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1
Geodesic Fiber Tracking in White Matter using
Activation Function
Temesgen Bihonegn1, Sumit Kaushik1, Avinash Bansal1, Lubom´
ır Vojt´
ıˇ
sek2, and Jan Slov´
ak1
1Department of Mathematics and Statistics, Masaryk University,Czech Republic
2Brain and Mind Research Programme, Central European Institute of Technology, Masaryk University, Czech
Republic
Abstract—The geodesic ray-tracing method has shown its
effectiveness for the reconstruction of fibers in white matter
structure. It provides multiple solutions and is robust to noise
and curvatures of fibers. The choice of the metric tensor has a
significant impact on the outcome of this method. The existing
metrics are not sufficient in the construction of white matter
tracts as geodesics. We propose a way to choose the appropriate
conformal class of metrics where the metric gets scaled according
to tensor anisotropy. We used the logistic functions, which are
commonly used in statistics as cumulative distribution functions.
To prevent deviation of geodesics from the actual paths, we
propose a hybrid ray-tracing approach. Furthermore, we em-
ploy diagonal projection of 4th order tensor to perform fiber
tracking in crossing regions. Results from synthetic and real data
experiments elucidate the method.
Keywords
Diffusion Tensor Imaging, Ray-tracing, Metric Tensor, Fiber
Tracking, Geodesic Equations
I. INTRODUCTION
DTI (Diffusion Tensor Imaging) has become a clinical
standard for studying and diagnosing neuro diseases. It is
the non-invasive approach to obtain information on the neural
architecture. Fiber tracking methods broadly comprise of two
classes, probabilistic [1] [2] [3], and deterministic [4] [5] [6]
[7] [8]. Probabilistic fiber tracking traverses all possible trajec-
tories and provides a simulated distribution of the fiber tracts,
which can be used in brain connectivity studies. Deterministic
tractography methods are primarily based upon streamline
algorithms where the local tract direction is defined by the
principal eigenvector of the diffusion tensor. These approaches
have been used to construct white matter anatomical connec-
tions in the human brain. In this work, we are considering the
latter class.
Earlier classical streamline based techniques [9] showed
ineffectiveness in the reconstruction of highly curved fibers.
Other difficulties with these methods appear in the isotropic
(slightly anisotropic) regions where direction information is
redundant. Apart from that, these methods are also sensitive
to noise and fiber crossings.
To overcome the problems mentioned above [10] [11] [12]
[13] propose methods based on geodesics in Riemannian
geometric space. These geodesics follow the shortest path
locally between two points lying on the manifold. This path
is optimal for the underlying actual fiber tracts. One class of
such methods is based on Hamilton-Jacobi (HJ) formalism.
These methods are sensitive to local changes and provide a
single solution.
In the works [10] [12] authors proposed to use the inverse
of diffusion tensor as the metric tensor of the geometric space.
Fuster and others [14] introduce modification of inverse metric,
called adjugate tensor, which better explained Brownian mo-
tion on Riemannian space and overcame the issue with inverse
diffusion tensor.
In [15] [16] [17] Sepasian et al comes up with multi-valued
ray-tracing method for anisotropic medium. Ray-tracing meth-
ods are based on the assumption that, locally in the medium,
a wave or particle follows a path corresponding to the least
action. Consequently, the directions of the path vary. These
methods are capable of producing multiple geodesics between
point and region in the medium.
Local variations of geodesics from underlying fibers are
taken under consideration using Euler-Lagrange equations, but
while traversing they deviate from the actual underlying path.
The conformal rescaling or adaptive Riemannian metric is
chosen for tractography in [33] and segmentation in their
subsequent work [21]. Similar to their work, authors in [34]
evaluated adjugate instead of D−1with or without sharpening.
The choice of adjugate tensor as a metric does not resolve
minimizing the Riemannian cost in all anisotropic or nearly
anisotropic regions.
