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Pattern Recognition

0031-32

doi:10.1

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journal homepage: www.elsevier.com/locate/pr

A Riemannian scalar measure for diffusion tensor images

Laura Astola a,�, Andrea Fuster b, Luc Florack a

a Department of Mathematics and Computer Science, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, The Netherlandsb Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, 5600 MB Eindhoven, The Netherlands

a r t i c l e i n f o

Keywords:

Riemann geometry

Diffusion tensor imaging

Ricci scalar

Finsler geometry

High angular resolution diffusion imaging

03/$ - see front matter & 2010 Elsevier Ltd. A

016/j.patcog.2010.09.009

esponding author.

ail address: [email protected] (L. Astola).

e cite this article as: L. Astola, et a0.1016/j.patcog.2010.09.009

a b s t r a c t

We study a well-known scalar quantity in Riemannian geometry, the Ricci scalar, in the context of

diffusion tensor imaging (DTI), which is an emerging non-invasive medical imaging modality. We

derive a physical interpretation for the Ricci scalar and explore experimentally its significance in DTI.

We also extend the definition of the Ricci scalar to the case of high angular resolution diffusion imaging

(HARDI) using Finsler geometry. We mention that the Ricci scalar is not only suitable for tensor valued

image analysis, but it can be computed for any mapping f : Rn-RmðmrnÞ.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Diffusion tensor imaging (DTI) is a non-invasive magneticresonance imaging technique that measures the intra-voxelincoherent motion of water molecules in tissue [1,4]. Here wefocus on its applications to study the brain white matter. In thiscontext, DTI is being used in clinical research, for example tolocalize major white matter tracts in the vicinity of tumors, thusassisting in surgical planning. DTI also enables the assessment ofwhite matter maturation in preterm infants. At each voxel, theinformation from diffusion weighted imaging (DWI) measure-ments is stored in a so-called diffusion tensor, which can berepresented by a 3�3 symmetric and positive definite matrix.One can construct different types of scalars based on thesediffusion tensors. Although a tensor contains more informationthan a scalar, scalar measures are indispensable for theirsimplicity and coordinate independent interpretation.

Indeed several scalar measures have been proposed in the DTIliterature. Typically they capture certain features of the tensorvalued image, giving some insight into the underlying tissuestructure. This in turn can indicate the presence of white matter-related pathologies. The most popular scalar measures are themean diffusivity (MD) and the fractional anisotropy (FA) [5].

In this paper we consider a well-known scalar quantity inRiemannian geometry, the Ricci scalar, in the analysis of DTIimages. In 2D image processing the Ricci scalar has been used forcurvature analysis [6]. The goal of this research is to evaluatewhether the Ricci scalar can provide additional information onwhite matter structures compared to the established scalarmeasures. We also extend this measure beyond DTI, to high

ll rights reserved.

l., A Riemannian scalar me

angular resolution diffusion imaging (HARDI), a framework whichhas certain advantages over DTI. We found promising preliminaryresults on simulated and phantom data showing negative valuesof the Ricci scalar at voxels with crossing structures.

This paper is organized as follows. In Section 2, we derive themetric tensors that are essential for computing curvatures fromthe DTI data, and show the definition of the Ricci scalar in detail.In Section 3 we give an intuitive physical interpretation of thisscalar measure and include a pseudo-code for the computation.Section 4 contains a brief survey of the popular scalar measures inDTI up to this date. In Section 5 we show results from severalexperiments using simulated, phantom and real DTI data. InSection 6 we take an extended definition of Ricci scalar for Finslerspaces and connect this to HARDI via higher- (than second) ordertensors. Finally in Section 7 we draw some conclusions anddiscuss the direction of future work.

2. Theory

In human tissue the random thermal motion of watermolecules is restricted by the surrounding microstructures.Therefore the range of displacement of an average particle canhave a directional bias. In DTI a second-order model is assumedand this range is determined by the diffusion tensor D. This tensorD can be computed from a collection of signals Si frommeasurements that are sensitive to molecular displacements inspatial direction va, using the following Stejskal–Tanner equation

Sa ¼ S0expð�b � vTaDvaÞ, a¼ 1,2,3, ð1Þ

where S0 is the non-weighted signal and b a known scalar.Because of the positivity of the measurements and theirsymmetry w.r.t. the origin, D is a symmetric, positive definitesecond-order tensor in dimension three. The physical unit of

asure for diffusion tensor images, Pattern Recognition (2010),

www.elsevier.com/pr

dx.doi.org/10.1016/j.patcog.2010.09.009

mailto:[email protected]



