Biomedical Engineering Reference
In-Depth Information
algorithm (or other segmentation algorithms) many times (e.g., 100 times in this
work) to generatemultiple outputs requires a dedicated user and a significant effort.
In order to avoid the need for a dedicated user, we developed “virtual operators”
that automatically initialize the algorithm being tested for a particular image. The
construction of virtual operators requires an initial investment of time from several
users. However, once the virtual operators are constructed, they can be used as
often as required to automatically initialize the segmentation algorithm without
further user interaction.
Mathematically, a virtual operator for our TRUS-generated prostate image
i
is denoted VO
i
and consists of four probability distributions,
p
VO
(
x, y
), one for
c,i
each control point
c
=1
,
2
,
3
,
4. Each
p
VO
c,i
(
x, y
) describes the spatial distribution
of selections made by the average user for control point
c
in image
i
. A separate
virtual operator should be constructed for each type of image because the choice
of control points depends on the image.
3.2. Constructing and Using Virtual Operators
The construction of a virtual operator requires four steps, as outlined by Ladak
et al. [38] and as summarized here. First, each user
u
(
u
1
,
2
,...,U
, where
U
is
the total number of users) selects each control point
c
in image
i
. This is repeated
a total of
R
times by each user, yielding a population of
R
×
U
selections per
control point when all user selections are pooled together.
Second, the parameters of a “population” distribution,
p
pop
c,i
(
x, y
), are com-
×
puted to describe the spatial distribution of the
R
U
selections for control point
c
in image
i
.
p
pop
c,i
(
x, y
) is analytically represented by a bivariate Gaussian distri-
bution. The parameters of
p
pop
c,i
(
x, y
) are
x
pop
c,i
, the mean
x
coordinate of control
, the mean
y
coordinate; and the variances
σ
pop
Xc,i
2
and
σ
pop
Y c,i
2
point
c
;
y
pop
c,i
along the major and minor principal axes of the distribution (
X
pop
c,i
and
Y
pop
c,i
,
respectively). The variances and principal axes are calculated using principal
component analysis [40].
Third, the parameters the of “individual” distributions,
p
ind
u,c,i
(
x, y
), which
describe the spatial distribution of user
u
selections of control point
c
in image
i
,
are computed from the
R
selectionsmade by that user.
p
ind
u,c,i
(
x, y
) is also described
by a bivariate Gaussian distribution. Each of the
U
distributions
p
ind
u,c,i
(
x, y
) for
control point
c
in image
i
can have different mean
x
and
y
coordinates,
x
ind
u,c,i
and
u,c,i
, variances
σ
ind
Xu,c,i
2
and
σ
ind
Y u,c,i
2
, and principal axes
X
ind
y
ind
u,c,i
and
Y
ind
u,c,i
;
the differences arise due to inter-observer variability.
Fourth, the parameters of the bivariate Gaussian distributions
p
VO
c,i
(
x, y
) are
computed from the individual distributions
p
ind
u,c,i
(
x, y
) and the population distri-
(
x, y
).
σ
V
Xc,i
2
denotes the variance of
p
VO
bution
p
pop
c,i
(
x, y
), and is computed
c,i
as the average of the variances
σ
ind
Xu,c,i
2
for all
U
users. Similarly,
σ
V
Y c,i
2
is
