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Title: Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI
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Title: Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI
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Creator: Barmpoutis, Angelos
Jian, Bing
Vemuri, Baba C.
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Symmetric Positive 4th Order Tensors & their

Estimation from Diffusion Weighted MRI


Angelos Barmpoutis1, Bing Jian, and Baba C. Vemuri

University of Florida, Gainesville FL 32611, USA,
{abarmpou, bjian, vemuri}@cise.ufl.edu


Abstract. In Diffusion Weighted Magnetic Resonance Image (DW-MRI)
processing a 2"d order tensor has been commonly used to approximate
the diffusivity function at each lattice point of the DW-MRI data. It is
now well known that this 2"d-order approximation fails to approximate
complex local tissue structures, such as fibers crossings. In this paper
we employ a 4th order symmetric positive semi-definite (PSD) tensor
approximation to represent the diffusivity function and present a novel
technique to estimate these tensors from the DW-MRI data guaranteeing
the PSD property. There have been several published articles in litera-
ture on higher order tensor approximations of the diffusivity function
but none of them guarantee the positive semi-definite constraint, which
is a fundamental constraint since negative values of the diffusivity co-
efficients are not meaningful. In our methods, we parameterize the 4th
order tensors as a sum of squares of quadratic forms by using the so called
Gram matrix method from linear algebra and its relation to the Hilbert's
theorem on ternary quartics. This parametric representation is then used
in a nonlinear-least squares formulation to estimate the PSD tensors of
order 4 from the data. We define a metric for the higher-order tensors
and employ it for regularization across the lattice. Finally, performance
of this model is depicted on synthetic data as well as real DW-MRI from
an isolated rat hippocampus.


1 Introduction

Data processing and analysis of matrix-valued image data is becoming quite
common as imaging sensor technology advances allow for the collection of matrix-
valued data sets. In medical imaging, during the last decade, it has become
possible to collect magnetic resonance image ('.. I; 1) data that measures the
apparent diffusivity of water in tissue in vivo. A 2nd order tensor has commonly
been used to approximate the diffusivity profile at each image lattice point in a
DW-MRI [4]. The approximated diffusivity function is given by

d(g) gTDg (1)

where g [gl g g3]T is the magnetic field gradient direction and D is the
estimated 2"d-order tensor. This approximation yields a diffusion tensor (DT-
MRI) data set Di, which is a 3D matrix-valued image, where subscript i denotes















location on a 3D lattice. These tensors Di are elements of the space of 3 x 3
symmetric positive-definite matrices. Mathematically, these tensors belong to
a Riemannian symmetric space, where a Riemannian metric assigns an inner
product to each point of this space. Using this metric, one can perform various
computations, e.g. interpolation, geodesics, geodesic PCA [3, 7, 11].
Use of higher order tensors was proposed in [9] to represent more complex
diffusivity profiles which better approximate the diffusivity of the local tissue
geometry. To date however, none of the methods reported in literature for the
estimation of the coefficients of higher order tensors preserve the positive defi-
niteness of the diffusivity function.
The use of a 4t -order covariance tensor was proposed by Basser and Pajevic
in [5]. This covariance tensor is employed in defining a Normal distribution of
2"d order diffusion tensors. This distribution function has been employed in
[6] for higher-order multivariate statistical analysis of DT-MRI datasets using
spectral decomposition of the 4t -order covariance matrix into eigenvalues and
eigentensors (2nd order). However, 2nd order tensors are used to approximate the
diffusivity of each lattice point of a MR dataset, failing to approximate complex
local tissue structures, such as fiber crossings.
In this paper we approximate the diffusivity profile using 4t -order tensors.
We propose a novel parametrization of these positive-definite higher order ten-
sors as a sum of squares of quadratic (2"d-order) forms. This parametrization
is enforced by employing the Gram matrix method in conjunction with the
Hilbert's theorem on ternary quartics [8]. We present an efficient algorithm which
estimates 4t -order symmetric positive semi-definite diffusion tensors from dif-
fusion weighted MR images. We also propose a distance measure for the space
of higher-order tensors that can be computed in closed form, and employ it to
regularize the estimated data across the lattice. Finally, we present experimental
results using real diffusion-weighted MR data from an isolated rat hippocampus.
The motivation for processing and analyzing the hippocampus lies in its impor-
tant role in semantic and episodic formation, which is particularly vulnerable
to acute or chronic injury [1, 15]. Based on knowledge of hippocampal anatomy,
complex local tissue structures such as fiber crossings are commonly present at
the anatomical regions of stratum lacunosum-moleculare, hilus, molecular layer
(see fig. 3(d) region 4) and stratum lucidum (fig. 3(d) region 5). The techniques
being developed here can approximate accurately such crossings and complex
fiber structures and thus could prove useful in improving the sensitivity and
specificity of diffusion MRI for detecting and monitoring hippocampal diseases.
The rest of the paper is organized as follows: In section 2, we present a novel
parametrization of the 4t -order tensors that is used to enforce the positivity
semi-definiteness of the estimated tensors. In section 2.1, we present a method to
estimate 4t -order tensors from diffusion-weighted MR images. Furthermore, in
section 2.2 we propose a distance measure for the space of 4t -order tensors, and
we employ it for regularization of the estimated tensor field. Section 3 contains
the experimental results and comparisons with other methods using simulated














