=Paper=
{{Paper
|id=None
|storemode=property
|title=Anisotropy of HARDI Diffusion Profiles Based on the L2-Norm
|pdfUrl=https://ceur-ws.org/Vol-715/bvm2011_50.pdf
|volume=Vol-715
}}
==Anisotropy of HARDI Diffusion Profiles Based on the L2-Norm==
https://ceur-ws.org/Vol-715/bvm2011_50.pdf
Anisotropy of HARDI Diffusion Profiles Based
on the L2 -Norm
Philipp Landgraf1 , Dorit Merhof1 , Mirco Richter1
1
Institute of Computer Science, Visual Computing Group, University of Konstanz
philipp.landgraf@uni-konstanz.de
Abstract. The fractional anisotropy (FA) value for Diffusion Tensor
Imaging is widely used to determine the anisotropy of diffusion in a
given voxel. As the FA value is based on the tensor’s eigenvectors it is
not possible to calculate this quantity for HARDI diffusion profiles. In
this paper we introduce an anisotropy index for HARDI data that utilizes
the L2 -norm as the most natural notion of distance for square-integrable
functions on the two-sphere such as HARDI diffusion profiles and show
that it is the limit of the generalized fractional anisotropy (GFA) index.
Our index is well-defined and rotationally invariant and thus resolves the
unsatisfactory issues with the GFA index.
1 Introduction
The gray matter in the human brain comprises about 10 billion neuronal cells
that use their axons to transmit information. These axons form bundles called
neuronal fiber tracts that connect functional areas.
The localization of white matter tracts is of great interest for neurological re-
search about brain structure and function, as well as for neurosurgery to preserve
important tract systems during surgical intervention.
The noninvasive and in vivo reconstruction of fiber tracts is possible with
Diffusion MRI. This is a medical imaging modality that is sensitive to the random
thermal movement of water molecules. This information allows for inference
about the structure of the tissue in the human brain because diffusion tends to
be hampered in directions orthogonal to fiber bundles.
A widely used model to describe the measured diffusion is the diffusion ten-
sor [1] which is the covariance matrix of a three-dimensional Gaussian distribu-
tion and is used to model the diffusion orientation distribution function (ODF).
The fractional anisotropy value (FA) is a scalar measure derived from the
diffusion tensor and quantifies the anisotropy of diffusion. As diffusion tends
to be more anisotropic in white matter this value can be applied to distinguish
white matter from gray matter within the brain. Tracking algorithms therefore
often operate with an FA threshold to prevent fiber tracking algorithms from
leaving regions of white matter as in [2].
However, the diffusion tensor can only resolve a single diffusion direction
per voxel. To overcome this deficit high angular resolution diffusion imaging
240 Landgraf et al.
(HARDI) techniques such as Q-Ball imaging [3] or higher order tensors [4] have
been developed that increase the number of gradients and reconstruct an appar-
ent diffusion coefficient (ADC) profile at each voxel.
The FA value is based on the eigenvalues of the diffusion tensor. Therefore,
it is not possible to calculate the FA value of a general ADC profile due to its
lack of eigenvalues. Several anisotropy indices for HARDI diffusion profiles can
be found in the literature, such as the indices of Frank [5] and Chen [6].
In this paper we generalize the popular GFA index [3] that for instance
was employed to investigate the genetic effects on brain fiber connectivity [7].
Nevertheless it has some theoretical limitations which are resolved in this work.
2 Materials and Methods
This section starts with the definition of the GFA index and its drawbacks.
Thereafter the L-index is defined.
2.1 The GFA index
The GFA index defined by Tuch [3] is a straightforward extension of the FA
value to HARDI ODFs. After picking directions {xi }i=1,...,n one can define
√ ∑
n
n i=1 (f (xi ) − ⟨f ⟩n )2
GFAn (f ) = ∑n (1)
(n − 1) i=1 f (xi )2
∑n
where ⟨f ⟩n = n1 i=1 f (xi ) is the mean value.
This index is automatically scaled to the unit interval and maps isotropic
ODFs to zero. In the special case of diffusion tensors, GFA reduces to the FA
value if the chosen directions happen to be the eigenvectors of the tensor.
The GFA index has, however, some severe drawbacks. First of all, it is not
well-defined as it strongly depends on the number and choice of the directions in
which the function is evaluated. Secondly, GFA(f ) is not rotationally invariant
as one would expect from an anisotropy index.
2.2 The L-Index
The basic idea of our anisotropy index is to measure how spherical the ADC-
profile is. The ODF is expressed as a linear combination of spherical harmonics
that are an orthonormal basis of the Hilbert space L2 of square-integrable func-
tions. The natural measure of distance in this space is the L2 -norm ∥·∥2 .
