Biomedical Engineering Reference
In-Depth Information
log
,
K
i
K
α
−
α
i
σ
, and
p
out
(
I
(
x
))
p
in
(
I
(
x
))
With
e
(
x
)=
ψ
=(
ψ
1
,...,ψ
n
),we
obtain the following system of coupled gradient descent equations:
=
i
=1
(
α
i
−
α
)
K
i
i
=1
d
dt
=
σ
2
α
,h,θ
(
x
))
ψ
(
R
θ
x
+
h
)
e
(
x
)
dx
+
δ
(
φ
,
K
i
Ω
dh
dt
=
δ
(
φ
α
,h,θ
(
x
))
∇
φ
α
,h,θ
(
x
)
e
(
x
)
dx,
(8)
Ω
dθ
dt
=
δ
(
φ
α
,h,θ
(
x
)) (
∇
φ
α
,h,θ
(
x
)
·∇
θ
Rx
)
e
(
x
)
dx.
Ω
In all equations, the Dirac delta function
δ
appears as a factor inside the integrals
over the image domain Ω. This allows to restrict all computations to a narrow
band around the zero crossing of
φ
. While the evolution of translation and pose
parameters
h
and
θ
are merely driven by the data term
e
(
x
), the shape vector
α
is additionally drawn toward each training shape with a strength that decays
exponentially with the distance to the respective shape.
5.
EXPERIMENTAL RESULTS AND VALIDATION
5.1. Heart Segmentation from Ultrasound Images
Figures 4-6 show experimental results obtained for the segmentation of the
left ventricle in 2D cardiac ultrasound sequences, using shape priors constructed
from a set of 21 manually segmented training images.
The segmentation in Figure 4 was obtained by merely imposing a small con-
straint on the length of the segmenting boundary. As a consequence, the segmen-
tation process leaks into all darker areas of the image. The segmentation of the left
ventricle based on image intensities and purely geometric regularity constraints
clearly fails.
The segmentation in Figure 5 was obtained by constraining the shape op-
timization to the linear subspace spanned by the eigenmodes of the embedding
function of the training set. This improves the segmentation, providing additional
regularization and reducing the degrees of freedom for the segmentation process.
Nevertheless, even within this subspace there is some leakage into darker image
areas.
The segmentation in Figure 5 was obtained by additionally imposing a non-
parametric statistical shape prior within this linear subspace. While the subspace
allows for efficient optimization (along a small number of eigenmodes), the non-
parametric prior allows to accurately constrain the segmentation process to a sub-
manifold of familiar shapes (see also Figure 2). This prevents any leakage of the
boundary and enables the segmentation of the left ventricle despite very limited
and partially misleading intensity information.
