Image Processing Reference
In-Depth Information
(a) Linear filter
A linear filter is a filter defined by a local linear function. In other words, the
output value for each voxel is calculated by a linear combination (the weighted
sum) of density values on an input image in the neighborhood of each voxel.
Formally it is represented by Eq. 3.3.
Linear filter LF
[
W
]:
F
=
{
f
ijk
}→
G
=
{
g
ijk
}
P
Q
R
g
ijk
=
w
pqr
·
f
i−
[
P/
2]+
p,j−
[
Q/
2]+
q,k−
[
R/
2]+
r
(3.3)
p
=1
q
=1
r
=1
Here a neighborhood of
P
R
voxels is employed. A set of filter parameters
w
pqr
is called a
weight matrix
or
mask
, and these are often illustrated by a 3D
array of the size
P
×
Q
×
R
. Since a weight can take
0
, the generality is not
lost by considering only this parallelepiped neighborhood.
×
Q
×
(b) Difference filter
The local function of this type of filter contains the calculation of the difference
between density values of an input image. The simplest form is the calculation
of the difference of two voxels adjacent to each other. Most edge detection
filters such as Laplacian are operated by the difference filter as shown in
Section 3.3.
(c) Local statistics filter
The local statistics filter introduced in the previous section is also considered
to define a local function.
3.2 Smoothing filter
Most smoothing filters in 2D image processing can easily be applied to a 3D
image. We present a few examples.
3.2.1 Linear smoothing filter
If a negative value is not contained in a weight matrix in Eq. 3.3, the filter has a
smoothing effect when used on a local function. The weight values are selected
according to individual applications. If all weights are
1
, the filter is called a
uniform weight smoothing filter
and represented by UF. This is equivalent to
the filter of which the output at each voxel is the sum (or average) of density
values in the neighborhood of an input image. A uniform weight smoothing
filter with the
3
×
3
×
3
neighborhood is most frequently used due to the
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