L. Astola et al. / Pattern Recognition ] (]]]]) ]]]–]]]2

diffusion is m2/s and so we must assign the inverted unit s/m2 tothe inverse g¼D�1 of the diffusion tensor. In this way we haveencoded in the tensor g the information of the average time neededfor particles to diffuse in certain directions. Thus g can be seen as ametric tensor [7,8] with large diffusion in a certain directioncorresponding to a short distance in the metric space. Such a tensordefines the inner products, i.e. the position dependent lengths andangles in the image that are induced by the diffusion. A number ofauthors have incorporated tools from Riemannian geometry in theanalysis of diffusion tensor images [9–14].

We use Einstein’s summation convention, meaning thatwhenever the same Latin index appears in subscript and super-script, a sum is taken over them as in the following example:

aiui :¼

Xi

aiui: ð2Þ

While in Euclidean space with standard Cartesian coordinates, aninner product of two vectors v, w is

/v,wS¼ dijviwj, ð3Þ

where dij denotes the (components) of the identity matrix, on thetangent space of a Riemann manifold the inner product of twotangent vectors v,w is

/v,wS¼ gijviwj, ð4Þ

and is thus determined by the metric tensor gij. The explicit schemesto compute the so-called covariant derivative and the Riemanncurvature, which we will use in the following, are in the Appendix.

Since we have a metric tensor g¼D�1, i.e. an inner product/,Sg defined at each image voxel we can compute Riemanniancurvatures. The Riemann curvature vector is

RðX,YÞZ ¼rXrY Z�rYrXZ�r½X,Y �Z, ð5Þ

where rV U is the covariant derivative [8] of U in direction V and

½U,V � ¼Ui @

@xiðVÞ�Vi @

@xiðUÞ: ð6Þ

It is a measure of the non-commutativity of the covariantderivative. In a Euclidean space R(X,Y)Z¼0 for all X,Y ,ZARn.Having defined the Riemann curvature vector, we can computethe so-called sectional curvature with respect to a planedetermined by two non-collinear vectors X,Y:

/RðX,YÞX,YS :¼ gijðRðX,YÞXÞiYj, ð7Þ

which is an inner product of curvature vector and one of its inputvectors. In dimension two, this is actually the Gaussian curvature[15]. By choosing a vector V, and taking the average of thesectional curvature w.r.t. every plane that contains vector V, weobtain the Ricci curvature in direction V, which indicates whetherthe geodesic with initial points in a small neighborhood of a given

Fig. 1. Left: a surface with positive Ricci scalar. Middle: a surface with Ricci

Please cite this article as: L. Astola, et al., A Riemannian scalar mdoi:10.1016/j.patcog.2010.09.009

point p, with initial direction V, tend to merge towards or divergeaway from the geodesic that goes through p with tangent vector V

[12]. This Ricci curvature can be computed as follows:

RicðVÞ ¼XXi?V

/RðXi,VÞXi,VS, ð8Þ

where Xi spans the orthonormal basis V?. Finally, by taking thesum of the Ricci curvatures in every spatial direction we end upwith the Ricci scalar

R¼Xa

RicðVaÞ, ð9Þ

where in general Va, a¼ 1, . . . ,n span an orthonormal basis oftangent space; n¼2 in dimension two. Alternatively, one cancompute the Ricci scalar simply as

R¼ gikgjl/RðUi,UjÞUk,UlS, ð10Þ

where for example in 3D (which is the dimension of interest here)by choosing the standard orthonormal basis U1¼X, U2¼Y, andU3¼Z. Details on how to compute this are in the Appendix.

Unlike the metric, the Ricci tensor is not positive definite,allowing for both positive and negative values of the Ricci scalar.This is a major difference with respect to the usual DTI scalarmeasures, which are typically positive. In dimension three theRicci scalar does not completely characterize the curvature butrepresents instead the average of the characterizing curvatures.

