A Fourth Order Tensor Statistical Model for Diffusion Weighted MRI

Application to Population Comparison

Theodosios Gkamas

, F

elix Renard

, Christian Heinrich

and St

ephane Kremer

ICube UMR 7357, University of Strasbourg – CNRS, F

eration de M

edecine Translationnelle de Strasbourg (FMTS),

300 bd S

ebastien Brant – CS 10413, 67412 Illkirch cedex, France

Gipsa-lab, 11 rue des Math

ematiques, Grenoble Campus BP 46, 38402 St Martin d’H

eres cedex, France

Keywords:

Diffusion-weighted MRI, Fourth Order Tensor, Non Euclidean Metric, Nonlinear Dimension Reduction,

Permutation Testing.

Abstract:

In this communication, we propose an original statistical model for diffusion-weighted magnetic resonance

imaging, in order to determine new biomarkers. Second order tensor (T2) modeling of Orientation Distribution

Functions (ODFs) is popular and has beneﬁted of speciﬁc statistical models, incorporating appropriate metrics.

Nevertheless, the shortcomings of T2s, for example for the modeling of crossing ﬁbers, are well identiﬁed. We

consider here fourth order tensor (T4) models for ODFs, thus alleviating the T2 shortcomings. We propose

an original metric in the T4 parameter space. This metric is incorporated in a nonlinear dimension reduction

procedure. In the resulting reduced space, we represent the probability density of the two populations, normal

and abnormal, by kernel density estimation with a Gaussian kernel, and propose a permutation test for the

comparison of the two populations. Application of the proposed model on synthetic and real data is achieved.

The relevance of the approach is shown.

1 INTRODUCTION

Diffusion-Weighted Magnetic Resonance Imaging

(DW-MRI) is a unique way to probe the diffusion of

water molecules in the human brain in vivo. The cor-

responding data provide information about the under-

lying white matter ﬁber tracts. Diseases such as mul-

tiple sclerosis, Alzheimer’s disease, and stroke can

be monitored using DW-MRI (Horsﬁeld and Jones,

2002).

Data stemming from DW-MRI require mathemat-

ical modeling. Diffusion Tensor Imaging (DTI) is a

popular description tool for such data (Jones, 2011).

The Orientation Distribution Function (ODF) of a

given voxel in white matter is accounted for by a six

parameter model, also known as second order tensor

(T2) model. The major limitation of the T2 model is

its inability to account for crossing ﬁbers. Neverthe-

less, accounting for crossing ﬁbers is important since

about half of the brain voxels are hosting crossings.

To this end, the T2 ODF model has been generalized

to a fourth order tensor (T4) model, comprising 15 pa-

rameters. This model encapsulates the T2 model as a

particular case, and is able to account for ﬁber cross-

ings (Weldeselassie et al., 2012). This versatility has

led us to rely on T4 modeling (Ozarslan and Mareci,

2003; Weldeselassie et al., 2012; Tuch, 2004).

The goal of the present study is to propose an orig-

inal statistical model related to T4 parameterization,

in order to provide more efﬁcient diagnosis and fol-

low up tools for pathologies of interest. In particular,

we aim at early diagnosis and at the determination of

new biomarkers. Many statistical description models

have been devised in the T2 framework (see e.g. (Ar-

signy et al., 2006)). The main concern is to account

for the particular geometry of the underlying space in

order to propose powerful tools. A popular model is

the log-Euclidean one (Arsigny et al., 2006). The con-

tribution of the present work is to propose a dedicated

metric and corresponding statistics in the T4 frame-

work.

A T4 ODF is parameterized by a vector in R

This working space will be equipped with a suited

metric. Besides, a 15-dimension space will be

sparsely ﬁlled with data: for a given voxel of interest,

we have a number of points corresponding to the num-

ber of patients (abnormal data) and controls (normal

data), which is orders of magnitude below the number

of points required to ﬁll R

. Reduction of dimen-

sion of this space will improve the robustness of the

subsequent statistical tests. We resort to a nonlinear

dimension reduction procedure (Isomap (Tenenbaum

277

Gkamas T., Renard F., Heinrich C. and Kremer S..

A Fourth Order Tensor Statistical Model for Diffusion Weighted MRI - Application to Population Comparison.