diffusion MRI data and real MR data from an isolated rat hippocampus. In
section 4 we conclude.



2 Diffusion tensors of 4th order


The diffusivity function can be modeled by Eq. (1) using a 2"d-order tensor.
Studies have shown that this approximation fails to model complex local struc-
tures of the diffusivity in real tissues [10] and a higher-order approximation
must be employed instead. A 4th-order tensor can be employed in the following
diffusivity function

d(g) E D (2)
i+j+k=4

where g [g9 g2 g3] is the magnetic field gradient direction. It should be noted
that in the case of 4th-order symmetric tensors there are 15 unique coefficients
Di,j,k, while in the case of 2"d-order tensors we only have 6.
In DW-MRI the diffusivity of the water is a positive quantity. This prop-
erty is essential since negative diffusion coefficients are nonphysical. However
there is no guarantee that the estimated coefficients Di,j,k by the above process,
will form a positive semi-definite tensor. Therefore, we need to develop a new
parametrization of the 4th-order tensor, which enforces the positive semi-definite
property of the estimated tensor.
Regarding gi in (2) as variables, the equivalence between symmetric tensors
and homogeneous polynomials is straightforward. Moreover if a symmetric tensor
is PSD, then its corresponding polynomial must be nonnegative for all real-
valued variables. Hence here we are concerned with the positive definiteness of
homogenous polynomials of degree 4 in 3 variables, or the so called ternary
quartics. In this work we propose a novel parametrization of the symmetric 4th
order PSD tensors, using the Hilbert's theorem on positive ternary quartics, was
first proved by Hilbert in 1888 (see [13] for modern exposition):
Theorem 1. Every positive real ternary quartic is a sum of three squares of
quadratic forms.
Assuming the most general case, a PSD ternary quartic can be expressed as
a sum of N squares of quadratic forms as.

d(g) (vTq)2 + ... + (vTqN)2 VTQQTv TGv (3)

where v is a properly chosen vector of monomials, (e.g. [g2 g9 92 9192 g9
g2gs]T), Q [qll ...qN] is a 6 x N matrix by stacking the 6 coefficient vectors
qi and G QQT is the so called Gram matrix.
Using this Gram matrix G expression, Eq. (2) can be written as d(g)
vTGv, and the correspondence between the 4th-order tensor coefficients Di,j,k














of Eq. (2) and the Gram matrix G can be established as follows:

D/ 4,0,0 a b D3s,1,o D3,o,1 d
a Do,4,0 c D1,3,0 e Dos,
G b c DO,0,4 f D1,o0,3 0Do,1,3
S D3,1,0 1o D3, f Do ,2,2 2a D2,1, d -D12 e ()
D3,0,1 e o,s0,3 D2,1,1 -d D2,0,2- 2b D,1,2 f
S d 0,3,1 20,1,3 21,2,1 e D1,1,2 f Do2,2 2c

where a, b, c, d, e, f are free parameters, i.e for any choice of those parameters
the obtained Gram matrix represents the same 4th-order tensor [12]. According
to Theorem 1, if N 3 (i.e. Gram matrix G has rank 3) then the whole space
of PSD ternary quartics is spanned. For some specific choices of the parameters
a, b, c, d, e, f of Eq. (4), the rank of matrix G becomes 3 [12]. Powers and Reznick
in [12] worked on finding fundamentally different choices of those parameters that
yield the same given PSD ternary quartic, i.e. in how many different ways can
a ternary quartic be expressed as a sum of squares of three quadratic forms.
However, given a Gram matrix G we can uniquely compute the coefficients
Di,j,k of the tensor (see Eq. (4)). Therefore, we can employ the Gram matrix
method for the estimation of the coefficients Di,j,k of the diffusion tensor from
MR images using the following two steps: 1) first we estimate a Gram matrix G
from the MR signal of the given images, and then 2) we uniquely compute the
coefficients Di,j,k of the 4th-order tensor by using formulas obtained from Eq.
(4). Note that although the estimated matrix G is not unique, the coefficients
Di,j,k are uniquely determined.
In the following section we will employ this Gram matrix method to enforce
the positive semi-definite property of the estimated diffusion tensors from the
diffusion weighted MR images.


