Image Processing Reference
In-Depth Information
1
0.8
a
0.6
0.4
0.2
0
5
4
3
2
1
0
1
2
3
4
5
Fig. 9.3. A Gaussian function (
black
), plotted with the
χ
function (
red
) with
2
=
π
2
.
Blue
lines
indicate the distance
a
used when estimating the
||E||
∞
norm
that will approximate
χ
is given in the same graph. Both
χ
and
G
have
ω
as input
variables to show that they are in the frequency domain. The value at the origin
G
(0) = 1 guarantees that subsampling of at least the constant signal will be perfectly
accomplished, in that the signal value will be unchanged. We will vary
σ
to obtain
the function
G
that is least dissimilar to
χ
.
To obtain a dissimilarity measure, we define first the error signal:
e
(
ω
)=
G
(
ω
)
−
χ
(
ω
)
Because the width of the function
G
depends on the constant
σ
, the norm of the
error signal will depend on
σ
. We will use the
k
L
norm:
k
=
∞
k
dx
1
/k
f
−∞
|
f
(
x
)
|
to minimize the dissimilarity
min
σ
e
k
=min
σ
G
−
χ
k
(9.19)
The higher the value of
k
the higher the impact of extremes will be on
f
k
.If
we take
k
towards the infinity, then only the maximum of
|f |
will influence
f
k
,
yielding:
||
f
||
∞
=max
|
f
|
(9.20)
For this reason
·
∞
is also called the max norm.
