Image Processing Reference
In-Depth Information
of memory cost, increase in computation speed, and development of the algo-
rithm to calculate squared Euclidean distance have changed the significance
of specific distance functions and various devices relating to them.
Remark 4.21 (Weighting in distance functions).
One important prob-
lem in calculating distance on a digitized image is how to give weights
to distances to adjacent voxels. For example,
w
111
=
√
3
,
w
112
=
√
2
,
w
122
=
1
, if we employ weights equal to the Euclidean distance to vox-
els in the 26-neighborhood. Suxes here show the locations according to
Fig. 4.2
. We will present below a mathematical formulation to optimize
these weights[Verwer91].
U
U
{
r
1
,
r
2
,...,
r
e
}
Let us consider a set of fundamental vectors
=
,and
assign a distance (weight)
d
i
r
i
. Consider next a path
P
i
toavector
as a
sequence of basic vectors
{
r
i
1
,
r
i
2
,...,
r
im
}
,where
m
is the length of the
sequence and we define the weight
|
P
i
|
of the path
P
i
as
=
m
|
P
i
|
d
ik
(= sum of distances assigned to
(4.69)
k
=1
fundamental vectors included in the path
P
i
)
The distance between two grid points (or voxels)
u
and
v
is obtained as
d
gc
(
u
,
v
)=min
(= the minimum of the above weight of (4.70)
a path between
{|
P
i
|}
u
and
v
)
Then let us find the set of weights
{
d
1
,d
2
,...,d
e
}
that minimizes the difference
(error) of
d
gc
(
. Results will
depend on the selection of the fundamental vector set. Let us assume first a
suitable set of fundamental vectors such as vectors from a point
P
to its 26-
neighborhood or to its
5
u
,
v
) from the Euclidean distance between
u
and
v
5
neighborhood. Following are examples of
integer weights recommended in [Verwer91] as realizing relatively small errors.
×
5
×
3
×
3
neighborhood:
d
100
=
4
,d
110
=
6
,d
111
=
7
,d
100
=
14
,
d
110
=
21
,d
111
=
25
3
×
5
5
neighborhood:
d
100
=
9
,d
110
=
13
,d
111
=
16
,d
210
=
20
,
d
211
=
22
,d
221
=
27
or
d
100
=
17
,d
110
=
25
,d
111
=
31
,d
210
=
39
,
d
211
=
43
,d
221
=
53
where suxes show the locations of voxels in the neighborhood (Fig. 4.2
×
5
×
U
)
and numerical values mean recommended values of weights.
The margin for error will be reduced if fractions or large values are em-
ployed. If squared Euclidean distance is acceptable, this type of complicated
distance may be not necessary.
Search WWH ::
Custom Search