2.1 Estimation from DWI

The coefficients Di,j,k of a 4th order diffusion tensor can be estimated from
diffusion-weighted MR images by minimizing the following cost function:
M
E(Q, So) = (, S- v- QQTvi)2 (5)
i=1

where M is the number of the diffusion weighted images associated with gradient
vectors gi and b-values bi; Si is the corresponding acquired signal and So is the
zero gradient signal. Using the magnetic field gradient directions gi we construct
the 6-dimensional vectors vi r- '. g,2 gT gigi2 gigi3 gigg3]T. In Eq. (5), the
4th order diffusion tensor is parameterized using the Gram matrix G QQT,
where Q is a 6 x N matrix and N > 3 is a predefined constant. In our experiments
we used N = 3, which is justified by Theorem 1. Having estimated the matrix
Q that minimizes Eq. (5), the coefficients Di,j,k can be computed directly from
the Gram matrix using the relation described by the matrix of Eq. (4). So can















either be assumed to be known or estimated simultaneously with the coefficients
Di,j,k by minimizing Eq. (5).
Starting with an initial guess for So and Q, we can use any optimization
method in order to minimize Eq. (5). For the optimization schemes that employ
the gradients of Eq. (5) with respect to the unknown coefficients of Q, the
gradient is given by the following equation
M
VQ E(Q, So) 4 biSoe-b VTQQTV(S S bVTQQ )TViQ (6)
i= 1
Now given Q at each iteration of the optimization algorithm we can update So
by again minimizing Eq. (5). The derivative of this equation with respect to the
unknown So is
M
Vs E(Q, So) -2 (S Sobe-vTQbi Q )v bi v (7)
i= 1
By setting Eq. (7) equal to zero, we derive the following update formula for So
M M
So = S e-ivqTi/ C 2bivTQQTV (8)
i= 1 i 1
In our experiments we used the well known Lavenberg-Marquardt (LM) nonlin-
ear least-squares method, which has advantages over other optimization meth-
ods, in terms of stability and computational burden.
As pointed out earlier, although the coefficients Di,j,k are uniquely estimated,
the Gram matrix parametrization G = QQT is not unique, i.e. there exist dif-
ferent matrices Q which parameterize the same Gram matrix. For example there
are infinitely many matrices Q that yield the same G, due to the orthogonality
property (RRT I) of the rotation matrices R, where I is the identity matrix.
Thus, in the case that Q is of size 6 x 3, for any 3 x 3 orthogonal matrix R we
have (QR)(QR)T QQT. In order to reduce this infinite solution space to a
finite set of solutions, which theoretically can be handled by the optimization
techniques, we use the well known QR decomposition of real square matrices to
uniquely decompose any given 6 x 3 matrix Q in the form Q TA ], where
all matrices are of size 3 x 3 and specifically T is lower triangular, and R is an
orthogonal matrix. Then by setting R I we reformulate Q as Q [T] and
thus the infinitely non-unique issue is replaced by a countably non-uniqueness
issue, which can be handled by the optimization algorithm. Note that using this
formulation there are only 15 unknown parameters in matrix Q, which is equal
to the number of the unknown coefficients Di,j,k of the estimated tensor.

2.2 Distance measure
In the previous section we discussed about estimating PSD 4th-order tensors from
DW-MRI data. For the estimation, we minimize the energy function expressed














in Eq.(5). We can add a regularization term in this energy function, in order to
perform smoothing of the estimated tensors across the lattice in the DW-MRI
data set. The regularization term is given by

