Biomedical Engineering Reference
In-Depth Information
where
θ
max
is the angle of the largest slope allowed, which is used to satisfy the
continuity constraint. The value of
θ
max
is dependent on the value of
J
and the
desired smoothness of the segmented prostate. For a smooth segmented prostate
or using a large
J
, a small
θ
max
should be selected, which can be determined
empirically. In our experiments with
J
=90, we used
θ
max
=15
◦
.
We define
E
s
(
j
)
,E
e
(
j
) to be the square differences between estimated dis-
tances
R
s
(
j
) and
R
'
e
(
j
) and original distances
R
s
(
j
) and
R
e
(
j
):
E
s
(
j
)=[
β
s
(
j
)
−
1]
2
R
s
(
j
)
,
(17)
E
e
(
j
)=[
β
e
(
j
)
−
1]
2
R
e
(
j
)
.
(18)
It can be shown that
β
s
(
i
) and
β
e
(
i
) can be determined using Eqs. (19) and (20)
so that Eqs. (7) and (8) are satisfied, and
E
s
(
i
) and
E
e
(
i
) are kept to a minimum.
According to Eqs. (17) and (18),
E
s
(
i
) and
E
e
(
i
)) attain their minimum when
β
s
(
i
) and
β
e
(
i
)) are as close to 1 as possible:
≤
tan(
θ
max
)
,
R
s
(
J
)
−
(
J
−
j
)
·
tan(
θ
max
)
R
s
(
j
)
if
R
s
(
J
)
−
R
s
(
j
)
J−j
1
R
s
(
J
)
−
R
s
(
j
)
J−j
β
s
=
>
tan(
θ
max
)
,
(19)
if
R
s
(
J
)+(
J
−
j
)
R
s
(
j
)
R
s
(
J
)
−
R
s
(
j
)
J−j
if
<
tan(
θ
max
)
,
≤
tan(
θ
max
)
,
R
e
(
J
)
−
R
e
(
j
)
J−j
1
if
R
e
(
J
)
−
(
J
−
j
)
·
tan(
θ
max
)
R
e
(
j
)
R
e
(
J
)
−
R
e
(
j
)
J−j
β
e
=
>
tan(
θ
max
)
.
(20)
if
R
e
(
J
)+(
J
−
j
)
R
e
(
j
)
R
e
(
J
)
−
R
e
(
j
)
J−j
if
<
tan(
θ
max
)
,
The start and endpoints on the initial segmentation slice,
X
s
(0)
,X
e
(0); the origin
of the coronal cross-sectional plane,
O
; and the estimated start point of the seg-
mented prostate contour in the
j
th slice,
X
s
(
j
), will form two adjacent triangles,
(
X
s
(0)
,O,X
s
(
j
)) and (
X
s
(
j
)
,O,X
e
(0)), in the coronal cross-sectional plane
(see Figure 14d). The desired coordinates of
X
s
(
j
) can be determined using the
cosine rule. Similarly, the coordinate of
X
e
(
j
) can be determined from triangles
(
X
e
(0)
,O,X
e
(
j
)) and (
X
e
(
j
)
,O,X
s
(0)).
After determining the start and endpoints,
X
s
(
j
)
,X
e
(
j
),
j
=0
,
1
,
−
1,
on each slice of the re-sliced 3D prostate image, these coordinates can be inserted
into the propagated contour as new vertices to obtain a new initial contour. From
this contour, a new prostate boundary can be refined by deformation using the
DDC model.
···
,J
4.2.3. Choice of coronal cross-sectional plane
The use of the continuity constraint as described above requires selection of
a coronal cross-sectional plane. While multiple planes may be selected, the use
