Biomedical Engineering Reference
In-Depth Information
the T-Surfaces framework is applied for offset generation and its utility for Dual
approaches is considered [18].
The following section presents some background on deformable surfaces and
related works. The T-Surfaces framework is developed in Section 2.1. The key
points behind utilization of isosurfaces to initialize T-Surfaces are considered in
Section 2.2. Sections 3 and 4 discuss the isosurface generation methods inside
the T-Surfaces context. We present our in Section 5. The experimental results are
given in Section 6. Finally, the proposed method is compared with related ones
and we discuss new perspectives for our method through Level Sets and vector
diffusion approaches (Section 7). Conclusions are given in Section 8. Appendix
A focuses on diffusion methods for image processing.
2.
BACKGROUND AND PREVIOUS WORKS
The use of isosurface methods in initialization of deformable surfaces requires
special considerations to guarantee efficiency [19, 20, 17, 21].
To explain the concepts, let us consider the following balloon-like model for
closed surfaces [3]:
v
:
+
×
[0
,
1]
×
[0
,
1]
→
3
;
v
(
t, r, s
)=(
v
1
(
t, r, s
)
,v
2
(
t, r, s
)
,v
3
(
t, r, s
)) ;
∂
2
v
∂s
2
−
ω
01
∂
2
v
∂r
2
∂
4
v
∂r
2
∂s
2
+
ω
20
∂
4
v
∂s
4
∂
4
v
∂r
4
∂v
∂t
−
ω
10
+2
ω
11
+
ω
02
=
F
(
v
)
−
kn
(
v
);
(1)
Initial Estimation :
v
(0
,r,s
)=
v
0
(
r, s
)
where
n
(
v
) is the normal (unitary) field over the surface
v
,
F
is the image force
field (may be normalized), and
k
is a force scale factor. The parameters
ω
ij
control
the smoothness and flexibility of the model.
By using the internal pressure force (
kn
(
v
)), the model behaves as a balloon-
like object that is inflated passing over regions in which the external force is
weaker. Consequently, the model becomes less sensitive to initialization, which
is an advantage over more traditional active models [22, 6].
For shape-recovering applications, numerical methods must be considered
in order to solve Eq. (1). If Finite Differences are used, the continuous surface
v
(
r, s
) is discretized, generating a polygonal mesh. During mesh evolution, self-
intersections must be avoided. In general, this is a challenge if we aim to use
isosurfaces to define the initial estimation
v
0
(
r, s
) in Eq. (1). This happens be-
cause isosurfaces are in general both not smooth and irregular for 3D medical
images. Also, they may be far from the target in some points. Figure 1 depicts a
bidimensional example with such difficulties.
Traditional deformable models [6, 3, 1], including the one defined by the
Eq. (1), cannot efficiently deal with self-intersections. This is due to the non-local
