Biomedical Engineering Reference
In-Depth Information
R ( β )
= {
Ω |
(
β ) <
}
where the limits of integration are given by
,which
includes the points interior to the superellipsoid and its complement with respect to
the domain
:
x
S
x ;
1
Ω
. The marginal distribution f describes the additive noise in the image
(
)
I
and is assumed to be independent and identically distributed.
Maximizing (27) is done by alternately estimating the intensity parameters and
then updating the boundary using gradient ascent. Assuming the intensities
x
(
,
)
I F
I B
are known, the gradient of (27) with respect to the parameters
β
is given by
log
T
f
(
I
(
x
(
u
))
I F )
C
(
u ;
β )
L
( β )=
n
(
u ;
β )
d S
(28)
(
(
(
))
)
f
I
x
u
I B
∂β
C
( β )
where the integral is over the superellipsoid surface C
( β )= R ( β )
parameterized
d S ,andwhere C ( u ; β )
∂β
by u with surface area element n
(
u ;
β )
is the Jacobian matrix
( β )
giving the change in the superellipsoid boundary C
with respect to parameters
β
. Finally, the gradient step used to update
β
is given by
β ( t + Δ t ) = β ( t ) +
T
( β )
L
Δ
t
(29)
where
t is a constant time step.
Above, it has been demonstrated how to find the optimal superellipsoid assuming
the intensity parameters are known. Since they are in fact unknown, they must be
estimated. The use of a robust image intensity estimate is important for dealing
with various sources of noise. The median is a good choice and can tolerate as
much as 50 % outliers before breaking down . However, when estimating the image
intensities, it's possible that more than 50 % of the voxels will be outliers. This can
happen at branch points or when adjacent vessels are very close to each other. For
additional robustness, the skipped median can be used, which censures very large
residuals from the estimate of the median.
Since the maximum likelihood intensity estimator under i.i.d. Laplacian noise
gives the population median, a Laplacian noise model is assumed when maximizing
(27). The resulting maximum value can be used to determine a robust likelihood
ratio test (LRT) statistic for the null hypothesis that no vessel is present. Hence, it
becomes possible to measure goodness of fit in a principled manner.
Δ
B.3
Traversing Vessels Using the Superellipsoid
This section describes the general procedure for tracing vessels using the proposed
model. The procedure is outlined in the following pseudo-code:
Seed extraction is done on a regular grid over the 2-D intensity projection image
as described in [ 2 ]. The model is fit to each candidate seed, and candidates are then
prioritized by goodness of fit. For each seed, there are two directions, 180 apart,
that are traversed sequentially. The inner traversal loop involves alternately shifting
the centerpoint of the model, and then reestimating the parameters. Specifically,
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