Biomedical Engineering Reference
In-Depth Information
tion of T-Surfaces (an implicitly defined surface inside the object) [10]. For
instance, when segmenting an object with approximately 7000 triangular ele-
ments immersed in a 128
×
128
×
25 synthetic volume, our method takes 33
seconds against the 9 minutes and 12 seconds with the traditional method (see
http://virtual01.lncc.br/˜rodrigo/links/tese/teste2.html). In this case, the grid res-
olution was 6
×
6
×
2. Besides, the computational complexity of our method can
be compared with that of using just a T-Surfaces initialized through the traditional
method. Let us suppose that the inner volume of a structure of interest encom-
passes
C
tetrahedrons. In this case, the T-Surfaces has to pass over
O
(
C
) simplices
to get the target surface when initialized by the implicit surface. Therefore, if
C
is
O
(
N
), where
N
is the number of simplices of the domain subdivision, then the
computational complexity of both our segmentation approach and the traditional
T-Surfaces is the same.
The proposed method is adaptive in the sense that we can increase the T-
Surfaces grid resolution where it is necessary. As the T-Surfaces grid controls the
density of the polygonal surfaces obtained, the number of triangular elements gets
larger inside these regions. That density increasing is not due to boundary details
but because the outer scale corresponding to the separation between the objects is
too fine (like in Figure 7). This is a disadvantage of our approach. Such a problem
would be avoided if we could define significant areas along the surfaces, and then
apply the refinement only in the regions around them. However, it is difficult to
automatically perform this task.
As a consequence, polygonal meshes generated by the T-Surface method may
not be efficient for some further application. For instance, for finite-element pur-
poses, small triangles must be removed. Consequently, filtering mesh procedures
must be applied in order to improve the surface. Mesh smoothing and denois-
ing filtering methods, like that one proposed in [35], could also be useful in this
post-processing step.
We tested the precision of our approach when segmenting a sphere immersed
on a uniform noise specified by the image intensity range 0-150. We found a mean
error of 1.58 (pixels) with standard deviation of 2.49 for a 5
×
5
×
5 grid resolution,
which we consider acceptable in this case. This error is due to the projection of
T-Surfaces as well as the image noise. Following [9, 10], when T-Surfaces stops,
we can discard the grid and evolve the model without it to avoid errors due to the
projections. However, for noisy images, the convergence of deformable models
to the boundaries is poor due to the non-convexity of the image energy [26].
Anisotropic diffusion applied to 3D images can improve the result, as already
stated in Section 6.3. Vector diffusion techniques can also be applied to generate an
image force field that improves the convergence of the model toward the boundary
when the grid is turned off (see Appendix A).
We will now consider the following question: Would it be possible to im-
plement the reconstruction method, the user interaction approach, and the offset
generation through Level Sets? The relevance of it will be not only the possibility
