Figure 1.
Example of whole brain fiber tractography and fiber tract clustering.
Fiber tractography is performed to generate streamlines that approximate the underlying axonal pathways of the white-matter architecture (left). Tracts are color-coded according to their orientation with red = left-right, green = anterior-posterior and blue = inferior-superior. Clustering methods can be used to cluster the fiber tracts and to group similar tracts into fiber bundles or set of tracts (right). By employing a white matter atlas that consists of several white matter bundles (middle), the automatic extraction can be improved to retrieve anatomically correct fiber bundles.
Figure 2.
Overview of the cluster analysis technique.
The size of the whole brain tractography dataset is reduced by extracting a random sample (1). For this reduced random sample, similarity matrices and Local Outlier Factors (LOFs) are computed, which are subsequently used during the clustering process (2). The reduced random sample is then automatically partitioned into a user defined number of N independent partitions (3). A first-pass partial clustering as well as outlier elimination is performed for each partition (4) before resulting clusters of all partitions are joined together (5). Resulting prototype clusters are generated during the second pass clustering (6). Outliers are then reassigned to the nearest prototype cluster in order to obtain the final clustering and the hierarchical cluster tree (7). During the last step, remaining tracts that were not part of the random sample in (1) are finally assigned to the nearest prototype cluster (8). The last two steps (step 7 and 8) are only performed for tracts that are sufficiently similar to a prototype cluster. To integrate anatomical information into the cluster analysis, a white matter atlas can be used as input during steps 4 and 6–8 (for details see section “Integration of anatomical information into the cluster analysis”).
Figure 3.
Determination of the number of cluster representatives.
To determine the number of representatives for a cluster
we use a two stage approach. As long as the number of tracts
in cluster
is smaller than
,
is selected in dependency of
using either a linear or a nonlinear function (stage 1). If
is larger than
,
is set to a constant value (stage 2).
Figure 4.
Color-coded visualization of Local Outlier Factors (LOFs) for a set of points in 2D.
For the points, LOFs have been computed that indicate how outlying the points are compared to their k-most similar tracts. The points are color-coded with the values of their respective LOFs, whilst yellow denotes low and red high LOF values.
Figure 5.
Influence of Local Outlier Factors on intra-cluster distances.
Given one cluster and the set of tracts
, the influence of LOFs on the intra-cluster distances between
and the exemplary tracts
is unveiled. In the example, the LOF of
,
is approximately one, the LOF of
is slightly increased and the LOF of
is considerably elevated. Since a reciprocal relation is used for the computation of intra-cluster distances compared to inter-cluster distances, high LOFs result in reduced distances between tracts – an attraction effect. Therefore, the LOF-corrected distance between
,
is considerably reduced, while the correction only slightly reduces the distance between
,
. Since
,
are not outlying (LOF
), the LOF correction has almost no effect on the distance between
,
.
Figure 6.
Influence of Local Outlier Factors on inter-cluster distances.
Given clusters –
and the shortest distances
,
,
between the clusters, LOF-related effects on the inter-cluster distances are emphasized. Due to the cluster's surroundings, the LOFs for tracts in
and
are approximately 1, whereas the LOF for tracts in the solitary clusters
,
is increased and higher than 1. Hence, the LOF-based correction has almost no effect on the distance between clusters
,
. The distance between
and
will be increased due to
's high LOF. This repulsion effect is even more pronounced between clusters
and
, because tracts in both clusters have high LOF values.
Figure 7.
Assignment and reassignment of tracts in CATSER.
Given the prototype clusters ,
, the solitary cluster
and the carefully selected subset of representatives
, the closest cluster to
is
since
. The representatives with the shortest distance are denoted as
and
.
is then (re-) assigned to the closest cluster
only if the distance
is smaller than
, whilst
is the standard deviation of the distances between all representatives in
.
Figure 8.
Effects of different weighting factors.
The weighting factors guide the clustering by modulating the distance between the clusters and
according to their anatomical correspondence in the atlas. While a weighting factor
has no effect, a weighting factor
will decrease the cluster distance (attraction). Contrary, a weighting factor
will increase the distance between the clusters (repulsion).
Figure 9.
The figure shows a single fiber tract that traverses through two classes of an atlas.
A single fiber tract (yellow) traverses through the inferior fronto-occipital fasciculus (IFO, in blue) and the forceps major (Fmaj, in red). The remaining atlas classes that share no spatial volume with the tract are displayed in grey. The class membership of the tract for the IFO is and for the Fmaj
.
Figure 10.
The four cases that determine the weighting factor for the atlas guidance.
The figure shows the four cases that determine the weighting factor. In order to present and visualize the four cases, the classes of the atlas that correspond to the left and right cingulum bundle are shown in pink and cyan along with two tracts that represent two clusters (shown in red and blue).
Figure 11.
Example for an atlas class in the probabilistic white matter atlas.
The maximum intensity projection of the probabilities is shown as a pseudo color image for the gyrus part of the left cingulum (CGC). Regions with high probability (
) are colored in red, while regions with low probability (
) are shown in blue.
Figure 12.
Volume renderings for a selection of fiber bundles defined in the white matter atlas.