1 dist(D, Dj,) (9)
j ict 7

where ]j is the set of lattice indices whose distance from lattice index 'j' is 1.
In the regularization term defined in Eq. (9) we need to employ an appropriate
distance measure between the tensors Di and Dj. Here we use the notation D
in order to denote the set of 15 unique coefficients Di,j,k of a 4th-order tensor.
We can define a distance measure between the 4th-order diffusion tensors D1
and D2 by computing the normalized L2 distance between the corresponding
diffusivity functions dl (g) and d2 (g) leading to the equation,


dist(DI, D)2 1 [di(g) d(g)]2dg (10)
47 J 2



1[(A4,0,0 + A0,4,0 + Ao,0,4 + A2,2,0 + A0,2,2 + A2,0,2)2 +
315
4[(A4,0,0 + A2,2,0)2 + (A4,0,0 + A2,0,2)2 + (0,4,0 + 2,20)2 +
(A0,4,0 + Ao,22)2 + (0A,0,4 + Ao,2,2)2 + (0A,0,4 + A2,0,2)2]
23(A 0o ,4,0 0,0,4) 6(,2,0 0,2,2 2,0,2) +
2(A4,0,0 + A0,4,0 + AO,0,4)2 + (A2,1,1 + A0,3,1 + AO,1,3)2 +
(A1,2,1 + A3,0,1 + A1,0,3)2 + (A1,1,2 + A3,1,0 + A1,3,0)2 +
2 [(A3,1,0 + 1,3,0)2 + (A3,o,1 + A1,0,3)2 + (Ao,3,1 + A0,1,3)2] +
2(A2,1,o + 'A2'0 + 'A2 + A2 A2 2
2,01 1,3, 0,3,1 + 1,0,3 A0,1,3)]

where, the integral of Eq. (10) is over all unit vectors g, i.e., the unit sphere S2
and the coefficients ,Aij,k are computed by subtracting the coefficients of the
tensor D1 from the corresponding coefficients of the tensor D2.
As shown above, the integral of Eq. (10) can be computed analytically and
the result can be expressed as a sum of squares of the terms A,Aj,k. In this
simple form, this distance measure between 4th-order tensors can be implemented
very efficiently. Note that this distance measure is invariant to rotations in 3-
dimensional space since it was defined as an integral over all directions g.
Another property of the above distance measure is that the average element
(mean tensor) D of a set of N tensors Di, i 1 ... N is defined as the Eu-
clidean average of the corresponding coefficients of the tensors. This property
can be proved by verifying that D minimizes the sum of squares of distances
E dist(D, Di)2. Similarly, it can be shown that geodesics (shortest paths) be-
tween 4th-order tensors are defined as Euclidean geodesics.















3 Experimental results


In this section we present experimental results on our method applied to sim-
ulated DW-MRI data as well as real DW-MRI data from an isolated rat hip-
pocampus.
In order to motivate the need of the PSD constraint in the 4th-order estima-
tion process, we performed the following experiment using a synthetic dataset.
The synthetic data was generated by simulating the MR signal from a single fiber
using the realistic diffusion MR simulation model in [14]. Then, we added differ-
ent amounts of Riccian noise to the simulated dataset and we estimated the 4th
order tensors from the noisy data by: a) minimizing Ei (S,-Soexp(- 1.1,gi)))2
without using the proposed parametrization to enforce PSD constraint and b)
our method, which guarantees the PSD property of the tensors. (S, is the MR
signal of the ith image and So is the zero-gradient signal).
It is known that the estimated 4th-order tensors represent more complex
diffusivity 1" pi.il -. with multiple fiber orientations which better approximate the
diffusivity of the local tissue geometry compared to the traditional 2"d-order
tensors [9]. Studies on estimating fiber orientations from the diffusivity profile
have shown that the peaks of the diffusivity profile do not necessarily yield the
orientations of the distinct fiber bundles [10]. One should instead employ the
displacement probability profiles The displacement probability P(R) is given
by the Fourier integral P(R) = E(q)exp(-27riqR)dq where q is the reciprocal
space vector, E(q) is the signal value associated with vector q divided by the
zero gradient signal and R is the displacement vector. In our experiments, we
numerically estimated the displacement probability 1".. '". -. from the 4th-order
tensors.


20 --Positive semi-definite DT4
, --Non positive-definite DT4
S15

.T 10
2 15 --- ------------
0)



0 16.6 12.5 8.3 6.2
SNR

Fig. 1. Comparison of the fiber orientation errors for different amount of noise in the
data, obtained by using: a) our parametrization to enforce positivity (red) and b)
without enforcing positivity of the estimated tensors (in blue).