If the mean value ⟨f ⟩ of f defined by
∫ π ∫ 2π
1
⟨f ⟩ = f (θ, φ) dφ dθ (2)
2π 2 0 0
Anisotropy of HARDI Diffusion 241
and the corresponding constant function (which we also denote by ⟨f ⟩) is taken
as the closest approximation of f by a sphere, the L2 distance ∥f − ⟨f ⟩∥2 be-
tween f and ⟨f ⟩ contains the information about how close f resembles a sphere.
Normalizing with the L2 norm of f yields the L-index
∥f − ⟨f ⟩∥2
L(f ) = (3)
∥f ∥2
3 Results
The main result of this paper is that the L-index is the limit of the GFAn index
as n increases if one chooses the xi as rectilinear grid points in the [0, π) × [0, 2π)
domain.
Theorem 1. limn→∞ GFAn (f ) = L(f )
Proof. Let f : [0, a] × [0, b] → R be a real valued function on a rectangle. Define
(n)
grid points xij := (i na , j nb ), i, j = 0, . . . , n for each n ∈ N Furthermore let
∑ ab ( (n) )2
n ∑ n
Φ(f, n) = f xij (4)
i=1 j=1
n2
Now consider the characteristic functions χA(n) : [0, a] × [0, b] → {0, 1} for the
ij
(n)
sets Aij = [(i − 1) na , i na ) × [(j − 1) nb , j nb ). The characteristic function χA of a
set A is defined by χA (x) = 1 if x ∈ A and χA (x) = 0 otherwise. Then we have
∫ a∫ b∑ n ∑ n
(n)
Φ(f, n) = f (xij )2 χA(n) (x, y) dy dx (5)
ij
0 0 i=1 j=1
20 20
40 40
60 60
80 80
100 100
120 120
20 40 60 80 100 120 20 40 60 80 100 120
(a) FA value (b) L-index
Fig. 1. (a) FA values of a single slice from the dataset, (b) L-index values from the
same slice capped at 0.5 for enhancement of contrast.
242 Landgraf et al.
∫a∫b
which converges to ∥f ∥22 = 0 0 f (x, y)2 dydx as the sum describes a step func-
tion that converges to f 2 .
A similar argument shows that limn→∞ ⟨f ⟩n = ⟨f ⟩. Together this yields the
convergence because
√ √
n Φ(f − ⟨f ⟩n , n) ∥f − ⟨f ⟩∥2
lim GFAn (f ) = lim √ = (6)
n→∞ n→∞ n−1 Φ(f, n) ∥f ∥2
So basically GFAn (f ) is an approximation to the true anisotropy value L(f )
using the rectangle rule for numerical integration. Note that GFAn (f ) ∈ [0, 1]
implies L(f ) ∈ [0, 1] as well. Furthermore, this index is invariant under scaling
with a scalar, i.e. L(cf ) = L(f ) as one would expect from a measure of shape.
Additionally, it is well defined and rotationally invariant since it is defined by
integration over the sphere and thus resolves the drawbacks of the GFA index.
3.1 Application to human brain data
The Diffusion MRI dataset used in this work was provided for the IEEE Visuali-
sation Contest 2010. It is courtesy of Prof. B. Terwey, Klinikum Mitte, Bremen,
Germany and was acquired on a Siemens 3T Verio MR scanner. 30 gradient
directions and two averages per gradient were acquired with b = 1000 s/mm2 .
ADC-profiles were reconstructed with maximal order M = 6 as in [8] with
regularization parameter λ = 0.5. Subsequently, diffusion tensors were recon-
structed by a least squares fit to calculate FA values.
Figure 1 shows the FA values (a) and L-index values (b) of a single slice in the
dataset. Note that due to the fact that the distribution of the L-index is relatively
narrow the values have been thresholded at 0.5 for contrast enhancement. From
the histograms over the whole dataset for FA values and the L-index, we conclude
that 0.5 is a reasonable threshold (Fig. 2).
4
x 10
7000 2
1.8
6000
1.6
5000 1.4
1.2
4000
1
3000
0.8
2000 0.6
0.4
1000
0.2
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(a) FA value (b) L-index
Fig. 2. Histogram of the FA values (left) and L-index (right) over the entire dataset.
For the sake of clarity, zero voxels were left out.
Anisotropy of HARDI Diffusion 243
The correlation coefficient corr(L-index, FA) = 0.9576 shows a strong (posi-
tive) linear correlation between the L-index and the FA value.
4 Discussion
We showed that the L-index resolves the unsatisfactory issues with the GFA
index not being well-defined nor rotationally invariant. Moreover the GFA index
can be interpreted as approximation of the L-index by numerical integration.
Due to the strong linear correlation between the FA value and the L-index
the latter can be used to segment white matter in the brain or serve as a stopping
criterion for tracking algorithms based on HARDI data.
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