Image Processing Reference
In-Depth Information
Median filter MED
:
F
=
{
f
ijk
}→
G
=
{
g
ijk
}
g
ijk
= the median of
{
f
pqr
;(
p, q, r
)
∈N
((
i, j, k
))
}
(3.8)
N
((
i, j, k
)) = the neighborhood
This filter changes the density distribution of an input image into a
smoother one by replacing the density value
f
ijk
of a voxel (
i, j, k
)bythe
median of density values in its neighborhood. Extraordinary density values
can be eliminated without being affected by its absolute value, and the filter
can preserve edges to some extent. Computing cost is relatively high because
sorting of density values in the neighborhood is performed at every voxel in
an input image.
Remark 3.3 (Order statistics filter).
The concept of the median filter is
easily extended to the one that outputs the
k
-th order statistics (or the
k
-th
largest density value in the neighborhood). This we call
order statistics filter
OS
[
k
]. Formal description is given as follows.
Order statistics filter OS
[
m
]:
F
=
{
f
ijk
}→
G
=
{
g
ijk
}
g
ijk
=the
m
-th largest of
{
f
pqr
;(
p, q, r
)
∈N
((
i, j, k
))
}
(3.9)
N
((
i, j, k
)) = the neighborhood
((
i, j, k
)) contains
n
(= odd) voxels,
OS
[
m
] reduces to the median filter, if
m
=[(
n
+
1
)
/
2
]. If
m
=
1
and
m
=
n
,we
call those filters
maximum filter
and
minimum filter
, and denote by
MAX
and
MIN
, respectively. Both
MAX
and
MIN
filter and their repetitive ap-
plication are very important in the distance transformation and the fusion
presented in Chapter 5 in detail. The basic structure and computation of an
algorithm essentially do not depend on the dimensionality of an input image.
The 1D order statistics filter has been studied in detail in the field of signal
processing [Nodes82, Bovik83, Arce87, McLoughlin87]. However, some of the
features such as edge preservation characteristics and the existence of ker-
nels (an invariant component to the iterative application) may not always be
extended to 3D image processing.
Assuming the neighborhood
N
3.2.3 Edge-preserving smoothing
The smoothing operation changes the spatial distribution of the density values
in an input image and makes it smoother. It also eliminates or weakens a
significant variation in density values such as in the edges and borders of a
3D figure. Because of this, there is always a trade-off between noise reduction
by smoothing and the edge detection (detection of abrupt change) in density
values.
A method to deal with this diculty is by smoothing while preserving the
edge. The basic idea is to roughly estimate the possible existence of edges
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