Graphics Reference
In-Depth Information
Figure 21.11.
Box and wavelet basis functions.
()
=
f
x
1
for x
elsewhere
0
£
<
1
,
=
()
=
0
,
, ,...,
(
)
j
j
f
x
f
2
x
-
i
for
i
=
0 1
2
-
1
,
ji
,
and
1
2
()
=
y
x
1
for
0
£
x
<
,
1
2
=-
1
for
£
x
<
1
,
=
()
=
0
elsewhere
,
(
)
j
j
y
x
y
2
x
-
i
for
i
=
0 1
, ,...,
2
-
1
.
ji
,
See Figure 21.11. Let V
j
à L
2
(
R
) be the vector subspace with basis (f
j,i
}
i
. Clearly,
0
1
2
VVV
ÃÃ Ã....
If W
j
is the orthogonal complement of V
j
in V
j+1
, then
j
+
1
j
j
VVW
=≈
and the {Y
j,i
}
i
are an orthogonal basis of W
j
. It follows that
j
+
1
0
0
1
j
VVWW
=≈ ≈ ≈≈
...
W
.
Definition.
The functions f
j,i
(x) are called
scaling functions
. Each function Y
j,i
(x) is
called a
Haar wavelet
and for each fixed j, the collection of these wavelets is called the
one-dimensional Haar wavelet basis functions
.
Now the functions f
j,i
and Y
j,i
are not unit vectors in L
2
(
R
). To make them unit
vectors one must multiply them by 2
j/2
. We did not do this here is because it would
cause some ugly coefficients in Example 21.9.1 below and obscure what is going on.
Nevertheless, this is normally done because orthonormal bases are desirable, so that
the terms “scaling function” and “wavelet” usually refer to the normalized versions of
the functions we are using here.
The next example shows how wavelets are used to represent functions up to a
desired resolution.