In this work, the contributions are as follows:
1. We present a method to choose the appropriate conformal
class of metrics where the metric gets scaled according
to tensor anisotropy. We use the idea that the rotational
information is related to the anisotropy of the tensor, and
logistic function can be exploited to capture it. In par-
ticular, the rotational information is misleading in nearly
isotropic regions in the presence of noise. The metric
tensor is rescaled, according to this information. We
compare various scalar anisotropies under the activation
function.
2. Ray-tracing method deviates from the geodesics path.
This problem is countered by feeding back the principal
eigenvector direction of underlying interpolated tensor
to ODE solver. This also enables the hybrid ray-tracing
method to perform better in high curvature regions.
3. We employ diagonal projection of 4th order tensor to
perform fiber tracking in crossing regions. The diagonal
2
components of the flattened 4th order tensor are second
order tensors and lie in Riemannian space. We showed
that these components have potential to resolve fiber
crossings even at small angle intersections.
This paper is organized as follows. In Section II, we review
the geodesic-based fiber tracking approach. In Section III,
we introduce a modified ray-tracing method, which enables
us to find multiple geodesics by shooting rays from point
to region. We describe the use of activation function with
diagonal component approach for crossing fibers by rescaling
the classical metric, which we call β-Scaled metric tensor. We
also comment on various choices of metrics suitable for local
interpolation of tensor data. Section IV shows the results of
our tracking approach on synthetic and real brain diffusion
data.
II. BACKGROU ND
In geodesic ray-tracing, a small deviation of the geodesics
from the direction of diffusion is preferred. It makes the
geodesics robust to noise, but if this deviation is big, it needs
a sharpening of diffusion tensor [15] [18]. It can be done by
powering the tensor. However, it causes artificial increase or
decrease in volume, which is not required as the diffusion
process is physical, and diffusion quality must be preserved.
This is partially done by the normalization of the tensor. The
sharpening strategy seems to result in better tractography. For
more details, we refer the readers to [18].
The works [19] [20] explain the choice of inverse diffusion
tensor as a metric in the context of DTI. It does not work
for all cases. Another approach for modification of metric has
been indicated by Hao [21], which has a similar effect as the
adjugate metric proposed by Fuster [14]. The two approaches
are build upon the conformal rescaling of the tensor. They
use adjugate tensor with sharpening to track high curvature
fibers. The main idea to use inverse diffusion tensor as the
metric tensor is to ensure that path is shorter if diffusion is
stronger along the high anisotropic direction. This provides the
minimization of the path, in essence, which can be treated as a
geodesic. The ray-tracing method works under the assumption
to consider a bundle of the rays together and provides a multi-
valued solution. In this work, we consider a cone formed on
the base of the ellipsoid.
Sepasian et al introduced a modified ray-tracing by adjusting
the direction of geodesics based on computing Ricci curvature
tensor from the metric tensor and its derivatives [22]. It pro-
vides a measure of the degree of deviation determined by the
Riemannian metric tensor from Euclidean space. DTI model
fails in the regions where fibers are merging, intersecting, and
kissing. The second order tensor in DTI lies in the Riemannian
space, which is well studied in [20] [23] [24] [25] [26].
The geodesic methods employing the Hamilton-Jacobi equa-
tion (HJ) fail in highly curved regions comparing to fast
marching techniques. In [27], geodesics are considered as a
function of position and direction. In isotropic regions, the rays
may deviate from the actual path [18] [28] [29]. Sharpening
is helpful in these cases, as mentioned above.
During traversal, geodesic rays tend to deviate so that
there is non-uniformity in their distribution across the regions,
which causes less dense fibers. Ray density can be altered by
changing the mesh size of interpolation. Anisotropic diffusion
in Euclidean space is similar to the Brownian motion of water
molecules in an isotropic medium in Riemannian Space [14].
