=Paper=
{{Paper
|id=None
|storemode=property
|title=Evaluation of Local Filter Approaches for Diffusion Tensor-Based Fiber Tracking
|pdfUrl=https://ceur-ws.org/Vol-715/bvm2011_73.pdf
|volume=Vol-715
}}
==Evaluation of Local Filter Approaches for Diffusion Tensor-Based Fiber Tracking==
https://ceur-ws.org/Vol-715/bvm2011_73.pdf
Evaluation of Local Filter Approaches for
Diffusion Tensor-Based Fiber Tracking
D. Merhof1 , M. Buchfelder2 , C. Nimsky3
1
Visual Computing, University Konstanz, Konstanz
2
Department of Neurosurgery, University of Erlangen-Nuremberg, Erlangen
3
Department of Neurosurgery, University of Marburg, Marburg
dorit.merhof@uni-konstanz.de
Abstract. Diffusion tensor imaging (DTI) allows investigating white
matter structures in vivo which is of great interest for various applications
in neuroanatomy and neurosurgery. In order to reconstruct white matter
tracts from DTI data, fiber tracking approaches are used which are com-
monly based on streamline propagation techniques. In order to provide
more reliable tracking results, local regularization filters were presented
previously, which can be applied during the fiber tracking process and
move forward as the fibers are constructed. In this work, Gaussian and
least-squares local filters are evaluated and compared against higher or-
der integration schemes. The results provide a more differentiated view
on the prospects and limitations of local regularization filters.
1 Introduction
Diffusion tensor imaging (DTI) is a magnetic resonance acquisition technique
which provides information about the location of white matter tracts within the
human brain in vivo. DTI measures the diffusion of water, which is anisotropic in
tissue with a high degree of directional organization such as the white matter of
the human brain. In this way, information about the location of neuronal fibers
is provided which is of great interest for various applications in neuroanatomy
and neurosurgery.
The reconstruction of white matter tracts from DTI data is commonly solved
by tracking algorithms [1, 2] which use streamline techniques known from flow
visualization. Respective algorithms propagate lines through the tensor field,
based on the local diffusion characteristics which reflect the course of the under-
lying white matter tracts. The resulting fiber bundles approximate the location
of neuronal fibers and are therefore a powerful tool for white matter reconstruc-
tion.
In recent years, local regularization techniques have been developed [3] with
the aim to provide more reliable tracking results by preserving directional con-
sistency and at the same time cross noisy regions and gaps in the data.
In this work, local filter approaches are evaluated and compared against
higher order integration schemes, which also show the desirable feature of in-
creased stability and tracking accuracy.
Diffusion Tensor-Based Fiber Tracking 355
2 Material and Methods
2.1 Image Data
The DTI datasets used in this work were acquired on a Siemens MR Magne-
tom Sonata Maestro Class 1.5 Tesla scanner using a field of view of 240mm
resulting in a voxel size of 1.875 ×1.875×1.9 mm3 . For each of the six diffusion
weighted datasets (gradient directions (±1,1,0), (±1,0,1) and (0,1,±1)) and the
reference dataset, sixty slices with no intersection gap and an acquisition matrix
of 128×128 pixels were measured.
2.2 Fiber Tracking
Fiber tracking algorithms based on streamline propagation have several steps in
common such as seed point selection, fiber propagation and termination strate-
gies [4]. Starting from predefined seed points, streamline integration is used to
propagate the fiber until a termination criterion is reached. For this purpose, a
threshold based on fractional anisotropy (FA) [1, 2] is commonly applied to stop
fiber propagation if the anisotropy of diffusion decreases.
In this work, fiber tracts were computed using first, second and fourth order
Runge-Kutta integration and trilinear tensor interpolation. A constant tracking
step size of 0.5 mm was employed. FA was used as termination threshold for
fiber propagation, and a length threshold of a minimum of 100 tracking steps
was used in order to exclude short fibers from the tracking result.
2.3 Local Regularization Filters
Local regularization filters [3] are applied during the fiber tracking process and
move forward as the fibers are constructed. In order to put more weight on the
tensors in the direction of the tracked fiber, a moving filter window of ellipsoidal
shape centered around the current end point of the fiber is employed (Fig. 1).
The size of the filter window is adjusted depending on the reliability of the local
tensor data.
During fiber tracking, a local regularization approach is applied to the tensors
in the filter window in order to approximate the local tensor. The resulting
regularized tensor is used for tract propagation.
In order to weight the tensors within the filter window, a Gaussian weighting
function G(s−p; D p ) is introduced [3], which assigns a weight to a point position
Fig. 1. Local filter window domain (yellow). The filtered tensor
at the center p is computed from samples at different locations s
within the filter window.
356 Merhof, Buchfelder & Nimsky
s in the vicinity of the center point p. The weighting function has the effect that
tensors located close to the center of the filter window have a higher impact on the
regularization result than tensors at the border of the filter window. The shape
of the Gaussian filter window is adapted from the tensor ellipsoid associated
with the local tensor D p at the center point p.
