Biomedical Engineering Reference
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Based on the aforementioned definition, we propose the region-based energy func-
tional as follows:
θ i )) dx ,
n
E )=
D ( p ( N ( x ))
p ( R i |
(3)
i =1
R i
where p ( N ( x )) is the PDF of the local region around voxel I ( x ), p ( R i |
θ i ) is the
PDF of the i th global region with statistical parameters θ i . D ( · ) is the dissimilarity
measure between two distributions. Equation (3) can be derived from Zhu's [16]
united regions competing segmentation framework. From another viewpoint, it
is also a special case of the Mumford-Shah functional, where the image fidelity
term is replaced by the dissimilarity between the local and global regions. The
region intensity smooth constraint is discarded to allow strong noise to be present.
Intuitively, the region-based energy functional is a weighted volume; when the
optimal contour is obtained, each voxel has a minimum weight that measures the
dissimilarity between it and the region it belongs to. Obviously, this happened
only when a voxel is correctly classified with regard to a region. For the sake of
simplicity and interpretation, we first consider the bipartitioning segmentation and
will later extend the scheme to the multiphase case.
We use the geodesic length constraint [8] as the edge-based energy. The edge
energy is equivalent to finding the minimal-length smooth curve in Riemannian
space and given by
E (Γ) =
g ( I (Γ)) ds,
(4)
where ds is the Euclidean arc length, and g ( · ) is the function indicating boundaries
in the image. When g ( · )=1, the geodesic length constraint reduces to a Euclidean
curve smooth constraint. A very common choice of g ( · ) is:
1
1+
g ( I (Γ)) =
,
(5)
I (Γ) 2
I (Γ) 2 is the norm of the image intensity gradient.
where
2.2. Representation with a Level Set
In this study, the level set technique [5] is used because of its topological flex-
ibility, numerical stability, and parameterization-free nature, in contrast to para-
metric representation. The key idea is that hypersurface representation with the
implicit function provided it is Lipschitz continuous. A common choice of level
set function φ is the signed distance function. It is defined as follows:
R , exterior of Γ ,
φ ( x )= D ( x
,
Γ )
if
x
φ ( x )= 0
if
x Γ ,
(6)
R + , interior of Γ,
φ ( x )= D ( x
,
Γ )
if
x
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