Biomedical Engineering Reference
In-Depth Information
Figure 8.16:
Segmentation using anisotropic measure
V
2
from the second DT-
MRI dataset. (left) Marching Cubes isosurface with iso-value 1.3. (middle) Result
of flood-fill algorithm applied to the volume and used for initialization. (right)
Final level set model.
8.6 Direct Estimation of Surfaces in
Tomographic Data
The radon transform is invertible (albeit, marginally so) when the measured
data consists of a sufficient number of good quality, properly spaced projections
[73]. However, for many applications the number of realizable projections is
insufficient, and direct grayscale reconstructions are susceptible to artifacts.
We will refer to such problems as
underconstrained
tomographic problems.
Cases of underconstrained tomographic problems usually fall into one of two
classes. The first class is where the measuring device produces a relatively dense
set of projections (i.e. adequately spaced) that do not span a full 180
◦
. In these
cases, the sinogram contains regions without measured data. Considering the
radon transform in the Fourier domain, these missing regions of the sinogram
correspond to a transform with angular wedges (pie slices) that are null, making
the transform noninvertible. We assume that these missing regions are large
enough to preclude any straightforward interpolation in the frequency domain.
The second class of incomplete tomographic problems are those that consist
of an insufficient number of widely spaced projections. We assume that these
sparse samples of the sinogram space are well distributed over a wide range of
angles. For this discussion the precise spacing is not important. This problem