DOI: 10.5220/0005252602770282

In Proceedings of the International Conference on Pattern Recognition Applications and Methods (ICPRAM-2015), pages 277-282

ISBN: 978-989-758-077-2

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

et al., 2000)) which is now widely used. Isomap has

already been used for group comparison with classi-

cal statistical tests in the T2 framework (Verma et al.,

2007). In the reduced space (the space of features),

we propose here an original statistical discriminative

test between the two populations. This test allows to

detect the voxels of interest, permitting the discrimi-

nation between normal and abnormal and discovering

voxels where the populations are signiﬁcantly differ-

ent (i.e. biomarker).

This communication is organized as follows. The

proposed model is presented in section 2. We detail

in particular the choice of the metric, the reduction

of dimension, and the statistical test in the reduced

space. Experimental results on synthetic data and on

real data are shown in section 3. Conclusion is drawn

in section 4.

2 PROPOSED STATISTICAL

MODEL

This section presents the considered data parameteri-

zation, the chosen data normalization procedure, and

the proposed statistical model.

2.1 T4 ODF Parameterization

A second order semi-deﬁnite Cartesian tensor writes

f (g

g) = g

g, where g

g is a unit vector of R

cor-

responding to a given direction in space and D

D is a

matrix containing the coefﬁcients of the tensor. Such

a model may be used to represent ODFs (T2 ODF

model).

The generalization to fourth order semi-deﬁnite

Cartesian tensors writes:

f (g

g) =

∑

i+ j+k=4

i, j,k

(1)

where (g

, g

) are the components of the unit vec-

tor g

g and the D

i, j,k

’s are the coefﬁcients of the T4 ten-

sor. Such a model may be used to represent ODFs

(T4 ODF model). The coefﬁcients are found by mini-

mizing a quadratic cost function under constraints that

imposes also the positivity constraint to the estimated

tensor model (see (Weldeselassie et al., 2012) for de-

tails, see Fig. 1 for a T2–T4 comparison).

2.2 Data Normalization

DW-MRI data sets have to be normalized in a com-

mon reference space. Such a spatial normalization is

particularly cumbersome for diffusion data, since care

must be taken of the underlying ﬁbers.

Figure 1: ODF proﬁles for a given brain area. T2 model

(left), T4 model (right). As expected, the T4 model captures

more detail and variability than the T2 model.

Considering a T2 model, we compute Fractional

Anisotropy (FA) maps which are used to determine

a nonlinear transformation between the source and

the template data. The estimated transformation is

applied on the raw data. The mapped raw data are

then reoriented using the local rotation component

extracted from the transformation. This amounts to

reorientation of the underlying ﬁbers (see (Tao and

Miller, 2006; Duarte-Carvajalino et al., 2013)). In

practice, we used the patch proposed in (Duarte-

Carvajalino et al., 2013), and available in the FSL

toolkit (Jenkinson et al., 2012).

2.3 T4 Parameter Space Metric

We have to deﬁne a metric in R

, the T4 parameter

space. An Euclidean distance would not be conve-

nient, since it would not take into consideration the

speciﬁcity of the corresponding underlying ODF.

A point in R

corresponds to an ODF on the unit

sphere. The metric in R

should thus correspond to a

distance between two positive valued functions on the

unit sphere. Such a positivity constraint has led to log-

based distances (see e.g. (Tarantola, 2005)), as is for

example the case in the Kullback-Leibler divergence

or in the T2 log-Euclidean framework (Arsigny et al.,

2006).

Following (Tarantola, 2005), we deﬁne the dis-

tance between two ODFs d

and d

as:

D (d

, d

) =

Z Z



log

(θ, φ)



sinθ dθ dφ (2)

where θ and φ are the angular parameters on the

sphere. This distance induces a metric in R

, with d

and d

issued from two points in R

. This distance

is original in the DW-MRI context, to the best of our

knowledge. Other (i.e. non log-based) distances have

been proposed for T4s (see e.g. (Barmpoutis et al.,

2007; Du et al., 2014)).

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278

The computation is achieved by discretization in

θ and φ of the equation above. In practice, a T4

ODF proﬁle d

is a regression model of noisy data.

Moreover, the underlying non noisy data are not nec-

essarily perfectly adequate with a genuine T4 proﬁle.