3. Interpretation

The Ricci scalar is a so-called intrinsic curvature, meaning thatit is not measured using extrinsic concepts such as the radii ofosculating circles that refer to the ambient Euclidean space thatcontains the manifold itself. It measures how much the volume ofa small ball on the manifold differs from the volume of a smallEuclidean ball with the same radius. For example, given an initialpoint on a surface, we can compute the geodesics of unit lengthwith all possible initial directions. By connecting the end points,we obtain a closed curve, whose length depends on the Ricciscalar of the surface. For illustration see Fig. 1. Let us consider themonkey saddle surface in Fig. 1 as a ‘‘warped’’ or distorted R2

space, i.e. the flat Euclidean plane. We can compute the metrictensors and visualize the corresponding ellipsoids that quantifythe distortion in x- and y-directions as in Fig. 2. We see that inthose areas where neighboring tensors have different orientationsthe Ricci scalar is negative. This is what we expect to happen alsoin higher dimensions. The Ricci scalar would then be a naturalindicator of inhomogeneities of tensors. In diffusion tensorimaging, such inhomogeneities can correspond to crossing/passing fiber bundles.

scalar zero. Right: a surface with everywhere non-positive Ricci scalar.

easure for diffusion tensor images, Pattern Recognition (2010),


Fig. 2. Left: the ellipsoids describing the metric tensors on the original domain. Right: Ricci scalars on the original domain.

Fig. 3. From left to right: a homogeneous isotropic tensor field R¼FA¼0, a homogeneous anisotropic field R¼0, FAa0, an inhomogeneous isotropic field Ra0, FA¼0, and

an inhomogeneous anisotropic field Ra0, FAa0.

L. Astola et al. / Pattern Recognition ] (]]]]) ]]]–]]] 3

4. Scalar measures in DTI

We briefly review the literature on scalar measures in diffusiontensor image analysis. Let l1,l2 and l3 be the eigenvalues of thediffusion tensor D. In [16], the trace

trðDÞ ¼ l1þl2þl3, ð11Þ

and anisotropy indices

l1=l2, l1=l3, l2=l3 ð12Þ

were introduced in the context of DTI. The mean diffusivity (MD)is defined as

l ¼ trðDÞ=3, ð13Þ

and it measures the average amount of diffusion in a voxel. Thefractional anisotropy (FA) [5] is defined as

FA¼

ffiffiffi3

2

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl1�lÞ2þðl2�lÞ2þðl3�lÞ2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2

1þl22þl

23

q : ð14Þ

The FA of an isotropic tensor is zero, and for a tensor with nonzerofirst eigenvalue and approximately vanishing second and thirdeigenvalues, the FA approaches value 1. If we represent thediffusion tensor with an ellipsoid, with semi-axes as theeigendirections with lengths proportional to eigenvalues, thismeans that a sphere has FA zero and an elongated cigar-shapedellipsoid has FA close to one. See Fig. 3 for an illustration of thedifferences between FA and the Ricci scalar.

As in [5], one can decompose a diffusion tensor into itsisotropic and anisotropic parts. In [5], this decomposition is doneby solving the eigenvalues, but alternatively this can be done as

Please cite this article as: L. Astola, et al., A Riemannian scalar medoi:10.1016/j.patcog.2010.09.009

follows. If we denote the isotropic and anisotropic parts of amatrix M as MI and MA, respectively, then

MI ¼1

3

M11þM22þM33 0 0

0 M11þM22þM33 0

0 0 M11þM22þM33

0B@

1CAð15Þ

and

MA ¼1

3

2M11�M22�M33 M12 M13

M12 2M22�M11�M33 M23

M13 M23 2M33�M11�M22

0B@

1CA:ð16Þ

The relative anisotropy (RA) [5] is the ratio

RA¼jMAjF

jMIjF, ð17Þ

where j jF denotes the Frobenius matrix norm. On the other hand,defining these quantities using eigenvalues makes it obvious thatthese are all rotational invariants and do not depend on the choiceof an orthonormal coordinate system. This is also evident from thefact that MI and MA are matrix representations of tensors. Anothermeasure introduced in the early years of DTI is the volume ratio(VR) [17]

VR¼l1l2l3

ðlÞ3: ð18Þ

In Fig. 4, we have plotted a surface that is swept by all possibletuples of ðl1,l2,l3Þ for which l1þl2þl3 ¼ c (i.e. the first octant ofa regular sphere), with samples of magnitudes of the VR and the




FA. Some additional invariant scalar measures that are derivedfrom the previous have been proposed in [18]. While FA and MDare the most popular scalar measures in clinical research, in[19,20] plotting the FA in the region of interest (ROI) on aðjMIjF ,jMAjF Þ�plane is seen to be useful.

