Biomedical Engineering Reference
In-Depth Information
Fig. 13.2.
Definitions for a conformal mapping
u
(
x
)
of a 3D surface
. The boundary of
is a
S
S
closed curve and is noted
∂
S
0
as possible in the parametrization and have the same norm. This can bee seen as an
approximation of the Cauchy-Riemann equations. For a piecewise linear mapping,
the least square conformal map can be obtained by minimizing the conformal energy:
∇
v
1
2
2
u
⊥
−
∇
E
LSCM
(
u
)=
ds
,
(13.2)
S
where
⊥
denotes a counterclockwise 90
◦
rotation in
. For a 3D surface with normal
vector
n
, the counterclockwise rotation of the gradient can be written as:
S
u
⊥
=
∇
n
×
∇
u
(see Fig. 13.2). Eq. (13.2) can be simplified and rewritten as follows:
v
ds
1
2
u
⊥
·
∇
u
⊥
+
∇
u
⊥
·
∇
E
LSCM
(
u
)=
∇
v
·
∇
v
−
2
∇
,
S
,
(13.3)
1
2
(
∇
=
u
·
∇
u
+
∇
v
·
∇
v
−
2
(
n
×
∇
u
)
·
∇
v
)
ds
.
S
Recalling that a “dot” and a “cross” can be interchanged without changing the result,
we have:
1
2
(
∇
E
LSCM
(
u
)=
u
·
∇
u
+
∇
v
·
∇
v
−
2
n
·
(
∇
u
×
∇
v
))
ds
.
(13.4)
S
We now derive the finite element formulation of the quadratic minimization problem
(13.2):
H
1
min
E
LSCM
(
u
)
,
with
U
(S)=
{
u
∈
(S)
,
u
=
u
D
(
x
)
on
∂
S
0
},
(13.5)
u
∈
U
(
S
)
where
∂
S
0
is a closed curve of
S
. We assume the following finite expansions for
u
=
{
u
,
v
}
:
)=
i
∈
I
u
i
φ
i
(
x
)+
i
∈
J
u
D
(
x
i
)
φ
i
(
x
)
u
h
(
x
(13.6)
where
I
denotes the set of nodes of
S
that do not belong to the Dirichlet boundary,
J
denotes the set of nodes of
S
that belong to the Dirichlet boundary and where
φ
i
are
