EXPONENTIAL TENSORS: A FRAMEWORK FOR EFFICIENT HIGHER-ORDER DT-MRI COMPUTATIONS

Abstract

In diffusion tensor magnetic resonance image (DT-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the 3D volume image. These tensors are symmetric positive definite matrices and the appropriate constraints required in algorithms for processing them makes these algorithms complex and significantly increases their computational complexity. In this paper we present a novel parameterization of the diffusivity function using which the positive definite property of the function is guaranteed without any increase in computation. This parameterization can be used for any order tensor approximations; we present Cartesian tensor approximations of order 2, 4, 6 and 8 respectively, of the diffusivity function all of which retain the positivity property in this parameterization without the need for any explicit enforcement. Furthermore, we present an efficient framework for computing distances and geodesies in the space of the coefficients of our proposed diffusivity function. Distances & geodesies are useful for performing interpolation, computation of statistics etc. on high rank positive definite tensors. We validate our model using simulated and real diffusion weighted MR data from excised, perfusion-fixed rat optic chiasm.

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EXPONENTIAL TENSORS: A FRAMEWORK FOR EFFICIENT HIGHER-ORDER DT-MRI
COMPUTATIONS
Angelos Barmpoutis and Baba C. Vemuri
University of Florida, Gainesville, FL 32611, USA
ABSTRACT
In Diffusion Tensor Magnetic Resonance Image (DT-MRI)
processing a 2ndorder tensor has been commonlyused to ap-
proximate the diffusivity function at each lattice point of the
3D volume image. These tensors are symmetric positive defi-
nite matrices and the appropriateconstraints required in algo-
rithms for processing them makes these algorithms complex
andsignificantlyincreases their computationalcomplexity. In
this paper we presenta novelparameterizationof the diffusiv-
ity function using which the positive definite property of the
function is guaranteed without any increase in computation.
This parameterization can be used for any order tensor ap-
proximations; we present Cartesian tensor approximations of
order 2, 4, 6 and 8 respectively, of the diffusivity function all
ofwhichretainthepositivitypropertyinthis parameterization
without the need for any explicit enforcement. Furthermore,
we present an efficient framework for computing distances
and geodesics in the space of the coefficients of our proposed
diffusivityfunction. Distances & geodesics are usefulfor per-
forming interpolation, computation of statistics etc. on high
rank positive definite tensors. We validate our model using
simulated and real diffusion weighted MR data from excised,
perfusion-fixedrat optic chiasm.
Index Terms— Biomedical imaging, Biomedical image
processing,Magneticresonanceimaging,Diffusionprocesses
1. INTRODUCTION
Data processing and analysis of matrix-valued image data is
becoming quite common as imaging sensor technology ad-
vances allow for the collection of matrix-valued data sets. In
medical imaging, in the last decade, it has become possible to
collect magnetic resonance image (MRI) data that measures
the apparent diffusivity of water in tissue in vivo. A 2ndorder
tensor has been commonly used to approximate the diffusiv-
ity profile at each lattice point of the image lattice [1]. The
approximated diffusivity function is given by gTDg, where
g =[g1g2g3]Tis thegradientunitvectorandD is a3×3ma-
trix. This approximation yields a diffusion tensor (DT-MRI)
data set Di, which is a 2D or 3D matrix-valued image, where
This research was in part supported by the NIH grants EB004752 and
NS42075. Authors thank Drs. T. Shepherd and E.¨Ozarslan for the data.
subscripti denoteslocationona 2Dor 3D lattice respectively.
ThesetensorsDiareelementsof thespaceof3×3 symmetric
positive-definite (SPD) matrices.
Mathematically, these SPD tensors belong to a Rieman-
nian symmetric space, where a Riemannian metric, which is
affine invariant assigns an inner product to each point of this
space. By using this metric, one can perform various com-
putations on the elements of the space [2, 3]. However im-
plementation of algorithms using this affine invariant frame-
work increases significantly the execution time of the algo-
rithms and complex algorithms may not be finish their execu-
tion within a reasonable time frame.
Recently, a Log-Euclideanmetric was proposed in [4] for
computing with tensors. In this work, the elements from the
space of SPD tensors, are mapped to a vector space of di-
mension ℜ6using the matrix logarithm map. Therefore, one
can use the Euclidean norm for computations in this space
and finally by using the inverse mapping,the data are mapped
back to the space of SPD matrices. This framework is quite
interesting and has advantages due to its high computational
efficiency in comparison to the affine invariant framework.
