Biomedical Engineering Reference
In-Depth Information
The self-intersections that may happen during evolution are easily resolved
by the T-Surfaces model. Thus, we can preserve the topology of
S
. However,
there is no correspondence between the points of
S
and the points of its
n
-offset
because the reparameterization depends on projection of the surface over the grid.
This is a disadvantage of this method compared with other offset generation
approaches [31].
The
n
-offset is smoother than the initial surface
S
due to the elastic forces
given by Eq. (3). The evolution can be seen as a curvature diffusion process in
which the velocity of each surface point depends on the surface curvature. This
kind of surface evolution has been explored in the context of implicit deformable
models (
Level Sets
) and applied for shape recovery and mesh generation problems
[30]. In Section 6.5 we will analyze the application of
n
-offsets for geometry
extraction.
6.
EXPERIMENTAL RESULTS
6.1. Noisy Images
The first point that will be demonstrated is the utility of image diffusion
methods in our work. We take a synthetic 150
×
150
×
150 image volume composed
of a sphere with radius 30 and an ellipsoid with axes 45, 60, and 30 inside a uniform
noise specified by the image intensity range 0-150.
Figure 8 shows the result for steps (1)-(4) from Section 5, applied to this
volume after gaussian diffusion (Figure 8a), and anisotropic diffusion (Figure 8d),
defined by the following equation:
∂I
∂t
=
div
∇
I
,
/K
]
2
(8)
1+[
∇
I
where the threshold
K
can be determined by a gradient magnitude histogram. In
this example,
K
was set to 300, and the number of iterations of the used numerical
scheme [32] to solve this equation was set to 4.
Figures 8b and 8e show the cross-section corresponding to slice 40. We ob-
serve that with anisotropic diffusion (Figure 8e) the result is closer to the boundary
than with the gaussian one (Figure 8b).
Also, the final result is more precise when pre-processing with anisotropic
diffusion (Figure 8f). This is expected because, according to Appendix A, Eq. (8)
enables one to blur small discontinuities (gradient magnitude below
K
) as well as
to enhance edges (gradient magnitude above
K
).
Another point becomes clear in this example: the topological abilities of T-
Surfaces enable correcting the defects observed in the surface extracted through
steps (1)-(4). We observed that, after a few iterations, the method gives two closed
components. Thus, the reconstruction is better.
