Suppose we are given as signal the projection of a function onto the space Vj+1Vj+1:
P
j
+
1
f
=
∑
k
s
j
+
1
,
k
ϕ
j
+
1
,
k
(
x
)
,
s
j
+
1
,
k
=
f
,
ϕ
˜
j
+
1
,
k
.
P
j
+
1
f
=
∑
k
s
j
+
1
,
k
ϕ
j
+
1
,
k
(
x
)
,
s
j
+
1
,
k
=
f
,
ϕ
˜
j
+
1
,
k
.
(1)Using the dual refinement equations, we have:
s
j
,
k
=
f
,
ϕ
˜
j
,
k
=
f
,
∑
l
h
˜
l
ϕ
j
+
1
,
2
k
+
l
=
∑
k
h
˜
l
-
2
k
s
j
+
1
,
l
,
s
j
,
k
=
f
,
ϕ
˜
j
,
k
=
f
,
∑
l
h
˜
l
ϕ
j
+
1
,
2
k
+
l
=
∑
k
h
˜
l
-
2
k
s
j
+
1
,
l
,
(2)where the coefficients sjksjk are called scaling coefficients, since they are related to scaling functions.
Similarly, the wavelet or detail coefficientsdjkdjk
are obtained as
d
j
k
=
f
,
ψ
˜
j
k
=
∑
k
g
˜
l
-
2
k
s
j
+
1
,
l
.
d
j
k
=
f
,
ψ
˜
j
k
=
∑
k
g
˜
l
-
2
k
s
j
+
1
,
l
.
(3)The coefficients sjksjk and djkdjk are obtained from sj+1,lsj+1,l by `moving average' schemes, using the filter coefficients {h˜l}{h˜l} and {g˜l}{g˜l} as `weights', with the exception that these moving averages are sampled only at the even integers, i.e. a downsampling is performed.
Such transform allows, once we have computed sJ,ksJ,k=f,ϕ˜J,k=f,ϕ˜J,k
for a fine level J∈NJ∈N, to compute sjksjk and djkdjk for all coarser levels j<Jj<J without evaluating the integrals.
Suppose now we are given the values of ff at n=2Jn=2J equispaced design points. The scaling functions ϕ˜J,k,k=0,...,2J-1ϕ˜J,k,k=0,...,2J-1, are compactly supported and localized
around 2-Jk2-Jk. Hence the coefficients f,ϕ˜J,kf,ϕ˜J,k are weighted and
scaled average of ff on a neighborhood of 2-Jk2-Jk which becomes smaller as JJ tends to infinity. Consequently, it makes sense to replace the integral f,ϕ˜J,kf,ϕ˜J,k by the (scaled) value of ff at 2-Jk2-Jk.
More complicate quadrature formulae have been developed
in [8], [6], [7].
With sj:={sjk;k=0,...,2j-1}sj:={sjk;k=0,...,2j-1} and dj:={djk;k=0,...,2j-1}dj:={djk;k=0,...,2j-1}, the forward (or analyzing) wavelet transform given
by Equation 2-Equation 3
can be rewritten as
s
j
=
H
˜
j
*
s
j
+
1
and
d
j
=
G
˜
j
*
s
j
+
1
,
s
j
=
H
˜
j
*
s
j
+
1
and
d
j
=
G
˜
j
*
s
j
+
1
,
(4)where H˜j*H˜j* denotes the Hermitian conjugate of H˜jH˜j.
The inverse (or synthesis) transform is found by using the primal refinement equations and the fact that Vj+1=Vj⊕WjVj+1=Vj⊕Wj.
P
j
+
1
f
=
∑
l
s
j
+
1
,
l
ϕ
j
+
1
,
l
=
∑
k
s
j
,
k
ϕ
j
,
k
+
∑
k
d
j
,
k
ψ
j
,
k
=
∑
k
s
j
,
k
∑
l
h
l
ϕ
j
+
1
,
2
k
+
l
+
∑
k
d
j
,
k
∑
l
g
l
ϕ
j
+
1
,
2
k
+
l
=
∑
l
ϕ
j
+
1
,
l
∑
k
h
l
-
2
k
s
j
,
k
+
∑
k
g
l
-
2
k
d
j
k
,
P
j
+
1
f
=
∑
l
s
j
+
1
,
l
ϕ
j
+
1
,
l
=
∑
k
s
j
,
k
ϕ
j
,
k
+
∑
k
d
j
,
k
ψ
j
,
k
=
∑
k
s
j
,
k
∑
l
h
l
ϕ
j
+
1
,
2
k
+
l
+
∑
k
d
j
,
k
∑
l
g
l
ϕ
j
+
1
,
2
k
+
l
=
∑
l
ϕ
j
+
1
,
l
∑
k
h
l
-
2
k
s
j
,
k
+
∑
k
g
l
-
2
k
d
j
k
,
(5)from which it follows that
s
j
+
1
,
l
=
∑
k
h
l
-
2
k
s
j
k
+
∑
k
g
l
-
2
k
d
j
k
.
s
j
+
1
,
l
=
∑
k
h
l
-
2
k
s
j
k
+
∑
k
g
l
-
2
k
d
j
k
.
(6)In matrix form, we have
s
j
+
1
=
H
j
s
j
+
G
j
d
j
.
s
j
+
1
=
H
j
s
j
+
G
j
d
j
.
(7)In the finite and classical setting, the matrices HjHj, GjGj, H˜jH˜j
and G˜jG˜j are of size 2j+1×2j2j+1×2j. Moreover, if the basis functions are compactly supported, the four filters (hlhl, glgl, h˜lh˜l, g˜lg˜l) have only a finite number of nonzero elements, and hence all these matrices are banded.
In case of the orthogonal Haar transform, H˜j*=Hj*H˜j*=Hj* and is of the form
H
˜
j
*
=
h
0
h
1
h
0
h
1
...
h
0
h
1
H
˜
j
*
=
h
0
h
1
h
0
h
1
...
h
0
h
1
(8)since only h0h0 and h1h1 are different from zero : h0=h1=1/2h0=h1=1/2.
The high-pass filter {gl}{gl} is such that g0=-1/2g0=-1/2
and g1=1/2g1=1/2. The forward transform Equation 2-Equation 3
reduces to
s
j
,
k
=
1
2
s
j
+
1
,
2
k
+
1
+
1
2
s
j
+
1
,
2
k
d
j
,
k
=
1
2
s
j
+
1
,
2
k
+
1
-
1
2
s
j
+
1
,
2
k
,
s
j
,
k
=
1
2
s
j
+
1
,
2
k
+
1
+
1
2
s
j
+
1
,
2
k
d
j
,
k
=
1
2
s
j
+
1
,
2
k
+
1
-
1
2
s
j
+
1
,
2
k
,
(9)and the reconstruction is given by
s
j
+
1
,
2
k
=
1
2
s
j
,
k
-
1
2
d
j
,
k
s
j
+
1
,
2
k
+
1
=
1
2
s
j
,
k
+
1
2
d
j
,
k
.
s
j
+
1
,
2
k
=
1
2
s
j
,
k
-
1
2
d
j
,
k
s
j
+
1
,
2
k
+
1
=
1
2
s
j
,
k
+
1
2
d
j
,
k
.
(10)