Both affine invariant and Log-Euclidean frameworks can
be employed for processing fields of 2ndorder tensors. Use
of higher order tensors was proposed in [5] to represent more
complexdiffusivityprofiles which better approximatethe dif-
fusivity of the local tissue geometry. However to date, none
of the reported methods in literature for the estimation of the
coefficients of higher order tensors preserve the positive defi-
niteness of the diffusivity function.
In this paper we propose a novel parameterization of the
diffusivity profile that guarantees the positive definite prop-
erty without the need of any further computation. We present
an efficientframeworkforcomputingdistances andgeodesics
in the space of the coefficients of our proposed diffusivity
function. The key contribution of our work is that we em-
ploy this framework for estimating higher (4th, 6thand 8nd)
order tensors from diffusion-weighted MR images. Note that
we are only interested for symmetric tensors and therefore
we consider only even orders. We compare our method with
other existing DTI methods showing high efficiency of the
proposedmethod. We presentvalidationofourframeworkus-
ing real diffusion-weightedMR data from excised, perfusion-
fixed rat optic chiasm [6].

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2. EXPONENTIAL DIFFUSION TENSORS
We define an Exponential Diffusion Tensor (EDT) of order
2 as a 3 × 3 symmetric matrix E, which will be used in the
following diffusivity function
d(g,E) = egTEg
(1)
The EDT matrix E is not necessarily an SPD matrix since
the diffusivity function 1 is positive for any symmetric ma-
trix. For example, in the case that E is the 3 × 3 zero matrix,
we have d(g) = e0= 1 ∀ g. If we use the standard diffusiv-
ity function gTDg, the previous example corresponds to the
diffusion tensor D = I. In this case we have gTDg = 1 ∀ g.
In eq. 1 the diffusivity function was defined by using a
2ndorder exponential tensor E. This function d(g,E) can be
generalizedby using higher order tensors. In the case of a 4th
order symmetric tensor we have 15 unique coefficients col-
lected into a vector?E =(E4,0,0, E0,4,0, E0,4,0, E2,2,0, E0,2,2,
E2,0,2, E2,1,1, E1,2,1, E1,1,2, E3,1,0, E3,0,1, E1,3,0, E0,3,1,
E1,0,3, E0,1,3). In the case of higherorder tensors we will use
the notation Ep1,p2,p3to indicate that it is the coefficient of
the term g1p1g2p2g3p3. By using this notation eq. 1 can be
generalized as
d(g,?E) = exp
?N
i=1
?
g1p1ig2p2ig3p3iEp1i,p2i,p3i
?
(2)
where in the case of 2nd, 4th, 6thand 8thorder N = 6, 15,
28 and 45 respectively.
2.1. Distance measure
We can define a distance measure between same order EDTs
?E1and?E2by computing the normalized L-2 distance of the
corresponding diffusivity functions d(g,?E1) and d(g,?E2),
1
?dg
vectors g). As an example, the distance between the 2nd
order EDT matrices E1 = 0 and E2 = (limx→−∞x)I is
dist(E1,E2)2= 1. This is true, since d(g,E1) = 1 and
d(g,E2) = 0 ∀ g. However, we need to define a metric that
assignsinfinitedistancebetweenthepurelyisotropicd(g,E1)
and the ’degenerate’case d(g,E2). Here we use the term ’de-
generate’ in order to highlight the correspondence between
d(g,E2) and the standard diffusivity function gTDg, where
D = 0. A distance measure that satisfies this property is
given by the following equation
given by dist(?E1,?E2)
2=
?[d(g,?E1) − d(g,?E2)]
2dg
where the integration is over the unit sphere (i.e. for all unit
dist(?E1,?E2)
2=
1
4π
?
[log(d(g,?E1)) − log(d(g,?E2))]
2dg
(3)
By analytically computing the integral, eq. 3 can be writ-
ten in the from of sum of squares, which is very fast to com-
pute. As an example in the case of 2ndorder EDTs eq. 3
can be evaluated using 8 additions and 10 multiplications , in
the 4thorder case using 47 add. and 33 mul.. Due to lack of
space we do not providethese formulas, since their derivation
is simple.
Note that the metric defined above is rotation invariant in
the case of any order exponential tensors. Furthermore, by
using this distance measure it is easy to prove that the mean
element?Eµis defined as the Euclidean average (?E1+ ... +
?EN)/N (or geometric mean of d1,...,dN) and the geodesic
(shortest path) between two elements?E1and?E2is defined as
Euclidean geodesic γ(t) = (1 − t)?E1+ t?E2, t ∈ [0,1].
In the following section we employ this distance measure
to define an anisotropy map of 2,4,6 & 8th-order EDTs.