Thereby, the precision in the T4 parameters and in the

corresponding ODF proﬁle is not inﬁnite. In addi-

tion, a small variation on low ODF values may have a

tremendous impact on D, because of a huge variation

of the logarithm. To avoid this harmful effect, we es-

timate the conﬁdence we have on ODF values by way

of a Monte Carlo study, thus yielding a standard de-

viation error for the ODF. This standard deviation is

determined off-line once for all, and each ODF value

at stake in the computation of D by a discrete sum is

corrected by addition of the standard deviation.

2.4 ODF Patches

In practice, it is hazardous to rely on the distance

D between two voxels, in particular because of po-

tential registration errors. This is why we resort to

a distance based on 3 × 3 × 3 voxel patches. The

distance between two patches is deﬁned as the sum

of the 3 × 3 × 3 distances between the enclosed cou-

ples of ODF proﬁles. Moreover, the matching patches

will be searched among all couples of patches in two

5 × 5 × 5 neighborhoods (one neighborhood for each

dataset). Another (popular) option would have been

to resort to some kind of data smoothing. We wanted

to avoid such a preprocessing, to avoid losing poten-

tial important information. Still another option would

have been to concentrate on skeletons of white mat-

ter ﬁber bundles (Smith et al., 2006), but still at the

expense of losing information.

2.5 Dimension Reduction

We now have a matrix of interpoint distances attached

to a voxel of interest (i.e. to a 5×5 × 5 ROI). Relying

on T4s, the underlying space would be R

, with s =

15 for a mere voxel (no patch), or s = 3 × 3 × 3 ×

15 = 405 for a 27-voxel patch. Given the number of

data points (in the order of 100) in usual studies, the

space is sparsely ﬁlled. Reducing the dimension will

provide more robust and reliable results.

We tested several nonlinear dimension reduction

methods, among which Isomap (Tenenbaum et al.,

2000), maximum variance unfolding (Weinberger and

Saul, 2006), and locality preserving projection (He

and Niyogi, 2003; He et al., 2005). There was no sig-

niﬁcant difference between the aforementioned meth-

ods, from a discrimination point of view. We ﬁnally

chose Isomap, which enables to reproduce the initial

structure of points in a reduced space, say R

or R

by preserving the geodesic interpoint distances. Scree

plots lead us to work in R

, as is the case in (Verma

et al., 2007).

2.6 Statistical Reduced Space Model

Each of the two populations (normal and abnormal)

corresponds to a set of points in the reduced space.

We represent the probability densities p

and p

cor-

responding to the populations using kernel density es-

timation, with a Gaussian kernel (Hastie et al., 2011).

One Gaussian kernel is attached to each point. There

is one parameter to be determined for each popula-

tion, which is the covariance matrix attached to each

point of this population. These matrices are deter-

mined using Scott’s rule (Scott, 1992).

The discrepancy between densities p

and p

has

to be quantiﬁed. The Kullback-Leibler divergence

and its symmetrized version are popular choices.

Nevertheless, in the case of a mixture of Gaussian

distributions corresponding to our kernel density es-

timation, there is no closed formula available and the

numerical estimation of the Kullback-Leibler diver-

gence is time consuming. We rely on another discrep-

ancy measure, noted P , which provides good results

with a low computation time (Sﬁkas et al., 2005). The

computation of P is about 150 times faster than the

computation of the symmetrical Kullback-Leibler dis-

tance in the present case. The discrepancy P writes:

P (p

, p

) =

− log







x) p

x) dx

x))

x +

x))







(3)

Considering the mixtures











x) =

∑

i=1

(a)



x; µ

(a)

, Σ

(a)



x) =

∑

j=1

(b)



x; µ

(b)

, Σ

(b)



(4)

a straightforward computation yields

x) p

x) dx

x =

∑

i, j

(a)

(b)

exp(k)

(2π)



(a)



(b)



(5)

where N

is the dimension of x

x and with

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279

V =





(a)



−1



(b)



−1



−1

(6)

k = µ

−1

µ − µ

(a)T



(a)



−1

(a)

−µ

(b)T



(b)



−1

(b)

(7)

µ = V





(a)



−1

(a)



(b)



−1

(b)



(8)

and from which we derive P (p

, p

) (we pick a in

{

1, 2

}

and b in

{

1, 2

}

). Indices i, j, a, b have been

dropped from V

V , k, and µ

µ for the sake of simplicity.

