Graphics Reference
In-Depth Information
is also a Riesz basis of
V
j
Z
.We
shall now check if the cardinal B-spline can act as a scaling function. Also, if
we would like to view the image in terms of signal space, then we should write
the functions in terms of a time variable
t
rather than of
x
. A scaling function
φ
is a function that generates a multiresolution analysis (MRA). We shall see
an
n
th order cardinal B-splines
N
n
satisfies all the conditions of an MRA, and
since a wavelet system can be defined in terms of a scaling function, cardinal
B-splines can be successfully used in wavelet systems. This helps to construct
different spline wavelets eciently that can be used effectively.
Consider a scaling function
φ
. The set of scaling functions based on integer
translates of the mother scaling function is
with the same bounds as of
B
for any
j
∈
L
2
(
IR
)
.
φ
r
(
t
)=
φ
(
t
−
r
)
,r
∈
Z, φ
∈
The subspace of
L
2
(
IR
) spanned by these functions is given by
V
0
=
span
r
{
φ
r
(
t
)
}
.
Hence, any function
f
(
t
)
∈
V
0
can be written as
f
(
t
)=
r
a
r
φ
r
(
t
)
.
Now instead of the mother scaling function, if we look at the scaling functions
at different resolutions, i.e., instead of
t
in the mother scaling function, if we
consider 2
j
t
, then
φ
j,r
(
t
)=2
j/
2
φ
(2
j
t
−
r
)
.
This helps us to write
f
(
t
)
V
j
as
f
(
t
)=
r
∈
a
r
φ
(2
j
t
+
r
)
.
Obviously,
φ
r
(2
j
t
)
V
j
=
span
r
{
}
.
To visualize the effect of
j
in the scaling function
φ
, we can think of ap-
proximation of a graylevel image by the scaling function. As an image is a
two-dimensional function, we can approximate row-wise and column-wise or
vice-versa. It is evident that as
j
=1
,
2
,
=
span
r
{
φ
j,r
(
t
)
}
,
φ
j,r
(
t
) becomes narrower and
narrower and hence it represents finer and finer details. On the other hand,
if
j
=
···
,
φ
j,r
(
t
) becomes wider and wider and hence it represents
coarser and coarser information. For narrower
φ
j,r
, the span is larger while
for wider
φ
j,r
, span is smaller. Thus,
V
j
s
represent the approximation spaces
and as
j
increases, the size of these approximation spaces increases.
Below we will explain the concepts of an MRA but before that, we will
examine what is meant by wavelets.
−
1
,
−
2
,
−···
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