Biomedical Engineering Reference
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tance. This is computed in terms of the minimum number of morphological
dilations needed to overlap the two regions [Figure 3.15(b)].
D
msk
is defined
as
D
(
W
i
,
W
j
)=
×
{
k
: d
k
(
W
i
)
∩
=
∩
d
k
(
W
j
)
=
}
K
msk
inf
W
j
0 ; W
i
0
(3.39)
msk
where d
k
denotes
k
-times dilation with a unit structuring element, and
K
msk
is a
constant.
msk
is closely related to Hausdorff distance [80] (Figure 3.16), redefined
in [81] using morphological operators as
D
H
(
A
,
B
)=
inf
{
k
:
A
⊆
d
k
(
B
)
;
B
⊆
d
k
(
A
)
}
(3.40)
olp
quantifies small-range displacement
between two regions/cells [Figure 3.15(c)] by the degree of their overlap, in terms of
shape
and
tonal dissimilarities. In order to emphasize overlap in nuclei (i.e., dark
regions with low intensity values), and to de-emphasize cytoplasm overlap (i.e.,
light regions with high intensity values), overlapping and nonoverlapping regions
are weighted by local tonal differences, as
The
tonal-weighted overlap distance
D
D
(
W
i
,
W
j
)=
olp
−
−
W
j
|
)
−
)
|
W
j
(
1
I
i
(
y
))
d
y
+
W
i
(
1
I
j
(
y
))
d
y
+
I
i
(
y
I
j
(
y
d
y
(3.41)
W
i
\
W
j
\
W
i
∩
K
olp
×
;
I
i
(
)
+
I
J
(
)
y
d
y
y
d
y
W
i
W
j
−
where the intensity images,
I
i
(
y
)=
I
i
(
y
,
t
)
and
I
j
(
y
)=
I
j
(
y
,
t
1
)
, are scaled such
∈
[
]
that
I
,and
K
olp
is a constant. The first two terms in the numerator of (3.41)
account for the distance due to uncovered regions in frames at time instants
t
and
t
0,1
−
1, respectively. The complement of intensity images are used to obtain higher
distances for uncovered low intensity regions (i.e., nuclei). The third term in the
numerator accounts for the intensity dissimilarity within the overlapping region.
Figure 3.16
Hausdorff (a, b) and
D
M
(c, d) distances for two cells A and B.
H
(
A
,
B
)=
min
(
35
,
29
)
,
D
M
=
min
(
11
,
11
)
.
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