The preceding formulas may be simpliﬁed, since Σ

(a)

does not depend on i and since Σ

(b)

does not depend

on j.

The discrepancy P is a statistic. We have to esti-

mate the corresponding p-value ν

, i.e. the probability

of getting a distance P larger than the reference one

P (p

, p

) under the hypothesis that both populations

are indiscernible. This will be achieved resorting to

permutation testing. Besides, we want to determine a

conﬁdence interval associated to the estimation of ν

This interval will depend on the number of permu-

tations (i.e. label shufﬂings) used. To address these

issues, we will resort to the Bayesian framework.

We shufﬂe N times the labels of the points in the

reduced space, and for each shufﬂing we compute

the value of P . Comparison of the n

sample of P

with the reference P (p

, p

) yields a binary value q

which is a sample of the Bernoulli distribution of (un-

known) parameter ν

. The problem is now the es-

timation of ν

from binary samples q

, . . . , q

. Us-

ing a uniform prior for ν, we derive the posterior

p(ν|q

, . . . , q

). The conﬁdence interval (or credibil-

ity interval) in ν is taken here as being the smallest in-

terval capturing 99 % of the a posteriori mass, which

is known as Highest Probability Density (HPD) inter-

val in the Bayesian framework. We thus obtain a con-

ﬁdence interval for the p-value. If needed, the length

of the conﬁdence interval may be reduced by using

more shufﬂings.

Finally, the voxels are sorted according to in-

creasing p-values. The most dissimilar voxels (the

biomarkers) appear on top of the list. We do not ad-

dress multiple comparison issues here, since we do

not provide a list of voxels considered as signiﬁcantly

different between populations. This is left for future

work.

3 EXPERIMENTAL RESULTS

In this section, we present experimental results based

on synthetic data and on real data. The pathology of

(a) (b)

Figure 2: Reduced space for (a) an Euclidean distance and

(b) the proposed distance in presence of an outlier (red

point). The outlier is well isolated in (b), as should be the

case.

interest for real data is Neuromyelitis Optica (NMO)

(see e.g. http://en.wikipedia.org/wiki/Neuromye-

litis optica).

3.1 Material

The proposed statistical model was implemented in

Python. The FSL toolbox (Jenkinson et al., 2012) was

used for the registration step.

Synthetic data were generated using a code writ-

ten by A. Barmpoutis et al (Barmpoutis et al., 2009).

Real data are composed of 22 normal (control)

and 36 abnormal DW-MRI datasets. Brain acquisi-

tion consisted in High Angular Resolution Diffusion

Imaging (HARDI) with 30 non colinear directions of

gradients. The size of the images is 128 × 128 × 41

for a resolution of 1.8 × 1.8 × 3.5 mm

. The b-value

is equal to 1000 s/mm

3.2 Results

Inﬂuence of the metric. In Fig. 2, we compare two re-

duced spaces obtained respectively with an Euclidean

distance (L

norm) and with the proposed distance,

both on the ODF proﬁles and for T4 parameteriza-

tion. The outlier is an ODF resulting from a tensor

for which one coefﬁcient was divided by a factor 60.

As expected, this outlier is separated from the rest of

the group in the case of the proposed metric, but not

in the case of the Euclidean metric.

Reduced space model and statistical test. We

represent the reduced space for three voxels in Fig.

3. The corresponding p-value HPD intervals are re-

ported in Table 1. The ﬁrst voxel is typical of a dis-

similar conﬁguration for both populations, whereas

the second and third voxels are typical of similar con-

ﬁgurations for both populations.

The ﬁrst voxel examined indicates that the two

ODF populations are different (see Fig. 3, case (a)). If

we resort to smoothing instead of using patches, the

populations do not differ anymore according to this

voxel. Information has been lost in the smoothing,

thus hampering the diagnosis potential.

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(a)

(b)

(c)

Figure 3: Representation of probability densities in the re-

duced space. Green dots: normal cases; red squares: ab-

normal cases. (a) dissimilar populations; (b) and (c) similar

populations. Left column: representation of the probability

density corresponding to the normal population; right col-

umn: representation of the probability density correspond-

ing to the abnormal population (blue: low density; red: high

density. The same color scale is used across all subﬁgures.).