2.2. Distance from the closest isotropic case
In the isotropic case the quantity d(g,?E) is the same con-
stant for every unit vector g, forming an isotropic sphere c =
log(d(g)) = c(g12+ g22+ g32)(K/2), where c ∈ ℜ and K
denotes the order of the symmetric tensor?E and is even. The
above equation is satisfied by: 2ndorder exponential diffu-
sion tensors of the form E = cI, 4thorder EDTs of the form
E4,0,0= E0,4,0= E0,0,4= c, E2,2,0= E0,2,2= E2,0,2=
2c, 6thorderEDTs of theformE6,0,0= E0,6,0= E0,0,6= c,
E4,2,0 = E4,0,2 = E2,4,0 = E0,4,2 = E2,0,4 = E0,2,4 =
3c, E2,2,2 = 6c, and 8thorder EDTs of the form E8,0,0 =
E0,8,0= E0,0,8= c, E6,2,0= E6,0,2= E2,6,0= E0,6,2=
E2,0,6 = E0,2,6 = 4c, E4,4,0 = E0,4,4 = E4,0,4 = 6c,
E4,2,2= E2,4,2= E2,2,4= 12c, where c is a scalar and the
rest of the elements of?E are equal to zero.
GivenanarbitraryKth-order?E, we cancomputetheclos-
est isotropy tensor coefficients?Eisoby finding the scalar c
that minimizes the distance of?E from the isotropic case. In
the 2nd-order case c = (E1,1+ E2,2+ E3,3)/3, in the 4th-
order case c = (E4,0,0+E0,4,0+ E0,0,4+ E2,2,0+E0,2,2+
E2,0,2)/9,inthe6th-ordercasec = (E6,0,0+E0,6,0+E0,0,6+
E4,2,0+E4,0,2+E2,4,0+E0,4,2+E2,0,4+E0,2,4+E2,2,2)/27
and in the 8th-order case c = (E8,0,0+ E0,8,0+ E0,0,8+
E6,2,0+E6,0,2+E2,6,0+E0,6,2+E2,0,6+E0,2,6+E4,4,0+
E0,4,4+ E4,0,4+ E4,2,2+ E2,4,2+ E2,2,4)/81.
The function fiso(?E) = dist(?E,?Eiso) maps the space of
?E to the space of non-negativereal numbers. The smaller the
value of the function, the closer is?E to the?Eiso. The behav-
ior of the function fisois similar to that of the well-known
fractionalanisotropy(FA) map of 2ndorderdiffusiontensors.
To illustrate this, we estimated the DT field and the EDT field
froma real dataset, and then we computedthe FA and the fiso
map respectively (Fig. 1a and 1b). Furthermore, we plot the
fisoas a function of FA in Fig.1d. The same figure also con-
tains the plot (in red) of the fitted function (−c)log(1 − FA)
for an estimated c = 0.318. The inverse of the above function
is 1 − exp(−(1/c)fiso(?E)) and can be used to map fisoto
values in the interval [0,1] as was done in Fig. 1c.

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Fig.1. ComparisonbetweenFA and4th-orderfisomapusing
rat optic chiasm data [6]. a) FA, b) fiso, c) fisomapped to
[0,1] using the fitted function of Fig. 1d, d) plot of fisovs.
FA
2.3. Estimation of EDT field from DWI
The coefficients of any order exponential diffusion tensor can
be estimated from diffusion weighted images (DWI) by min-
imizing the function E(?E,S0) =
M
?
i=1
(Si− S0e−bid(gi,?E))2,
where M is the number of diffusion weighted images associ-
ated with gradient vectors giand b-values bi, Siis the corre-
sponding acquired signal and S0is the zero gradient signal.
S0can either be assumed to be known or estimated simul-
taneously with the coefficients of?E. In our experiments we
minimized the above equation using simple gradient descent
algorithm, however any non-linear functional minimization
method can be used. One can use also additional regulariza-
tion terms in the above function in order to enforce smooth-
ness across the lattice.
2.4. Displacement probability profile
Studies on estimating the fiber orientation from the diffusiv-
ity profile has been shown that the peaks of the diffusivity
profile does not necessarily yield the orientations of the dis-
tinct fiber populations. One should instead employ the dis-
placement probability profiles should [6]. The displacement
probability P(R) is given by the Fourier integral P(R) =
?E(q)exp(−2πiq · R)dq where q is the reciprocal space
vided by the zero gradient signal and R is the displacement
vector.