Then, we computed the displacement probability profiles of the 4th-order
tensors estimated earlier with the two different methods, and we computed the















fiber orientations from the maxima of the probability 1'.. ii. The error angles
(mean and standard deviation) of the two methods for different amount of noise
in the data are plotted in Fig. 1. As expected, our method yields smaller er-
rors in comparison with the method that does not enforces the PSD property
of the tensors. When we increase the amount of noise in the data, the errors
observed by the later method are significantly increased, while our proposed
method shows clearly much smaller errors. This conclusively demonstrates the
need for enforcing the PSD property of the estimated tensors and validates the
accuracy of our proposed method.

Furthermore, in order to compare our proposed method with other existing
techniques that do not employ 4th-order tensors, we performed an other exper-
iment using synthetic data. The data were generated for different amounts of
noise by following the same method as previously using the simulated MR signal
of a 2-fiber crossing (see Fig. 2(a)) We estimated 4th-order tensors from the
corrupted simulated MR signal using our method and then we computed the
fiber orientations from the corresponding probability 1"IiIl'. For comparison
we also estimated the fiber orientations using the DOT method described in
[10] and the ODF method presented in [2]. For all three methods we computed
the estimated fiber orientation errors for different amount of noise in the data
(shown in Fig. 2(b)). The results conclusively demonstrate the accuracy of our
method, showing small fiber orientation errors (~ 6) for typical amount of noise
with signal to noise ratios (SNR): 12.5-16.6. Furthermore, by observing the plot,
we also conclude that the accuracy of our proposed method is very close to that
of the DOT method and is significantly better than the ODF method.





40
S-m-DT4
S -0-DOT
(10
S20 ODF




0
0 16.6 12.5 8.3 6.2
SNR

(a) (b)


Fig. 2. Fiber orientation errors for different SNR in the data using our method for the
estimation of positive 4th-order tensors and two other existing methods: 1) DOT and
2) ODF. In the experiment we used simulated MR signal of a 2-fiber crossing, whose
probability profile is shown in (a).














In the following experiments, we used MR data from an isolated rat hip-
pocampus. The diffusion weighted MR images of this dataset were acquired
using the following protocol. This protocol included acquisition of 22 images
using a pulsed gradient spin echo pulse sequence with repetition time (TR)
1.5 s, echo time (TE) = 28.3 ms, bandwidth = 35 kHz, field-of-view (FOV)
4.5 x 4.5 mm, matrix = 90 x 90 with 56 continuous 200-mn-thick axial slices (ori-
ented transverse to the septo-temporal axis of the isolated hippocampus). After
the first image set was collected without diffusion weighting (b ~ 0 s/mm2), 21
diffusion-weighted image sets with gradient strength (G) 415 mT/m, gradient
duration (6) = 2.4 ms, gradient separation (A) = 17.8 ms and diffusion time
(Ta) 17 ms were collected. Each of these image sets used different diffusion
gradients (with approximate b values of 1250 s/mm2) whose orientations were
determined from the 2nd order tessellation of an icosahedron projected onto the
surface of a unit hemisphere. The image without diffusion weighting had 36
signal averages (time =81 min), and each diffusion-weighted image had 12 av-
erages (time = 27 min per diffusion gradient orientation) to give a total imaging
time of 10.8 h per hippocampus. Temperature was maintained at 20 0.2C
throughout the experiments using the temperature control unit of the magnet
previously calibrated by methanol spectroscopy. Figures 3(a) and 3(b) show the
So image and the FA map respectively of a slice extracted from the 3D volume
of the above dataset.












(a) (b) (c) (d)

Fig. 3. Isolated rat hipppocampus. a) So, b) FA, c) White pixels indicate locations
where the estimated 4th-order tensor was not positive-definite, d) Manually labeled
image based on knowledge of hippocampal anatomy. The index of the labels is: 1)
dorsal hippocampal commissure, 2) fimbria, 3) alveus, 4) molecular layer, 5) mixture
of CA3 stratum pyramidale and stratum lucidum.


First, we estimated a 4th-order diffusion tensor field from this dataset by min-
imizing 1 (S, Soexp(- 1. '(gi)))2 without using the proposed parametriza-
tion to enforce positivity. As expected, some of the estimated tensors were not
positive. In Fig. 3(c) we show in white color the locations where those non-
positive-definite tensors were estimated. These tensors are mainly located in the















regions .1..-- -I hippocampal commissure", "fimbria" and "alevus", which cor-
respond to the regions 1, 2 and 3 respectively, shown in Fig. 3(d). Based on
knowledge of hippocampal anatomy, those regions are highly anisotropic with
FA ~ 0.9. Therefore, from the experimental results (Fig. 3(c)) we conclude that
highly anisotropic diffusivities are most likely to be inaccurately approximated
by a non-positive semi-definite tensor. Thus one needs to employ a method that
guarantees the PSD property.






