Biomedical Engineering Reference
In-Depth Information
The distance between
S
and
T
can be then defined as
1
|S|
1
/
2
2
d
dist
(
ϕ
(
S
)
,T
)
≡
S
(
δ
(
ϕ
(
x
)
,T
))
S
(
x
)
(12.9)
where
is a normalization factor. In practice the integral needs to be
numerically approximated. For triangulated surfaces like
|S|
:
=
d
S
S
S
and
T
a reasonable and
viable approximation is
1
n
S
∑
j
(
ϕ
(
S
)
,T
)=
(
)
2
dist
min
i
d
ji
(12.10)
where
d
ji
=
dist
(
ϕ
(
x
j
)
,
tri
i
)
is the distance from vertex
j
of
S
to triangle
i
in
T
,
n
S
(
n
T
) is the number of vertices
(triangles) of
). By using a tree search algorithm, it is possible to reduce the
computational complexity to
S
(
T
(see [23]).
This non-parametric registration by itself is in general ill-posed and multiple so-
lutions are expected. Some of them are clearly unphysical and need to be filtered
out. For this reason a regularizing term is introduced, forcing the solutions to be
“physically acceptable” by adding some regularizing properties (see e.g. [5, 72]).
In particular, we resort to a regularizing term stemming from a simplified physical
model of the vascular wall as an elastic thin membrane [73] accounting for traction
and bending internal forces. The membrane energy provides the regularizing term.
In this way, displacements
O
(
n
S
log
(
n
T
))
that would cause a large increase to the membrane
energy are heavily penalized (see [69]).
Additional constraints are required for preventing “flips” of triangles. Let
ϕ
(
·
)
A
i
=
area
(
tri
(
x
,
y
,
z
))
betheareaofthe
i
th
triangle before deformation and
x
,
y
,
z
its corresponding vertices.
Correspondingly, let
ϕ
(
A
i
)=
area
(
tri
(
ϕ
(
x
)
,
ϕ
(
y
)
,
ϕ
(
z
)))
be the area of the deformed
i
th
triangle.
Therefore, we add to the minimization of
J
the constraint of positive deformed
area
C
i
(
ϕ
)=
ϕ
(
A
i
)
>
0
.
(12.11)
The minimization problem has been solved by means of the L-BFGS procedure
(Limited memory BFGS - see [6]), that requires only the computations of gradients
and features (at least) a linear convergence even for non-smooth problems.