In order to approximate the regularized tensor D̄ p at position p using the
Gaussian filter window, the weighted average of the tensors within the filter
volume is computed.
Moving least-squares filter The local Gaussian filtering scheme is extended
by a least-squares approximation within the filter window [3]. Least-squares
approximation aims at minimizing an error functional E which measures the
quality of a polynomial fit of the data
∫ [ ]2
1
E(p) = G(s − p; D p ) · F (s − p) − D(s) ds3 (1)
V V
In order to solve this equation, a low degree polynomial F has to be found which
best approximates the data D(s) at different locations s within the filter window
centered around p. The quality of fit in a least-squares sense is assessed by the
functional E, which must be minimized. The impact of a point location on the
functional E is controlled by the filter function G(s − p; D p ).
For fiber tracking, the original, noisy tensor D p is approximated with the
polynomial fit F at position p.
3 Results
Fiber tracking was performed both with and without local filters. In Figure 2,
the result of fiber tracking without local filtering (left), with a Gaussian local
Fig. 2. Pyramidal tract in a healthy volunteer reconstructed with streamline integra-
tion (RK-1) without filtering (left), with a Gaussian local regularization filter (middle),
and with a moving least-squares filter (right).
Diffusion Tensor-Based Fiber Tracking 357
Table 1. Accuracy and computing time for oriented filters. Time: Computing time
for different integration schemes without filtering (left) and with a Gaussian local reg-
ularization filter and a moving least-squares filter (right) for a tracking of the whole
brain. Divergence: Average divergence between the original fiber and the reverse fiber
after 50 integration steps.
Time Divergence Time Divergence
RK-1 none 4 sec 8.61 mm RK-1 none 4 sec 8.61 mm
RK-2 none 11 sec 1.46 mm RK-1 Gauss 242 sec 5.91 mm
RK-4 none 18 sec 1.27 mm RK-1 MLS 464 sec 5.88 mm
regularization filter (middle) and with a moving least-squares filter (right) is
shown. In each case, streamline propagation using Euler integration (RK-1) was
employed. For both local filtering approaches, the moving filter windows were
scaled with a factor σ = 50 and tensor samples within the filter window were
taken at a distance of 1 mm for each dimension in 3D space. In the case of the
least-squares filter, a polynomial of degree 2 was applied for the approximation.
In order to assess the quality of local filtering, both techniques were compared
to fiber-tracking results without filtering. For this purpose, streamlines were
traced until a termination criterion was reached, and the final sampling point
of the fiber was used as the starting point for a reverse fiber, running in the
opposite direction. Depending on the quality of the tracking approach applied,
the reverse fiber closely follows the original fiber for a certain number of steps,
before it diverges from the course of the original fiber. As a consequence, the
distance between corresponding sampling points of the original fiber and the
reverse fiber indicates the quality of the respective tracking approach. In order
Fig. 3. Divergence between original fiber and reverse fiber for 50 integration steps,
averaged for all fibers in a tracking of the whole brain. Left: Comparison of different
integration schemes (RK-1, RK-2 and RK-4). Right: Comparison of RK-1 integration
without local filter (RK-1 none), with local Gaussian filter (RK-1 Gauss) and with
moving least-squares filter (RK-1 MLS).
358 Merhof, Buchfelder & Nimsky
to obtain significant results, approximately 300.000 streamlines within the whole
brain and associated reverse fibers were computed and the results were averaged.
The diagram in Figure 3 (right) shows the averaged divergence between original
and reverse fibers for streamline propagation without filtering, with a Gaussian
local regularization filter and with a moving least-squares filter. In Table 1, the
divergence after 50 integration steps as well as the computing times on a PC
equipped with an Intel Pentium 4, 3.4 GHz, and 2 GB RAM are provided.
4 Discussion
From the diagram in Figure 3 (right), it can be concluded that the Gaussian local
regularization filter provides similar accuracy to that of the moving least-squares
filter, i.e. the computational effort added by the least-squares approximation has
only a very slight impact on the accuracy. Taking into account the computing
time in Table 1 (right), which is much higher in the case of least-squares ap-
proximation, it can be concluded that the Gaussian local regularization filter is
much more suitable for practical application.
In principle, the local filters improve the tracking accuracy of the respective
integration scheme, as shown in Figure 3 (right) for streamline tracking based
on first order Runge-Kutta integration. However, for a higher-order integration
scheme, such as second or fourth order Runge-Kutta in Figure 3 (left), even
greater accuracy is obtained compared to the lower-order integration in combi-
nation with local filters. For this reason, it is more feasible to apply higher order
integration instead of a RK-1 integration scheme combined with local filters.
5 Conclusion
In this work, local regularization filters were evaluated and compared against
higher order integration schemes. For both the Gaussian local filter and the
least-squares local filter, significantly increased computing times were encoun-
tered compared to higher order integration schemes. In terms of tracking accu-
racy, higher order integration schemes outperform RK-1 integration with local
regularization filters. In summary, it can be concluded that higher order inte-
gration is preferable to local regularization filters.
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Diffusion Tensor MRI. In: Proc IEEE Vis; 2002. p. 387–394.
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