So far, all the scalars in this section have been pointwisemeasures, that is they contain information only on the (zerothorder) diffusion tensor in a particular voxel. One of the firstmeasures that gather information also from the neighborhood of aparticular voxel is the lattice index (LI) [21,22]. For example whenthe diffusion is nearly isotropic, the eigenvectors of neighboringtensors have no correlation. On the other hand, it is reasonable toexpect that in case of coherently organized tissue (at least at asufficient resolution), there will be such a correlation. Let usdenote the componentwise product of two tensors A and B as/A,BSF , a reference tensor as M and a neighboring tensor as Ma,then a basic LI is

LI¼

ffiffiffi3

8

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/MI ,Ma

I Spffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/M,MaS

p þ/MI ,Ma

I S/M,MS

!: ð19Þ

By weighting LIs with the inverse of voxel distance a localcontextual anisotropy measure can be computed in any ROI w.r.t.the reference voxel [22]. Based on LI, another lattice indexmeasure which is more robust to noise has been proposed in [23].In one of the first proposals to use Riemannian geometry [12] toproduce differential scalar measures for DTI, it is suggested totake the inner product between the main eigenvector of thediffusion tensor and the most coherent direction of local diffusion,i.e. the eigenvector of the Ricci tensor that corresponds to the

Fig. 4. Left: the lengths of the blue arrows represent the magnitude of VR correspondi

lengths corresponding to FA. Indeed the FA is largest when one of the eigenvalues is larg

interpretation of the references to color in this figure legend, the reader is referred to

Fig. 5. Left: a simulated crossing of tensors. Middle: ellipsoids representing the diffusio

one for a 3D object in Mathematica). Right: a temperature map (blue is negative and red

we obtained negative values, also when we varied the angle of crossing. (For interpretat

version of this article.)


greatest eigenvalue. This measures the degree to which theprincipal eigenvector determined by a single tensor is alignedwith the locally most coherent direction. By this direction wemean the direction along which the neighboring geodesics, whichare the analogues of straight lines in the curved space distorted byanisotropic diffusion, tend to stick together. In DTI fiber trackingand segmentation it is necessary to use the information on theneighboring tensors. Ricci scalar is a mean curvature measure andas such takes the whole neighborhood into account. The diameterof the neighborhood can be tuned by a proper scale selection forthe derivative operator [24].

5. Experiments

In order to explore the geometric significance of the Ricciscalar, we have experimented with simulated, physical phantomand real data.

5.1. Simulated data

To get insight in what the Ricci scalar can detect in a tensorfield, we refer to Fig. 5, where we have simulated a crossing oforiented sets of tensors, modeling homogeneous diffusion tensorscorresponding to two fiber bundles. In the center of the crossingregion of this tensor field, the Ricci scalar tends to be negative.Since the Ricci scalar involves second-order derivatives (see theAppendix), the minimum size of the region to be considereddepends on the scale of the Gaussian differential operator [24,25].

ng to triple of eigenvalues ðl1 ,l2 ,l3Þ. Right: as on the left-hand side, but with the

e compared to the others and the VR is largest when all eigenvalues are equal. (For

the web version of this article.)

n profiles on the image slice in the middle of crossing (color coding is the default

is positive) of the Ricci scalars on the previous plane. In the middle of the crossing

ion of the references to color in this figure legend, the reader is referred to the web




5.2. Phantom data

We computed Ricci scalars on a real phantom consisting ofcylinder containing a water solution, three sets of crossingsynthetic fiber bundles and three supporting pillars on theboundary. In Fig. 6 we see that in the region close to the crossingbundles Ricci scalars have relatively large negative values, despitethe noisy nature of the DTI data. This might be explained by thefact that crossings can be related to saddle-shaped structures [2],for which the Ricci scalar takes negative values (see Section 3).Due to the resolution, we did not obtain exactly the same resultsas with the simulated data, which are a more ideal representationof a crossing structure.

5.3. Real data

We have also experimented with real DTI data of a rat brain.We plotted the Ricci scalars in a temperature map, to emphasizethe differences in sign. We identified positive (negative) outliersof the Ricci scalar data with maximum (minimum) values of therest of the data. The Ricci scalar gives information about the localspatial variations in diffusion tensor orientations unlike FA, which

Fig. 6. Top left: a physical phantom, with three crossings. Top right: a mean of HARDI i

foremost crossing area, marked by yellow dashed lines, solid line estimating the la

corresponding to previous area. Blue (red) means negative (positive). Bottom left: diff

scalars on the same slice. A planar picture cannot fully show the three dimensional s

references to color in this figure legend, the reader is referred to the web version of th


will identify tensors with similar anisotropy even if theirorientations differ. This can be seen e.g. in the boxed region inFig. 7, which is known to have complex structure [26]. This regioncontains the so-called STN (subthalamic nucleus), which iscurrently thought to play a prominent role in Parkinson’s disease.