From left to right the bundles are: anterior thalamic radiation (ATR), cortico-spinal tract (CST
), the forceps minor (Fmin), gyrus part of the right cingulum (CGC
), hippocampal part of the right cingulum (CGH
), forceps major (Fmaj), inferior fronto-occipital fasciculus (IFO
), temporal part of the superior longitudinal fasciculus (SLFt
), uncinate fasciculus (UNC
).
Table 1.
Parameters for the three different outlier elimination strategies.
Figure 13.
Atlas-guided clustering results for one dataset, clustered with the combined distance measure.
In the top row all bundles are presented. Fiber bundles are shown for the left hemisphere (left image), bundles that cross the hemispheres (middle image) and bundles of the right hemisphere (right image). Different fiber bundles are displayed in distinct colors. In the middle image a tracking error is present that resulted in a fiber bundle connecting the prefrontal lobe and the corpus callosum (green bundle, marked with an arrow). The bottom row shows the clusters that correspond to the atlas classes that are shown in Figure 12.
Figure 14.
Spatial agreement of clustered fiber bundles and atlas classes for the combined distance measure.
Using the atlas-matched fiber bundles of the 46 datasets that were clustered with the three different methods (atlas-guided CATSER, CATSER, HAC), we determined the average spatial agreement between fiber bundles and atlas class. The results for each bundle and each clustering technique are shown above.
Figure 15.
Spatial agreement of clustered fiber bundles and atlas classes for the Hausdorff distance measure.
Using the atlas-matched fiber bundles of the 46 datasets that were clustered with the three different methods (atlas-guided CATSER, CATSER, HAC), we determined the average spatial agreement between fiber bundles and atlas class. The results for each bundle and each clustering technique are shown above.
Figure 16.
Inferior fronto-occipital fasciculus (IFO) of one dataset clustered with different methods.
The IFO of one dataset is shown, which was clustered with all three methods (atlas-guided CATSER, CATSER, HAC) and both similarity measures (CD, HD). Bundles are shown in the atlas space and are superimposed with the corresponding class of the atlas (in semi-transparent blue). The spatial agreement for the bundles clustered using CD is (from left to right): 0.58; 0.62; 0.69 and for the bundles clustered using HD: 0.75; 0.67; 0.73.
Figure 17.
Left uncinate fasciculus (UNC) of two datasets clustered with atlas-guided CATSER clustering.
Both bundles reside in the atlas space and are superimposed with the corresponding class of the atlas (in semi-transparent green). While the bundle on the left follow the anticipated trajectory of the UNC, the bundle on the right side contains additional tracts that leave the bundle and follow other paths. The spatial agreement for the left bundle is s and for the right bundle s
.
Figure 18.
Temporal part of the left superior longitudinal fasciculus (SLFt) for 15 datasets.
The datasets reside in the atlas space and were clustered with the atlas-guided CATSER clustering. The spatial agreement for the bundles in the top row is (from left to right): 0.62; 0.65; 0.67; 0.61; 0.68, in the middle row: 0.67; 0.60; 0.82; 0.64; 0.60 and in the bottom row: 0.64; 0.61; 0.74; 0.65; 0.69. In certain images, tracking errors have been observed that connect the SLFt to the external capsule (see arrows).
Figure 19.
Results of the experiments that were conducted to demonstrate the effects of outlier elimination.
One dataset was segmented into a set of bundles that are shown in distinct colors in the top left corner. Using the segmented fiber tracts, centroids of tracts were computed and used as a gold standard. Each point in the gold standard image (top right) represents the centroid of one fiber tract. The groups of centroids (gold standard) are show in the same color as their corresponding fiber bundles. All data is presented in a tableau view that shows the data in three different orientations: top left image = anterior-posterior view, bottom left image = left-right view, top right image = superior-inferior view. The centroids of the gold standard dataset were clustered using the tract centroid distance as similarity measure. Varying levels of artificial noise (0%, 33%, 66% and 99% additional noise) were used and different outlier elimination strategies were tested. The results for the clustering are displayed in the table (D1–D4).
Figure 20.
Performance analysis of the cluster analysis for the similarity measures CD (left) and HD (right).
The overall computation time is shown in the top row and the achieved speedup below. For the analysis, the whole clustering process was separated into three distinct parts and analyzed separately (see section “Performance analysis” in Materials and methods). In all plots, each part is highlighted in distinct colors, whereas blue denotes the computation of the distance matrix (part 1: “DM computation”), green the clustering (part 2: “CATSER”) and red the labeling of remaining tracts (part 3: “Assign tracts”). For both distance measures, we observed that part 1 and part 3 were the most time consuming stages during the clustering. By utilizing multiple CPUs during both parts, a high speedup was achieved and the computation time was significantly reduced. While the speedup is nearly optimal for HD, the speedup for CD is slightly reduced. However, the overall processing speed for CD is still 10 times faster.
Figure 21.
Overview of the three fundamental stages of the clustering and their sequential and parallel sub-stages.
The figure depicts the way how the data is processed during the stages of the clustering process. It is illustrated, which parts of the clustering are performed either in serial or in parallel and how the data is distributed across multiple threads.