III. RAY-TR ACI NG V IA ACTIVATION FUNCTION
A. Initial Shooting Direction
Ray-tracing method is used to find the trajectory of the par-
ticle moving in medium. To compute the geodesics using this
technique we need the initial position and direction. In DTI
tractography, we can restrict the initial shooting directions. Let
Rbe the radius of the base and σ∈(0,1), which adjusts the
base of the cone. The different values of σprovide different
bases (σ.R) of a cone. Directions are uniformly distributed
over a spherical section of the cone, as shown in Fig 1. This
is done to restrict the shooting direction and to ensure the ray
bundle remains densely packed.
(a) elliptic cone (b) cone shooting
Fig. 1: Initial shooting direction
The values of σapproaching to 1 causes bigger pertur-
bation. The more realistic way is to ray trace from point to
region or region to region because it is not possible to know
in advance if the initial and final points are connected [15].
B. Activation Function
The logistic or activation functions are known for their com-
mon use in deep learning methods and statistics as cumulative
distribution functions. One of their special cases is sigmoid
functions, which are differentiable over real domain values
and have positive derivatives at each point. To account for the
tensors with negligible difference between the maximal (λmax)
and minimal (λmin) eigenvalues, a smooth transition function
is applied. In our case, we tested the following three functions
with equal performance.
S1(x) = tanh(x)(1a)
S2(x) = 1
1+exp(−1
2x)(1b)
S3(x) = x
√1+x2(1c)
3
Hilbert Anisotropy [31] is given by:
HA =log(λmax
λmin
),(2)
where HA ≥0,H A =0 for full isotropic tensor.
HA is a scalar measure of anisotropy and is scale-invariant
(depends on the shape not the size of the tensor). It is also
invariant to rotation and it is a dimensionless number reflecting
microscopic diffusion at the level of tissues [30]. To choose
an appropriate metric, we scale the Riemannian metric by an
activation function, which is adapted according to the inherent
anisotropy.
Let β:=Si(x)for x=HA,i=1,2,3, then the β-Scaled metric
is given by
gβ=βpD−n,(3)
where, Dis the second order diffusion tensor, p≥1,n=2.
Recall, Dbelongs to the space S+(3)of positive definite 3×3
matrices (SPD). In particular, Dand all its powers are metric
tensors.
When p<1 it decreases the Riemannian cost in the isotropic
region and increase the cost in anisotropic region. So we
choose p≥1 to minimize the cost in anisotropic region.
Beside the Hilbert anisotropy, various scalar measures exist,
which can serve as description for the degree of anisotropy
of diffusion tensor. These measures can be composed with
the above functions. They include: mean diffusivity (MD),
fractional anisotropy (FA), relative anisotropy (RA) and ge-
ometric ones: geodesics anisotropy (GA), Hilbert anisotropy
(HA) [30].
Spectral metrics allow for proper scaling of the rotational
contribution according to the anisotropy. This is achieved
by using the combination of the activation function with
anisotropy scalar measure. In Fig 2, we compare the Rieman-
nian cost while considering these anisotropy measures under
the activation function. From left to right the tensor exhibit
high-low-high spectrum of anisotropies. The interpolation of
tensors in between two extreme anisotropic tensors is obtained
using Log-Euclidean metric shown in equation (11). Under this
metric we can observe that the anisotropy is not preserved.
There is variation in eigenvalues and rotational component of
interpolated tensors.
The Riemannian cost in anisotropic direction is given by
β λ −2
max. It is observed that Riemannian cost increases as tensors
achieve high isotropy in the middle of the spectrum and
after which a smooth descend is noticed for HA case. The
other scalar measures do not give linear interpolation as
shown in [30], see Fig 3 there. HA is the only one of all
above mentioned scalar anisotropy measure keeping affine
combinations invariant.
Fig. 2: The effect of interpolation between the tensors and the
Riemannian cost from anisotropy to isotropic region. HA gives
low Riemannian cost for anisotropic and high Riemannian cost
for isotropic region.