6. Ricci scalar for high angular resolution diffusion imaging

In the previous we considered the Ricci scalar in Riemanniangeometry. In DTI, we see from the expression vi

TDvi in Eq. (1) thatthe diffusion profile (a spherical surface with diffusion constant asthe radius) is a second-order polynomial on the sphere. Thismodel is insufficient in voxels that contain two or more bundles ofaxonal fibers with different orientation. To be able to model morecomplex shapes of diffusion profiles, we use Finsler geometry[27–32], which is a general framework that also includes Riemanngeometry as a special case. Instead of DTI, we consider highangular resolution diffusion images (HARDI) [33] that containmore angular measurements than the DTI, although it is possibleto use (typically up to sixth order) Finsler model also on DTI. InRiemannian space, to each point we can associate a second-ordermetric tensor. In Finsler space we can associate a convex norm

mages from the top of the cylinder on the left. Middle left: diffusion tensors in the

teral center of the crossing. Middle right: Ricci scalars in ‘‘temperature map’’

usion tensors slightly off the plane that includes the crossing. Bottom right: Ricci

ituation. Physical phantom by courtesy of Pim Pullens. (For interpretation of the

is article.)



Fig. 7. Left: temperature map of Ricci scalars on a slice of the rat brain DTI image. Blue (red) indicates negative (positive) values. Middle: fractional anisotropy. Right: mean

diffusivity. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


function to each point. First question is how to construct such aconvex function from a HARDI measurement. For this we computethe so-called orientation distribution function (ODF) [34,3] thatrepresents the actual diffusivity profile. Then, all we need to do isto convexify this profile. This can be done for example assuggested in [30], or by representing the diffusion profile as annth order polynomial and then taking a nth root as suggested in[32]. Whatever the method, as soon as we have a (strongly)convex [27] modification of the spherical diffusion profile, we cancompute directional metric tensors gij(y) that locally approximatethe profile [28,32]. Let F(x,y) be the convex function, wherex¼(x1,x2,x3) stands for spatial direction and y¼(y1,y2,y3) the unittangent vector originating from x. Then the directional metrictensor gij is computed as follows:

gij ¼1

2

@2Fðx,yÞ

@yi@yj: ð20Þ

The Ricci curvature can be then computed in a similar manner tothe Riemannian case, keeping in mind that the tangent vector y isfixed. The Ricci scalar Ric(x,y) becomes then [28]

Ricðx,yÞ ¼ RiiðyÞ, ð21Þ

where

Rikðx,yÞ ¼ 2

@Giðx,yÞ

@xk�yj @

2Giðx,yÞ

@xj@yk

þ2Giðx,yÞ@2Giðx,yÞ

@yj@yk�@Giðx,yÞ

@yj

@Gjðx,yÞ

@yk: ð22Þ

The G here is the so-called geodesic coefficient [28]

Giðx,yÞ ¼1

4gilðx,yÞ 2

@gjlðx,yÞ

@xk�@gjkðx,yÞ

@xl

� �yjyk: ð23Þ

A drawback is that the formulae become more complicated andthat the interpretation becomes more difficult due to the y-dependence.

7. Discussion and outlook

The work described in this paper is only a first approach to thestudy of the Ricci scalar in the context of DTI. This is a novel ideaand, as such, further research is needed in order to assess itspossibilities and limitations. Our experiments with simulated andreal phantom DTI data have shown large negative values of theRicci scalar at crossings. It is to be expected that the same willhold for real brain data, although more experiments should beperformed to confirm this. The development of a 3D visualizationtool for the Ricci scalar might be crucial here, given the three-


dimensional character of both DTI data and Ricci scalar. On theother hand, conventional DTI measures such as FA generally takesimilar values at crossings and isotropic regions, and are not ableto distinguish between them.

The Ricci scalar may therefore be useful in voxel classification.After the classification, the regions with single orientation areidentified as the regular DTI data (second-order tensors), andhigher-order models (e.g. fourth-order tensors) can be used inregions where inhomogeneous fiber population is anticipated. Thecomputational cost of DTI data processing would be greatlyimproved in this way.