.- r '-,li% .1 t -ri- -x"


... - ,-.d/ 2 2d-order DT













the-estim ated pobi l 44 horderh t wih eg le -DT
----''-s ---------- ---s.. : e a ; m'













Fig. 4. The estimated 4th order tensor field from an isolated rat hippocampus dataset
using our method. (a) top:So and bottom:FA, (b) the estimated displacement proba-
bility profiles of the 4th -order tensor field in the region of interest (ROI) indicated by
a green rectangle in (a). (c)Comparison of the estimated 2d -order tensors (top)and
the estimated probability profiles of the 4th -order tensors without (middle) and with
regularization (bottom) in a ROI indicated by a black rectangle in (b).



We computed the displacement probability profiles from: a) the 4th -order
tensor field estimated previously without the positive-definite constraint, and b)
the 4th-order tensor field estimated by our proposed method. In order to compare
the results of the above algorithms, in Fig. 5 we plot the corresponding proba
ability 1" -., -. from a region of interest in the .1.. I hippocampal commissure".
By observing this figure, we can say that the field of probability 1" Iil, -. is noisy





















Fig. 5. Left: The region of interest from the "dorsal hippocampal commissure", which
is magnified in the next plates of this figure. Comparison between the displacement
probability profiles computed from non-PSD 4th-order tensors (middle) and PSD ten-
sors estimated by our method (right).


if we do not enforce the PSD constraint (Fig. 5 middle). On the other hand the
1 ..i.1. -. obtained by our method (Fig. 5 right) are more coherent and smooth.
Note that this is a result of enforcing the PSD constraint, since in this experi-
ment we did use any regularization. This demonstrates the superior performance
of our algorithm and motivates the use of the proposed PSD constraint.
Finally Fig. 4(b) shows displacement probability profiles computed from the
estimated (by our method) 4th-order tensor field in another region of hippocam-
pus. This tensor field corresponds to the region of interest denoted by a green
rectangle in So and FA map shown in Fig. 4a. The X, Y, Z components of the
dominant orientation of each profile are assigned to R, G, B (red, green, blue)
components of the color of each surface. By observing Fig. 4(b) we can see several
fiber crossings in different regions of the rat hippocampus. One of those regions
is marked by a black rectangle and it is presented enlarged in Fig. 4(c). This re-
gion is consisted of a mixture of CA3 stratum pyramidale and stratum lucidum,
and it is most likely to contain fiber-crossings. As expected, in the center of this
region there are profiles presenting fiber crossings. These fiber crossings cannot
be resolved by using 2nd-order diffusion tensors estimated from the same dataset
(shown on the top of Fig. 4(c)). Finally in the bottom plate of Fig. 4(c), we show
an example using the regularization term defined in section 2.2. By comparing
the probability 1" ,i ,l. -. shown in this image with those of the middle plate of Fig.
4(c) we can see that the regularization of the estimated data removes some of
the noise in the dataset, and as a consequence some of the crossings are observed
more clearly (see at the center of the image).


4 Conclusions

In diffusion weighted MR imaging 2nd-order tensors have commonly been used
to approximate the diffusivity profile. However, this approximation fails to rep-
resent complex local tissue structures, such as fiber crossings. 4th-order tensors
were employed in this work, showing better approximation capabilities compared
to the 2"d-order case. We presented a method for estimating the coefficients of
4th-order tensors from diffusion-weighted MR images. Our technique guarantees
the positive semi-definite property of the estimated tensors, which is the main


f I f f 9# \ \
47J- 3s


* a 4 & a e eS
0 .P e .
004000 to
















contribution of our work. This property is essential since non-PSD diffusivity
pi"'1.i. -. are not meaningful from the point of view of physics of diffusion. To
date, there is no other reported work in literature which handles this constraint
for rank-4 tensors. We applied our proposed algorithm to a real MR dataset from
an isolated rat hippocampus. The superior performance of our method in the
experimental results demonstrates the need for employing the constraint and
motivates the use of our technique. The accuracy of our model was validated
by using simulated MR data of fiber crossings, and compared to other existing
methods. In our future work we plan to employ the methods proposed here to ex-
tend various techniques used for the 2"d-order tensor fields such as segmentation,
registration and fiber-tracking, to the space of higher-order tensors.


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