C. Governing Equations
The trajectory of a fiber pathway is computed iteratively
from the hybrid approach, position from ODE solver, and
direction equal to the principal eigenvector direction. The
geodesic method in Riemannian manifold which is used to
compute the trajectory of the fibers from ODEs is shown
below.
Let x(τ)be a smooth and differentiable parametrized curve
in the Riemannian manifold, τ= [0,T]. The Riemannian length
is given as follows:
L(x,˙x) = ZT
0
(gαβ ˙xα˙xβ)1/2dτ(4)
The geodesic is the curve that minimizes the length (4).
The technique of the Euler-Lagrange equations for solving
variational problems is explained, e.g., in [32].
Let ˙xγand ¨xγbe the first and second derivative with respect
to τ, respectively, of the geodesic for dimension γ=1,2,3.
The geodesics are given by the following system of equations
¨xγ+
3
∑
α=1
3
∑
β=1
Γγ
αβ ˙xα˙xβ=0,(5)
where Γγ
αβ are the so called Christoffel symbol, given by
Γγ
αβ =1
2
3
∑
σ=1
gγσ ∂gβ σ
∂xα+∂gασ
∂xβ−∂gβ α
∂xσ(6)
and gβ σ denotes the matrix component of the inverse diffusion
tensor, and gγσ represents an element of the original diffusion
tensor. We compute the solution of equation (5) for the
given initial position and multiple initial directions using
4
the standard ODE solvers, such as fourth-order Runge-Kutta
method. This gives us a set of geodesics connecting the given
initial point, which we integrate until they hit the boundary.
Depending on the equations (5) and (6), we need nine symbols
per dimension, for a total of 27 symbols. However, dealing
with torsion free connections allows to exploit additional
symmetries. The initial position is user-specified and directions
are computed by forming a cone with a base of the radius.
The choice of power of βis done experimentally. In our
work, we have compared the results on synthetic data with
D−1, adjugate and β-Scaled diffusion tensor. The experiment
shows our approach works irrespective of configuration in
terms of curvature and (an)isotropy of neighboring tensors.
Based on the observation that the ODE solver’s output
direction deviates from the actual fiber path, we used the
principal eigenvector of the underlying interpolated tensor as
input for the ODE solver. While picking up the principal
vector direction, there are always possibilities of choosing
two directions. At each iteration, we need to keep track of
following the direction consistent with traversing fiber. This
hybrid approach resulted in the traversal of geodesics in the
high curvature cases and is robust to noise as well.
Algorithm 1 Hybrid Ray-Tracing
Input: Initial position (x) and direction ( ˙x)
Output: Local geodesics
1: Define the mesh size locally over the physical grid with
size m=0.1
2: Compute interpolated inverse diffusion tensor locally.
3: Find local geodesics
a Give the position and direction to ODE solver.
b Compute Christoffel symbols using Algorithm 2.
4: Take the new position and replace directions with principal
eigen vector of the underlying interpolated tensor.
5: Repeat step 2 −4 until the geodesics leave the grid.
Algorithm 2 Compute Christoffel symbol Γi
jk
Input: the diffusion tensor Dand the indices i,j,k.
Output: Christoffel symbols
1: Adapt Dto gβaccording to (3)
2: Set Γi
jk =0
3: Loop through all three dimensions, i.e m=1,2,3
4: Γi
jk =Γi
jk +1
2gβ−1(m,k).D2(j,m,i) + D2(m,i,j)−
D2(i,j,m)
5: End of loop
Here, D2 is the second order difference of the metric
tensor gβin (6).
D. Rescaling of Metric Tensor
Illustration of Fig 3comes from [14] that advocate the use
of adjugate diffusion tensor instead of the inverse of diffusion
tensor as a metric tensor. The intuitive idea is to minimize the
Riemannian cost along the trajectory.
Fig. 3: An isotropic region (left) and anisotropic region (right)
Consider two tensors whose principal eigenvalues are equal.