The densities correspond to kernel density estimation with

a Gaussian kernel.

Table 1: P-value HPD intervals of the cases depicted in

Fig. 3. We used 1000 shufﬂes. It took approximately 3 min-

utes on a standard computer to process each voxel. Most of

the computation time was devoted to determining the patch

correspondence.

Fig.3 p-value width of

Cases HPD interval interval Decision

(a) [0, 0.0046] 0.0046 Dissimilar

(b) [0.35, 0.43] 0.08 Similar

T2-T4 comparison on Synthetic Data. We con-

sider a case of two ﬁbers crossing at 90 degrees. Both

ﬁbers have the same amplitude. The two populations

differ by a rotation angle of 5 degrees affecting both

ﬁbers likewise. Noise was added to the synthetic data.

The T4 model is able to discriminate the two popula-

tions, whereas the T2 model is not able to do so (this

ﬁber crossing is seen by the T2 model as an isotropic

ODF, independently of the rotation angle).

Comparison of the proposed Statistical Test with

the Hotelling Test on Synthetic Data. We consider a

Figure 4: Distribution of p-values for a NMO-related ROI.

As expected, a large number of voxels have low p-values

and are thus identiﬁed as abnormal.

case of crossing ﬁbers, where the angle between the

two ﬁbers differs from one population to the other (the

values considered are 90 degrees for the normal pop-

ulation, and 85 degrees for the abnormal population).

The amplitude of the ﬁbers are different as well (the

amplitude is 1 for the ﬁber common to both popula-

tions; for the ﬁber with a different orientation in both

populations, the amplitude is 0.35 for the normal pop-

ulation, and 0.45 for the abnormal population).

The proposed statistical test is able to discriminate

both populations. Besides, we perform a Hotelling

test. This test takes place in the reduced space. Each

population is modeled by a Gaussian distribution, and

means are compared, looking for a statistical differ-

ence. The Hotelling test was not able to discriminate

both populations.

T2-T4 comparison on Real Data. We compared

the results (similar/dissimilar) issued from T2 model

and from T4 model. Whenever the results disagreed,

it happened that the T4 model gave the correct answer

and that the T2 model was not able to capture all infor-

mation conveyed by the data. This was validated by

testing the residuals (the dissimilarity information is

contained in the T2 residual). Using a log-Euclidean

distance on second order tensors to reduce the dimen-

sion of the data yields globally the same results as

those issued by the metric we propose, when applied

on T2 proﬁles.

P-value Distribution on Real Data. On Fig. 4, we

represent the distribution of p-values (one p-value for

each voxel), for a region of interest (ROI). The ROI is

identiﬁed as being sensitive to the disease of interest

(NMO). This is conﬁrmed on the plot, where a peak

of low p-values appears.

The proposed statistical T4 model is robust and

sensitive. On the evaluation tests proposed, it showed

better performance than models relying on Euclidean

) norms or on T2 ODF proﬁles. The proposed

model is thus pertinent to extract biomarkers with a

view to early diagnosis in NMO. Other pathologies

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are concerned as well.

4 CONCLUSION

We proposed an original statistical model for fourth

order tensor ODF modeling of DW-MRI. This models

incorporates a suitable metric in the parameter space.

It relies on nonlinear dimension reduction, and on

an original statistic in the reduced space, allowing to

compare two populations and to extract biomarkers.

This approach has shown better ability to discriminate

two populations, as compared to models relying on

other metrics, on T2 ODF proﬁles, or on the Hotelling

test. It has thus a potential for early diagnosis.

Resorting to T4 ODF proﬁles will also enable us

to identify more precisely the changes effectively tak-

ing place between the two populations, as compared

to changes identiﬁed by T2 models where the proﬁles

are described less accurately. This is left for future

work.

REFERENCES

Arsigny, V., Fillard, P., Pennec, X., and Ayache, N. (2006).

Log-Euclidean metrics for fast and simple calculus on

diffusion tensors. Magnetic Resonance in Medicine,

56(2):411–421.

Barmpoutis, A., Bing, J., and Vemuri, B. (2009). Adap-

tive kernels for multi-ﬁber reconstruction. In Proceed-

ings of the Information Processing in Medical Imag-

ing conference (IPMI), volume 21, pages 338–349.