In the case of 2ndorder exponential diffusion tensors the
peak of the diffusivity profile coincides with the peak of the
displacement probability profile. Therefore fiber orientation
canbe estimatedas the eigenvectorassociatedwith the largest
eigenvalue of matrix E. In higher order case, instead of find-
ing the maxima of P(R) we can find the maxima of the ex-
N
?
approximates the reciprocal space using the icosahedral tes-
sellation of the unit hemisphere, where q is a scalar and qi
are unit vectors corresponding to the tessellation. In the case
of third-order tessellation we have N = 81, which is the ap-
proximation that we used in our experiments. By computing
vector, E(q) is the signal value associated with vector q di-
pression:
i=1
?∞
−∞
E(qqi)exp(−2πiqqi·R)4πq2dq, which
TABLE A: COMPARISON OF DTI FRAMEWORKS
Frameworks: Affine invar. Log-Euc. EDT framework
Distance map0.19 sec
Smoothing5.65 sec
–
TABLE B: PROPERTIES OF DTI FRAMEWORKS
Properties / Frameworks Affine inv. Log-Euc. EDT
Affine. Invariance
Rotation Invariance
Fast DTI processing
Unconstrained estimation
Use of higher order tensors
0.45 sec
0.64 sec
0.04 sec
0.10 sec
X
XX
X
X
X
X
X
the integral analytically in the above approximation we have
P(R) ≈
√π
4N
N
?
i=1
exp
?
−α
4β
??
2
β5/2−
α
β3/2
?
(4)
where α = (2πqi· R)2, β = 4πtd(qi,?E) and t is the effec-
tive diffusion time.
3. EXPERIMENTAL RESULTS
In this section we presentexperimentalresults usingsynthetic
and real data. All the synthetic data were generated by simu-
latingtheMRsignalfromsinglefibersorfibercrossingsusing
the realistic diffusion MR simulation model in [7].
In order to compare the time performanceof the proposed
framework with other existing frameworks (affine invariant
[3], and Log-Euclidean [4]) for processing tensor fields we
synthesized a single row of a 2nd-order DTI and EDT field
of size 10000 and then we applied two simple calculations
on every pair of tensors: a)computing their distance and b)
smoothingbyfindingtheiraverageusingtheabovementioned
frameworks. According to the times reported in Table A, our
frameworkis the fastest. In the case of smoothing,it is signif-
icantly faster than the Affine invariant frameworkand asymp-
totically faster than the Log-Euclidean. A comparisonof their
properties is presented in Table B. Note that only the EDT
framework can be used for higher-order approximations.
Furthermore, we estimated the 4th-order exponential ten-
sor field of a dataset acquired from excised, perfusion-fixed
rat optic chiasm [6]. Figure 2 shows the displacement prob-
ability profiles computed from the estimated field. The prob-
ability profiles demonstrate the distinct fiber orientations in
the central region of the optic chiasm where myelinated ax-
ons from the two optic nerves cross one another to reach their
respective contralateral optic tracts. These orientation maps
are consistent with other studies on this anatomical region of
the rat nervous system [6]. Furthermore the fisomap (Fig.
1c) has slightly brighter intensities in the central region, com-
pared to the FA map (Fig. 1a). This is because FA uses 2nd-
order approximation, which fails in approximating the fiber

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Fig. 2. Displacement probability profiles of a 4th-order EDT
field from a rat optic chiasm data set [6]. In the background
the distance from the closest isotropy fisois shown.
crossings in this region and produces estimations close to the
isotropy (darker intensities).
In our framework, after having estimated the coefficient
vectors?E, we can use algorithms developed for vectorfield
processing in order to compute statistics (average, principal
components),interpolateEDTfields etc. Figure3(top)shows
some examples of processes for resolving fiber crossings, in-
terpolatingandcomputingprincipalcomponentsusingthepro-
posed framework, for the entire image of Fig. 2.
Finally Fig. 3 (bottom) presents a comparison of 4th-
order DTI [5] and EDT in estimating fiber orientations us-
ing simulated MR signal [7] for different amounts of Riccian
noise in the data. The errors observed by using our method
are significantly smaller than those of 4th-order DTI, which
conclusively validates the accuracy of our proposed method.
4. CONCLUSION
In this paper a novel framework for efficient high-order DT-
MRI processingwas presented. This frameworkensuresposi-
tivedefinitenessofthediffusivityprofileandcanbeemployed
for higher order approximations. A metric was developed
for doing computations between same order exponential ten-
sors. The isotropic cases and the distance map from the clos-
est isotropies were also analyzed. Comparisons of the pro-
posed framework with other existing DTI frameworks were
presented demonstrating the high efficiency of the proposed
method. Finally the proposed framework was validated using
simulated and real data from a rat optic chiasm.
Fig. 3. Top: Uses of the proposed framework for: resolv-
ing fiber crossings (8th-order example), geodesic interpola-
tion and calculating PCA (4th-orderexample)from dataset of
Fig. 2. Bottom: Comparison of 4th-order DTI and EDT in
estimating fiber orientations for different SNR in the data.
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