In D−1case, Riemannian cost (4) of (traveling along) an
infinitesimal vertical line element scales by 1\λ. For adjugate
case i.e., dD−1, where d=det(D)the Riemannian cost for
isotropic tensor is proportional to λ2(size of the shaded
circle) and for anisotropic tensor it is λ2λ3(proportional to
the size of the shaded region). This method does not work if
λ>λ2,λ3. When the area of the orthogonal cross section in
the isotropic case becomes equal (i.e., same λ2,λ3) but their
principal eigenvalues are different, adjugate tends to give the
same Riemannian cost whereas our approach scales the metric
appropriately according to the scalar anisotropy. The scaling
coefficient takes zero value for isotropic and higher values for
anisotropic cases.
HA is zero irrespective of the size of the isotropic tensor.
This leads to same evaluation of Riemann costs for any
isotropic tensor. Such scaling suggests that the diffusion of
water molecules is uniform in all directions and hence the
Riemann cost as well. In Fig 4two cases are depicted. Fig 4(a)
shows the case 1, where isotropic tensors in the intersection
region are chosen with the smaller eigenvalues. Fig 4(e)
shows the case 2 with larger eigenvalues. In both of the cases,
the metrics D−1and adjugate induces different Riemann cost.
However, in both cases, the β-Scaled metric lowers the cost of
traversing irrespective of the eigenvalues of isotropic tensors.
E. Decomposition of 4th order tensor
Tuch [35] introduced the idea to use mono-exponential
model for diffusion of water molecules in the tissues using
multiple gradient directions:
S=S0exp(−bD(g)) (7)
For anisotropic diffusion this equation (7) is linear in the log
domain, thus,
log(S) = log(S0)−bD(g)
where,
D(g) =
3
∑
i1=1
3
∑
i2=1
3
∑
i3=1···
3
∑
in=1
Di1j2i3···ingi1gi2gi3···gin
Here, Di1···inare the coefficients of n-th order tensor, while
giare components of the unit gradient vector g,bis the
diffusion weighting factor, and Sand S0are drop in the signal
in presence and absence of diffusion gradients respectively.
Earlier methods based on the least square estimation do not
ensure positive diffusion profile. The methods proposed in
5
(a) case1 (b) D−1(c) Adjugate (d) β-Scaled, p=2
(e) case2 (f) D−1(g) Adjugate (h) β-Scaled, p=2
Fig. 4: Comparison of the three metric tensors with the two extreme cases
[36] [37] ensures positive semi-definiteness of the tensors.
We apply flattening of 4th order tensor, which gives 9 ×9
matrix, and eigen-tensors have the potential to reveal actual
fiber directions [38]. The diagonal components approach [39]
retains geometrical information of the full tensor. The diagonal
component of this matrix is symmetric positive definite tensor
[39]. In general, nth order tensor T(n)can be expressed as a
matrix of (n−2)order tensors:
T(n)=
T(n−2)
xx T(n−2)
xy T(n−2)
xz
T(n−2)
yx T(n−2)
yy T(n−2)
yz
T(n−2)
zx T(n−2)
zy T(n−2)
zz
(8)
For instance, the diagonal block element T(2)
xx in the fourth
order diffusion tensor is given by
Txx(xx)Txx(xy)Txx(xz)
Txx(xy)Txx(yy)Txx(yz)
Txx(xz)Txx(yz)Txx(zz)
=
Dxxxx Dxxxy Dxxxz
Dxxxy Dxxyy Dxxyz
Dxxxz Dxxyz Dxxzz
(9)
Another observation is that the flattening of 4th order tensor
using diagonal components (DC) can potentially reveal the
actual underlying fiber directions. This observation could be
quite useful in fiber tracking.