Barmpoutis, A., Vemuri, B., and Forder, J. (2007). Regis-

tration of high angular resolution diffusion MRI im-

ages using 4th order tensors. In Int. Conf. on Med-

ical Image Computing and Computer Assisted Inter-

vention (MICCAI), volume 4791 of Lecture Notes in

Computer Science, pages 908–915. Springer.

Du, J., Goh, A., Kushnarev, S., and Qiu, A. (2014).

Geodesic regression on orientation distribution func-

tions with its application to an aging study. NeuroIm-

age, 87:416–426.

Duarte-Carvajalino, J., Sapiro, G., Harel, N., and Lenglet,

C. (2013). A framework for linear and non-linear reg-

istration of diffusion-weighted MRIs using angular in-

terpolation. Frontiers in Neuroscience, 7.

Hastie, T., Tibshirani, R., and Friedman, J. (2011). The ele-

ments of statistical learning. Springer, second edition.

He, X. and Niyogi, P. (2003). Locality preserving projec-

tions. In Advances in Neural Information Processing

Systems 16 (NIPS), volume 16, pages 153–160.

He, X., Yan, S., Hu, Y., Niyogi, P., and Zhang, H.-J. (2005).

Face recognition using Laplacianfaces. IEEE Trans-

actions on Pattern Analysis and Machine Intelligence,

27(3):328–340.

Horsﬁeld, M. A. and Jones, D. K. (2002). Applications of

diffusion-weighted and diffusion tensor MRI to white

matter diseases – a review. NMR in Biomedicine,

15(7-8):570–577.

Jenkinson, M., Beckmann, C., Behrens, T., Woolrich, M.,

and Smith, S. (2012). FSL. NeuroImage, 62(2):782–

790.

Jones, D. K. (2011). Diffusion MRI: theory, methods, and

applications. Oxford University Press.

Ozarslan, E. and Mareci, T. H. (2003). Generalized dif-

fusion tensor imaging and analytical relationships be-

tween diffusion tensor imaging and high angular res-

olution diffusion imaging. Magnetic Resonance in

Medicine, 50(5):955–965.

Scott, D. (1992). Multivariate density estimation: theory,

practice, and visualization. Wiley.

Sﬁkas, G., Constantinopoulos, C., Likas, A., and Galat-

sanos, N. (2005). An analytic distance metric for

Gaussian mixture models with application in image

retrieval. In 15th International Conference on Arti-

ﬁcial Neural Networks (IEEE ICANN 2005), volume

3697 of Lecture Notes in Computer Science, pages

835–840. Springer.

Smith, S., Jenkinson, M., Johansen-Berg, H., Rueckert, D.,

Nichols, T., Mackay, C., Watkins, K., Ciccarelli, O.,

Cader, M. Z., Matthews, P., and Behrens, T. (2006).

Tract-based spatial statistics: voxelwise analysis of

multi-subject diffusion data. NeuroImage, 31(4):1487

– 1505.

Tao, X. and Miller, J. (2006). A method for registering

diffusion weighted magnetic resonance images. In

Int. Conf. on Medical Image Computing and Com-

puter Assisted Intervention (MICCAI), volume 4191

of Lecture Notes in Computer Science, pages 594–

602. Springer.

Tarantola, A. (2005). Elements for physics: quantities,

qualities, and intrinsic theories. Springer.

Tenenbaum, J. B., Silva, V., and Langford, J. C. (2000). A

global geometric framework for nonlinear dimension-

ality reduction. Science, 290(5500):2319–2323.

Tuch, D. S. (2004). Q-ball imaging. Magnetic Resonance

in Medicine, 52(6):1358–1372.

Verma, R., Khurd, P., and Davatzikos, C. (2007). On an-

alyzing diffusion tensor images by identifying mani-

fold structure using isomaps. IEEE Transactions on

Medical Imaging, 26(6):772–778.

Weinberger, K. and Saul, L. (2006). Unsupervised learning

of image manifolds by semideﬁnite programming. In-

ternational Journal of Computer Vision, 70(1):77–90.

Weldeselassie, Y., Barmpoutis, A., and Atkins, M. (2012).

Symmetric positive-deﬁnite Cartesian tensor ﬁber ori-

entation distribution (CT-FOD). Medical Image Anal-

ysis, 16(6):1121–1129.

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