We begin by generating two scalar volume datasets based on the
invariants described in Section 2. The first scalar volume dataset (
) is
formed by calculating the trace (
) of the tensor matrix for each
voxel of the diffusion tensor volume. It provides a single number that
characterizes the total diffusivity at each voxel within the
sample. Higher values signify greater total diffusion irrespective of
directionality in the region represented by a particular voxel. A
slice from this volume can be seen in Figure 6 (left).
The second scalar volume dataset (
) is formed by
calculating
invariants for each voxel and combining them
into
. It provides a measure of the magnitude of the anisotropy
within the volume. Higher values identify regions of greater spatial
anisotropy in the diffusion properties. A slice from the second
scalar volume is presented in Figure 6 (right). The measure
does not by definition distinguish between linear and planar
anisotropy. This is sufficient for our current study since the brain
does not contain measurable regions with planar diffusion anisotropy. We
therefore only need two scalar volumes in order to segment the DT
dataset.
We then utilize level set methods to extract smoothed models from the two derived scalar volumes. Our level set segmentation approach consists of defining a set of suitable pre-processing techniques for initialization and selecting/tuning different feature-extracting terms in the level set equation to produce a surface deformation. Within our segmentation framework a variety of operations are available in each stage. A user must ``mix-and-match'' these operations in order to produce the desired result. We only describe those operations needed to produce the models in this paper. A more detailed description of our segmentation methods may be found in [28].
Because level set models move using gradient descent, they seek local solutions, and therefore the results are strongly dependent on the initialization, i.e., the starting position of the surface. Thus, one controls the nature of the solution by specifying an initial model from which the surface deformation process proceeds. We are able to computationally construct reasonable initial estimates directly from the input data by combining a variety of techniques.
The first step involves filtering the input data with a low-pass Gaussian
filter (
) to blur the data and thereby reduce
noise. This tends to distort shapes, but the initialization need only
be approximate. Next, the volume voxels are classified for
inclusion/exclusion in the initialization based on the filtered
values of the input data (
for
and
for
). For grey scale images, such as
those used in this paper, the classification is equivalent to high and
low thresholding operations. These operations are usually accurate to
only voxel resolution, but the deformation process will achieve
sub-voxel results. The final step before the actual level set
deformation consist of performing a set of topological or logical
operations on the voxels to ``clean up'' the initialization
surface. This allows for the removal of undesired internal and
external structures, which is extremely helpful to obtain simple
models. It includes unions or intersections of voxel sets to create
the better initializations. Specifically the topological operations
consist of connected-component analyses (e.g. flood fill) to remove
small pieces or holes from objects.
The initialization described above positions the model near the desired solution while retaining certain properties such as consistent geometry, connectivity, etc. Given this rough initial estimate, the level set surface deformation process, as described in Section 3.1, moves the surface model toward specific features in the data.
Figures 6 and 6 present two models that we extracted
from DT-MRI volume datasets using our techniques.
Figure 6 contains segmentations from volume
, the
measure of total diffusivity. The image in the first row shows a marching cubes
iso-surface using an iso-value of 7.5. In the bottom we have
extracted just the ventricles from
. This is accomplished
by creating an initial model with a flood-fill operation inside the ventricle
structure shown in the middle image. This identified the connected
voxels with value of 7.0 or greater. The initial model was then refined and smoothed
with the level set method described in Section 3,
using a
value of 0.2.
Figure 6 again provides the comparison between direct
iso-surfacing and and level set model, but on the volume
. The image in the top-left corner is a marching cubes
iso-surface using an iso-value of 1.3. There is significant
high-frequency noise and features in this dataset. The challenge here
was to isolate coherent regions of high anisotropic diffusion. We
applied our segmentation approach to the dataset and worked with
neuroscientists from LA Childrens Hospital, City of Hope Hospital and
Caltech to identify meaningful anatomical structures. We applied our
approach using a variety of parameter values, and presented our
results to them, asking them to pick the model that they felt best
represented the structures of the brain. Figure 6
contains three models extracted from
at different
values of smoothing parameter
used during segmentation. Since
we were not looking for a single connected structure in this volume,
we did not use a seeded flood-fill for initialization. Instead we
initialized the deformation process with an iso-surface of value 1.3.
This was followed by a level set deformation using a
value of
0.2. The result of this segmentation is presented on the bottom-left side of
Figure 6. The top-right side of this figure presents a
model extracted from
using an initial iso-surface of
value 1.4 and a
value of 0.5. The result chosen as the
``best'' by our scientific/medical collaborators is presented on the
bottom-right side of Figure 6. This model is produced with an
initial iso-surface of 1.3 and a
value of 0.4. Our
collaborators were able to identify structures of high diffusivity in
this model, for example the corpus callosum, the internal capsul, the
optical nerve tracks, and other white matter regions.
We can also bring together the two models extracted from datasets
and
into a single image. Figure
6 demonstrates that we are able to isolate
different structures in the brain and show their proper spatial
inter-relationship. For example, it can be seen that the corpus
callosum lies directly on top of the ventricles, and that the white
matter fans out from both sides of the ventricles.
Finally, to verify the validity of our approach we applied it to the second
data set of a different volunteer. This data set has 20 slices of the 256x256
resolution. We generated the anisotropy measure volume
and performed the level-set model extraction using
the same iso-values and smoothing parameters as for
.
The results are shown in Figure 6.
Figure 4
![]() |
Figure 4. Segmentation using isotropic measure
for the first DT-MRI dataset. The first row is the marching cubes
iso-surface with 7.5.iso-value. The second row is the result of
flood-fill algorithm applied to the same volume and used for level set
initialization. The third row is the final level set model.
Figure 5
![]() |
Figure 5. Model segmentation from volume
. Top
left image is an iso-surface of value 1.3, used for initialization of
the level set. Clockwise, are the results of level set development
with corresponding
values of 0.2, 0.4 and 0.5.
Figure 6
![]()
Figure 6. Combined model of ventricles and (semi-transparent)
anisotropic regions: rear, exploded view (left), bottom view (right),
side view (bottom). Note how model of ventricles extracted from
isotropic measure dataset
![]() ![]() |
Figure 7
![]() |
Figure 7. Segmentation using anisotropic measure
from the second DT-MRI dataset. The first row is the marching cubes
iso-surface with iso-value 1.3. The second row is the result of
flood-fill algorithm applied to the volume and used for level set
initialization. The third is the final level set model.