Databases Reference
In-Depth Information
φ
(
1
)
φ
(
0
)
(
t
)
into the sum of a lower-resolution version of itself, namely,
(
t
)
, and the difference
g
g
φ
(
1
)
)
−
φ
(
0
)
(
t
(
t
)
. Let's examine this difference over an arbitrary unit interval
[
k
,
k
+
1
)
:
g
g
c
0
,
k
−
√
2
c
1
,
2
k
1
2
k
t
<
k
+
φ
(
1
)
g
)
−
φ
(
0
)
g
c
0
,
k
−
√
2
c
1
,
2
k
+
1
(
t
(
t
)
=
(32)
1
k
+
2
t
<
k
+
1
Substituting for
c
0
,
k
from (
27
), we obtain
−
1
1
1
2
√
2
c
1
,
2
k
+
√
2
c
1
,
2
k
+
1
k
t
<
k
+
φ
(
1
)
g
)
−
φ
(
0
)
g
(
t
(
t
)
=
(33)
1
1
1
√
2
c
1
,
2
k
−
√
2
c
1
,
2
k
+
1
k
+
2
t
<
k
+
1
Defining
1
√
2
c
1
,
2
k
+
1
√
2
c
1
,
2
k
+
1
d
0
,
k
=−
over the arbitrary interval
[
k
,
k
+
1
)
,
φ
(
1
)
g
)
−
φ
(
0
)
g
(
t
(
t
)
=
d
0
,
k
ψ
0
,
k
(
t
)
(34)
where
1
1
2
<
+
k
t
k
ψ
0
,
k
(
t
)
=
(35)
1
−
1
k
+
2
t
<
k
+
1
But this is simply the
k
th translate of theHaar wavelet. Thus, for this particular case the function
can be represented as the sum of a scaling function and a wavelet at the same resolution:
φ
(
1
)
g
(
t
)
=
c
0
,
k
φ
0
,
k
(
t
)
+
d
0
,
k
ψ
0
,
k
(
t
)
(36)
k
k
In fact, we can show that this decomposition is not limited to this particular example. A
function in
V
1
can be decomposed into a function in
V
0
—that is, a function that is a linear
combination of the scaling function at resolution 0 and a function that is a linear combination
of translates of a mother wavelet. Denoting the set of functions that can be obtained by a linear
combination of the translates of the mother wavelet as
W
0
, we can write this symbolically as
V
1
=
V
0
⊕
W
0
(37)
In other words, any function in
V
1
can be represented using functions in
V
0
and
W
0
.
Obviously, once a scaling function is selected, the choice of the wavelet function cannot be
arbitrary. The wavelet that generates the set
W
0
and the scaling function that generates the sets
V
0
and
V
1
are intrinsically related. In fact, from (
37
),
W
0
⊂
V
1
, and therefore any function
in
W
0
can be represented by a linear combination of
{
φ
1
,
k
}
. In particular, we can write the
mother wavelet
ψ(
t
)
as
ψ(
t
)
=
w
k
φ
1
,
k
(
t
)
(38)
k
or
w
k
√
2
ψ(
t
)
=
φ(
2
t
−
k
)
(39)
k