The Ricci scalar could also be useful in the so-called splittingtracking method in HARDI framework [35], by indicating thepotential bifurcation points of fiber bundles with large negativevalues. It goes without saying that DT-MRI images are not theonly possible applications, although they are especially suitablesimply because the metric tensors come with the data ‘‘for free’’.The physical interpretation of the generalized Ricci scalar as wellas its practical applications to the analysis of HARDI data is aninteresting problem for future research.

Appendix

To compute Riemannian Ricci scalars, we essentially need onlyto know the diffusion tensors and Riemann tensors. The necessaryingredients are then the diffusion tensors and their component-wise differentials up to order two. It may be convenient tocompute the Riemann tensors using Christoffel symbols gk

ij:

gkij ¼

1

2gkl @gil

@xjþ@gjl

@xi�@gij

@xl

� �, ð24Þ

where gijgjk ¼ dik. The components of the Riemann tensor can be

defined as

Rijks ¼/RðXi,XjÞXk,KsS¼ gms glikg

mjl þ

@

@xjgm

ik�gljkg

mil �

@

@xigm

jk

� �, ð25Þ

where Xl span the orthonormal basis on the tangent space. Wecomputed the necessary derivatives by applying Gaussianderivatives to the whole tensor volumes, storing the results, andthen taking the linear combinations indicated in Eqs. (24) and(25). Florack provides an alternative scheme consistent with thelog-Euclidean paradigm (see [24] and references therein). Whatmay look like a tedious task is really only book-keeping andfortunately in 3D there are only six independent components ofthe Riemann tensor. Although the expressions will get longer, wemay also use the fact that we can express the components ofRiemann tensor in terms of sectional curvatures [36], and thereare essentially only three of these in 3D.




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Laura Astola received her M.Sc. degree in computer assisted visualization in mathematics teaching from Helsinki University in 2000. She received her Licentiate’s degree inmathematics with a thesis on uniformly quasiregular mappings from Helsinki University of Technology in 2009. In 2010 she obtained her Ph.D. degree in appliedmathematics with a thesis on the applications of Riemann–Finsler geometry in diffusion weighted medical imaging from Eindhoven University of Technology. Currentlyshe works as a postdoc at the Center for Analysis Scientific computing and Applications (CASA), Eindhoven University of Technology.

Andrea Fuster received her M.Sc. degree in theoretical physics in 2001 from University of the Basque Country in Spain, having spent two years as an exchange student inUniversity of Groningen, The Netherlands, on grants from the Erasmus programme and the European Physical Society. She obtained her Ph.D. degree in 2007 from the FreeUniversity in Amsterdam, The Netherlands, with support from a Basque Government research grant and FOM (Foundation for Fundamental Research on Matter). The areaof her thesis was General Relativity. In particular, she studied a certain type of space-times with applications in Supergravity and Superstring Theory. In 2008 she moved toEindhoven University of Technology, Department of Biomedical Engineering, where she became a postdoc in the Biomedical Image Analysis group. Her research interestsfocus on tensor analysis and differential geometry in the context of (biomedical) image analysis.

Luc Florack received his M.Sc. degree in theoretical physics in 1989, and his Ph.D. degree cum laude in 1993 with a thesis on image structure, both from Utrecht University,The Netherlands. During the period 1994–1995 he was an ERCIM/HCM research fellow at INRIA Sophia-Antipolis, France, and INESC Aveiro, Portugal. In 1996 he was anassistant research professor at DIKU, Copenhagen, Denmark, on a grant from the Danish Research Council. In 1997 he returned to Utrecht University, where he became anassistant research professor at the Department of Mathematics and Computer Science. In 2001 he moved to Eindhoven University of Technology, Department of BiomedicalEngineering, where he became an associate professor in 2002. In 2007 he was appointed full professor at the Department of Mathematics and Computer Science, retaining aparttime professor position at the former department. His research covers mathematical models of structural aspects of signals, images, and movies, particularly multiscaleand differential geometric representations, and their applications to imaging and vision, with a focus on cardiac cine magnetic resonance imaging, high angular resolutiondiffusion imaging, and diffusion tensor imaging, and on biologically motivated models of ‘‘early vision’’.


10.1007/s10851-010-0217-3.3d

10.1007/s10851-010-0217-3.3d


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