Biomedical Engineering Reference
In-Depth Information
using the minimum curvature,
M
=
AbsMin(
k
1
,
k
2
) for preserving thin, tubular
structures, which otherwise have a tendency to
pinch off
under mean curvature
smoothing.
In previous work [41], the authors have proposed a weighted sum of mean
curvatures that emphasizes the minimum curvature, but incorporates a smooth
transition between different surface regions, avoiding the discontinuities (in the
derivative of motion) associated with a strict minimum. The
weighted curvature
is
k
1
k
1
+
k
2
k
2
k
1
+
k
2
2
HK
D
2
W
=
k
2
+
k
1
=
,
(8.48)
k
1
+
k
2
is the deviation from flatness [99].
For an implicit surface, the shape matrix [100] is the derivative of the normal
map projected onto the tangent plane of the surface. If we let the normal map
be
n
=∇
φ/
|∇
φ
|
, the derivative of this is the 3
×
3 matrix
where
D
=
∂
n
∂
x
T
∂
n
∂
y
∂
n
∂
z
N
=
(8.49)
.
The projection of this derivative matrix onto the tangent plane gives the shape
matrix
B
=
N
(
I
−
n
⊗
n
), where
⊗
is the exterior product and
I
is the 3
×
3
identity matrix. The eigenvalues of the matrix
B
are
k
1
,
k
2
and zero, and the
eigenvectors are the principle directions and the normal, respectively. Because
the third eigenvalue is zero, we can compute
k
1
,
k
2
, and various differential
invariants directly from the invariants of
B
. Thus the weighted-curvature flow is
computing from
B
using the identities
D
=||
B
||
2
,
H
=
Tr(B)
/
2, and
K
=
2
H
2
−
D
2
/
2. The choice of numerical methods for computing
B
is discussed in the
following section.
8.6.4 Implementation
The level set equations are solved by finite differences on a discrete grid, i.e.
a volume. This raises several important issues in the implementation. These
issues are the choice of numerical approximations to the PDE, efficient and
accurate schemes for representing the volume, and mechanisms for computing
the sinogram-based deformation in Eq. (8.47).