F. Crossing fiber reconstruction
Resolution at Fine Angles: We have shown experimentally
that these diagonal components produce small orientation
errors in comparison to the Cartesian tensor fiber orientation
distribution (CT-ODF) method [40]. These components can
potentially be used to track the fibers at crossings. The other
observation is about fuzziness in finding maxima using the CT-
ODF method, which is potentially better than DOT, QBI, etc
[41] [42]. These maxima provide the direction of underlying
fibers. The ODF appears at millimeter-scale whereas actual
diffusion occurs at micro-meter scale. The maximal of ODF
does not necessarily align with the actual underlying fiber
direction. The CT-ODF method does the misalignment cor-
rection. The method involves the computation of the maxima.
This method shows ambiguity in finding maxima, as shown
in Fig 5. In Fig 5(a), when the angle between fibers is less
than 70◦, the points labeled by arrows are supposed to be
the better choices for maxima than the middle one (point
labeled by red color). Our projection to second order tensors
mentioned above provides the two right results, while CT-ODF
fails. This effect disappears when the angle difference falls in
70 ≤θ≤90 range (see Fig 5(b)). In this range, both methods
are comparable. Outside this range, the CT-ODF method is
inefficient compared to the diagonal component approach. In
the next section, we use CT-ODF for reorientation and diag-
onal components for tracking fibers, particularly in crossing
regions. This method is extendable to higher-order tensors;
for instance, 6th order tensor has nine diagonal components
which can resolve nine directions. However, practically more
than 3 or 4 fibers per crossing seldom arise.
(a) angle between fibers 60◦(b) angle between fibers 75◦
Fig. 5: 4th order ODF with angle differences between the two
fibers
Resolving crossing fibers: When dealing with the crossing
regions, we enhance Algorithm 1 by working in two layers
corresponding to two projections of the fourth order tensor to
its diagonal components, see Algorithm 3.
G. Local Interpolation Effect
The interpolation step in the algorithm affects the flow of
geodesics while fiber tracking. Aside the elementary Euclidean
6
Algorithm 3 Reconstruction of fibers
Input: Reoriented 4th order tensor field
Output: Fiber reconstruction
1: Flatten the 4th order tensor
2: Extract two layers corresponding to the diagonal compo-
nents in 2D using equation (9)
3: Shoot the rays from initial point/region using Algorithm
1 in layer 1 and layer 2.
interpolation of the tensor data, there are smarter choices avail-
able, including the Log-Euclidean (LogE), Spectral Quaternion
(SQ), and spherical version of spectral quaternionic interpola-
tion (SlerpSQ) [39], detailed explanation can be also found in
[30].
Log-Euclidean Interpolation: In this geometry, the distance
between two tensors T1,T2∈S+(3)is given by
dLogE (T1,T2) = ||log(T1)−log(T2)|| (10)
It is based on the fact that the symmetric 3 ×3 matrices
are diffeomorphic to S+(3)via the exponential mapping. The
interpolation curve between two tensors is the geodesics curve
γLogE :[0,1]−→ S+(3), where the space S+(3)is a convex
subset of the Euclidean space R3×3of 3×3 matrices and it is
given for all 0 ≤t≤1 by:
γLogE (t) = exp(tlog(T1)+(1−t)log(T2)),(11)
Spectral Quaternion Interpolation: The basic idea of spec-
tral metric is to treat eigenvalues and eigenvectors of a SPD
matrix separately. The eigenvalue decomposition of the SPD
matrix in spectral geometry is T=RΛRTinto a rotation matrix
R∈SO(3)and a diagonal matrix Λcontaining the eigenvalues,
which provides a natural way of splitting the tensor. Thus
using the spectral decomposition of a positive definite matrix,
the interpolation curve is given by the equations
S(t) = R(t)Λ(t)R(t)T,(12)
R(t) = R1exp(tlog(RT
1R2)),(13)
Λ(t) = R1exp(tlog(RT
1R2)) (14)
The geometric interpretation of the interpolation curve is a
geodesic in the product space of the Lie group defined as G=
SO(3)×D+(3), where D+(3)is the group of diagonal matrices
with positive elements. In [39], both SQ and SlerpSQ has a
similar effect in interpolation, we choose SQ interpolation for
fiber tracking.
