Biomedical Engineering Reference
In-Depth Information
4.2.2. Slice-based segmentation using AR model-based continuity constraint
To ensure that the surface of the segmented prostate is smooth, a continuity
constraint must be imposed in the contour propagation phase of the slice-based
3D segmentation procedure. In [44], we described how to apply a so-called zero-
order autoregressive (AR) model to impose a continuity constraint on the start and
endpoints associated with the first and last propagated 2D contours. The following
is a description of the continuity constrained method used in conjunction with the
slice-based 3D prostate segmentation.
Suppose that the start and endpoints are represented by
X
s
(
j
) and
X
e
(
j
)
for slice
j
,
j
−
1.
R
s
(
j
)
,R
e
(
j
) are the radial lengths between
X
s
(
j
)
,X
e
(
j
) and
O
, the origin of a chosen coronal cross-sectional plane. Be-
cause
R
s
(
j
)
,R
e
(
j
) can be calculated, a linear equation model in Eqs. (3) and (4)
can be used to represent the smoothed radial length so that the computational cost
for the estimation of the model parameters is minimized:
=0
,
1
,...,J
R
s
(
j
)=
β
s
(
j
)
·
R
s
(
j
)=
X
s
(
j
)
,O
,
(11)
R
e
(
j
)=
β
e
(
j
)
·
X
e
(
j
)
,O
R
e
(
j
)=
.
(12)
In Eqs. (11) and (12),
is the Euclidean distance between
A
and
B
;
β
s
(
j
)
and
β
e
(
j
) are the coefficients used in the estimation of
R
s
(
j
)
,R
e
(
j
).
R
s
(
j
)
,R
e
(
j
)
are the radial lengths that satisfy the continuity constraint. In the following, we
discuss how the coefficients
β
s
(
i
) and
β
e
(
i
) are used to impose bounds on the rate
of change of
R
s
(
j
) and
R
e
(
j
) during the propagation phase of the segmentation.
The following two continuity constraints are imposed in the estimation of the
two rates of change,
R
s
(
j
)
,R
e
(
j
):
A, B
R
s
(
J
)=
R
e
(0)
,
(13)
R
e
(
J
)=
R
s
(0)
.
(14)
Empirical experiments demonstrated that most segmentation errors occurred
near the end of the propagation phase of the slice-based segmentation due to error
accumulation, as shown in Figure 15(d). Thus, a continuity constraint is imposed
by restricting the rate of change of both
R
s
(
j
) and
R
e
(
j
) with respect to
R
s
(
J
)
and
R
e
(
J
), i.e.,
R
e
(0) and
R
s
(0), to a maximum tan (
θ
max
) based on Eqs. (15)
and (16):
R
s
(
J
)
−
R
s
(
j
)
R
s
(
j
)
R
e
(0)
−
=
≤
tan(
θ
max
)
,
j
=1
, ..., J
−
1
,
(15)
J
−
j
J
−
j
R
e
(
J
)
−
R
e
(
j
)
R
e
(
j
)
R
s
(0)
−
=
≤
tan(
θ
max
)
,
j
=1
, ..., J
−
1
,
(16)
J
−
j
J
−
j
