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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

Smoothing 5.65 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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