Fig 6(a) and (b) show the spectral metric preserves
anisotropy, which is crucial in fiber tracking application. In
Fig 6(c), a tensor located at (12,13), the rectangular section is
considered. The tensor at this position is part of the vertical
fiber and is underlying to the uniform background and not
cross-section with horizontal fiber. Fig 6(d) is interpolation in
its neighbor using the LogE metric and Fig 6(e) done to the
SQ metric. The flow of interpolated tensors is more accurately
captured in Fig 6(c), which shows interpolation flows towards
the left. Spectral metrics are known for robustness with respect
to noise segmentation of curved fibers and presentation of
anisotropy.
(a) LogE (b) SQ
(c) Near the tensor at
(12,13) location
(d) LogE interpolation (e) SQ interpolation
Fig. 6: Local Interpolation in crossing fibers using LogE and
SQ interpolation
IV. EXP ER IM EN TS A ND RE SU LTS
A. Results on Synthetic Data
For the experiments, we generate synthetic tensor fields
with many configurations that have similar properties to many
white matter tracts in the brain. The synthetic images are
simulated using a signal generated with b-value 1500s/mm2
with a signal without gradient impulse S0=1. Total of 81
gradient directions are chosen, which are uniformly distributed
over the sphere. We use the adaptive kernel method to create
fibers as detailed in [40].
In Fig 8, deep inverted U-shaped is considered with four
points in the starting region and five shooting per point. It is
visible that the adjugate metric fails to trace the fibers as it
approaches the target region. β-Scaled metric tensor geodesics
follow the fibers well, and higher power of pproduces smooth
fibers and increases fiber density. Fig 7shows tractography
result on the layer of the diagonal components where the two
fibers cross closely (cf. Fig 5(a)).
Fig 9shows that the hybrid approach can trace in high
curvature fiber flows. On top of that, the 4th order tensor field
image is shown with 2nd order tensor field obtained by sum of
the diagonal components. This produces sharp images contrary
to DTI.
In Algorithm 1, the ODE solver method increases the deviation
of geodesic along the path. To overcome this problem, we
feedback the principal eigenvector direction of the underlying
interpolated tensor to ODE solver. This causes geodesic devi-
ation to disappear and leads to better performance under the
three metrics. We tested our method on different high curvature
fiber flows, as shown in Fig 9.
Fig 10 shows the S-shaped configuration corrupted with
Riccian noise. The hybrid method acts robust and stable even
in case where the fibers is poorly visible Fig 10(b). We show
results of geodesic tracking on synthetic data for crossing
fibers based on β-Scaled metric and diagonal component
approach.
7
(a) 4th order tensor (b) Diagonal sum
(c) β-Scaled tracking
Fig. 7: Fiber tracing using β-Scaled and DC when two fibers
are close.
(a) Deep inverted U-shaped
(b) adjugate (c) β-Scaled p=2
(d) β-Scaled p=4 (e) β-Scaled p=6
Fig. 8: FA images corresponding to (a),comparison of ray-
tracing with ODE solver using adjugate, and β-Scaled metric
tensors from (b) - (e) respectively.
(a) 4th order inverted deep U
shaped
(b) 4th order in Reflected S shape
(c) Diagonal sum (d) Diagonal sum
(e) β-Scaled tracing (f) β-Scaled tracing
Fig. 9: Ray-tracing using Hybrid approach and diagonal sum
2nd order in high curvature fiber flows. Here we consider with
10 points and 5 shoots per points inside the rectangular region.
.
Fig 11 depicts two linear fibers intersecting at small angle.
Fig 11(c) shows the two components in the intersection region.
These components are sharp and follows the trajectory of their
individual single fibers. Fig 11(d) shows the β-Scaled metric
tensor used for tracing the fiber bundle.
