Biomedical Engineering Reference
In-Depth Information
Figure 11.18: The image pixels (red solid lines) corresponding to co-volume
mesh. Triangulation (black dashed lines) for the co-volume method with degree
of freedom nodes (red round points) corresponding to centers of pixels (color
slide).
approach given in [25, 56] in such a way that our co-volume mesh will con-
sist of cells
p
associated only with DF nodes
p
of
T
h
, say
p
=
1
,...,
M
. Since
there will be one-to-one correspondence between co-volumes and DF nodes,
without any confusion, we use the same notation for them. In this way we
have excluded the boundary nodes (due to Dirichlet boundary data) and NDF
nodes.
For each DF node
p
of
T
h
, let
C
p
denote the set of all DF nodes
q
connected
to the node
p
by an edge. This edge will be denoted by
σ
pq
and its length by
h
pq
. Then every
co-volume p
is bounded by the lines (co-edges)
e
pq
that bisect
and are perpendicular to the edges
σ
pq
,
q
∈
C
p
. By this construction, the co-
volume mesh corresponds exactly to the pixel structure of the image inside the
computational domain
where the segmentation is provided. We denote by
E
pq
the set of triangles having
σ
pq
as an edge. In a situation depicted in Fig. 11.18,
every
E
pq
consists of two triangles. For each
T
∈
E
pq
let
c
pq
be the length of
the portion of
e
pq
that is in
T
, i.e.,
c
pq
=
m
(
e
pq
∩
T
), where
m
is a measure in
IR
d
−
1
. Let
N
p
be the set of triangles that have DF node
p
as a vertex. Let
u
h
be
a piecewise linear function on triangulation
T
h
. We will denote a constant value
