
The maximally possible power A is called polynomial approximation order (or pol. app. power) or number of vanishing moments. It describes the ability to represent polynomials up to degree A-1 with linear combinations of integer translates of the scaling function.
In the biorthogonal case, an approximation order A of
\phi
\tilde\psi
\tilde\psi
\tilde\phi
\psi
A sufficient condition for the existence of a scaling function is the following: if one decomposes
a(Z)=21-A(1+Z)Ap(Z)
1\le\supt\in[0,2\pi]\left|p(eit)\right|<2A-1-n,
holds for some
n\in\N
p(Z)=1
a(Z)=21-A(1+Z)A
a(Z)= | 14(1+Z) |
2((1+Z)+c(1-Z)). |
Expansion of this degree 3 polynomial and insertion of the 4 coefficients into the orthogonality condition results in
c2=3.