Fig 12 has two curved fibers intersecting. The Fig 12(b)
shows the diagonal components in the intersection region. Fig
12(c) represent horizontally orientated regions of fiber curves
whereas 12(d) indicates the vertically oriented regions and
12(e) is the result of β-Scaled metric tensor tracking. In Fig
12(c and d), the diagonal components are able to align along
with the correct running curved fibers.
In Fig 13, the image shows two linear fibers crossing at
sharp angles corrupted with noise. In this difficult case our
method is able to reconstruct the fibers and behaves robust.
8
(a) (b)
(c) (d)
(e) (f)
Fig. 10: (a) Reflected S-shaped fiber with Riccian noise 0.25,
(b) Signal corrupted with Riccian noise =0.30, (c) Ray-
tracing with Principal eigenvector direction using adjugate and
noise 0.25 (d) Ray-tracing with Principal eigenvector direction
using adjugate and noise 0.30, (e) Ray-tracing with principal
eigenvector direction using β-Scaled metrics with p=2 and
Riccian noise 0.25, (f) Ray-tracing with principal eigenvector
direction using β-Scaled metrics with p=2 and Riccian noise
0.30
B. Results on Real Data
We explain the results of our tracking algorithm applied to
real images of the human brain. The DW-MRI image consist
of total size 114 ×114 ×70 and each voxel is the size of 2 ×
2×2mm3. The real images are obtained by applying gradient
in 64 diffusion directions with diffusion weighting factor b=
1500s/mm2with single reference image (b = 0). We have used
generalized logistic function (1a) as activation to test on real
images.
In Fig 14, a rectangular section of the Dorsal Longitudinal
Fasciculus (DLF) is selected. The Fig 14(c) is corresponding
to FA scalar image.
In Fig 15, 10 points were randomly picked in rectangular
section on the top of the structure. Five geodesics are shot per
point in both directions. The results indicate that most of the
fibers trace the white matter structure in all three cases. Results
(a) 4th order tensor (b) Diagonal sum
(c) DC of rectangular section (d) Fibers using β-Scaled
Fig. 11: Fibers computed using the hybrid approach and
diagonal sum at crossing area
under β-Scaled metric tensor produce smoother geodesics.
V. CONCLUSION
We propose a new geodesic based tractography method by
using a β-Scaled metric tensor. This metric tensor is adapted
according to the inherent anisotropy property. We have shown
that the performance of adapted metrics by means of sigmoid
function as activation function composed with the Hilbert
anisotropy is better than performance of the classical metric
and it also performs better than adjugate metric for highly
curved fiber flows.
To increase the accuracy of the tracking approach, we iterate
local geodesics tracing via Runge-Kutta ODE solver in the
interpolated grid of tensor data, initiated by the principal
eigenvector direction, called the hybrid approach.
We also illustrate the potential of using diagonal compo-
nent approach for capturing crossing fibers. We resolve the
fiber tracking in intersecting fibers region by using flattened
4th order tensors components. These components are second
order tensors lying in Riemannian symmetric space. We have
shown that they have potential to effectively locate orientation
distribution functions (ODFs) maxima even at small angle
intersections.
In future work, we will use the spectral metric approach
for local interpolation. The experiment discussed in section
III-G suggests to use spectral metric for local interpolation to
preserve anisotropy which is crucial for fiber reconstruction.
ACK NOW LE DG ME NT
The first three authors have been supported by the grant
MUNI/A/0885/2019 of Masaryk University, Jan Slov´
ak grate-
9
(a) 4th order (b) zoom section
(c) first diagonal component (d) second diagonal component
(e) β-Scaled fiber tracking
Fig. 12: This figure illustrate β-Scaled fiber tracking can trace
in high curvature fiber flow using the hybrid approach and
diagonal sum at crossing area
fully acknowledges support from the Grant Agency of the
Czech Republic, grant Nr. GA17-01171S. We acknowledge
the core facility MAFIL supported by the Czech-BioImaging
large RI project (LM2018129 funded by MEYS CR) for their
support with obtaining scientific data presented in this paper.
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