J Math Imaging VisDOI 10.1007/s10851-010-0217-3

A New Tensorial Framework for Single-Shell High AngularResolution Diffusion Imaging

Luc Florack · Evgeniya Balmashnova · Laura Astola ·Ellen Brunenberg

© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract Single-shell high angular resolution diffusionimaging data (HARDI) may be decomposed into a sumof eigenpolynomials of the Laplace-Beltrami operator onthe unit sphere. The resulting representation combines thestrengths hitherto offered by higher order tensor decompo-sition in a tensorial framework and spherical harmonic ex-pansion in an analytical framework, but removes some ofthe conceptual weaknesses of either. In particular it admitsanalytically closed form expressions for Tikhonov regular-ization schemes and estimation of an orientation distributionfunction via the Funk-Radon Transform in tensorial form,which previously required recourse to spherical harmonicdecomposition. As such it provides a natural point of de-parture for a Riemann-Finsler extension of the geometricapproach towards tractography and connectivity analysis ashas been stipulated in the context of diffusion tensor imaging(DTI), while at the same time retaining the natural coarse-

L. Florack (�)Department of Mathematics & Computer Science andDepartment of Biomedical Engineering, Eindhoven Universityof Technology, Eindhoven, The Netherlandse-mail: L.M.J.Florack@tue.nl

E. Balmashnova · L. AstolaDepartment of Mathematics & Computer Science, EindhovenUniversity of Technology, Eindhoven, The Netherlands

E. Balmashnovae-mail: E.Balmashnova@tue.nl

L. Astolae-mail: L.J.Astola@tue.nl

E. BrunenbergDepartment of Biomedical Engineering, Eindhoven Universityof Technology, Eindhoven, The Netherlandse-mail: E.J.L.Brunenberg@tue.nl

to-fine hierarchy intrinsic to spherical harmonic decomposi-tion.

Keywords High angular resolution diffusion imaging ·Diffusion tensor imaging · Tikhonov regularization ·Orientation distribution function · Riemann-Finslergeometry

1 Introduction

High angular resolution diffusion imaging (HARDI) has be-come a standard MRI technique for mapping apparent wa-ter diffusion processes in fibrous tissues in vivo. The termi-nology HARDI is used here to collectively denote schemesthat employ generic functions on the unit sphere, includingTuch’s orientation distribution function (ODF) via classicalQ-Ball imaging [44], the higher order diffusion tensor modeland the diffusion orientation transform (DOT) by Özarslanet al. [36, 37], analytical Q-Ball imaging [14, 15], and thediffusion tensor distribution model by Jian et al. [27], etcetera.

A homogeneous polynomial expansion of HARDI dataon the sphere has been proposed in a seminal paper byÖzarslan and Mareci [36], which we shall refer to on sev-eral occasions further on. However, in this paper we con-sider an inhomogeneous expansion, including all (even) or-ders up to some fixed N (including N = ∞), and exploitthe redundancy of such a representation. The idea is to con-struct a polynomial series on the unit sphere in such a waythat higher order terms capture residual degrees of freedomthat cannot be represented by a lower order polynomial, akinto the hierarchical structure of a spherical harmonic expan-sion. (A different higher order tensor model for the unre-stricted case, with a specific focus on the representation of

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the apparent diffusion coefficient (ADC), has been proposedby Liu et al. and Jensen et al. in terms of formal expansionsin the b-parameter [26, 33].)

Interestingly, each homogeneous term in the new poly-nomial representation turns out to be self-similar under theLaplace-Beltrami operator on the unit sphere, mimickingthe behaviour of its spherical harmonic equivalent. Thisself-similar nature is the fundamental property underlyingmany operations on HARDI that have hitherto been con-sidered only in the context of a spherical harmonic decom-position, such as Tikhonov regularization and estimation ofan orientation distribution function (ODF) from the ADC[13, 14, 25, 38]. Regularization is of interest, since, unlikeDTI [10, 11, 30], a generic HARDI model accounts for ar-bitrarily complex diffusivity profiles, thus raising the con-comitant demand for regularization. ODFs are of interest fortheir putative connection to the complex architecture of fi-brous tissue.

The higher order tensor representation introduced byÖzarslan and Mareci (loc. cit.) does not exhibit self-similarity, making it somewhat problematic in a context ofregularization or ODF estimation. (In this context, a func-tion is called self-similar if it retains its form under regu-larization, except possibly for an overall attenuation factor.)The typical way to deal with this is to proceed via a spher-ical harmonic representation, which is inconvenient if theapplication context urges a tensorial approach, such as thegeometric rationale towards tractography and connectivityanalysis in classical DTI [4, 5, 16, 31, 38, 39]. Our poly-nomial representation can be seen as a formal refinementto the extent that degrees of freedom captured by a homo-geneous N -th order tensor are decoupled and hierarchicallyrearranged into self-similar monomials of degrees up to N ,obviating an explicit spherical harmonic detour altogether.(Spherical harmonics do play behind-the-scenes, as the spanof our homogeneous terms will be seen to coincide with thatof the spherical harmonics of corresponding order.)

2 Theory

We consider a higher order tensor representation of theform1

u(y) =∞∑

k=0

ui1...ik yi1 . . . yik . (1)

1Summation convention applies to identical index pairs, i.e. wheneveran identical index symbol occurs twice, once in lower and once in up-per position, one tacitly sums over its possible values. Thus (1) is meantto be read as u(y) = ∑∞

k=0∑n

i1=1 . . .∑n

ik=1 ui1...ik yi1 . . . yik . For k = 0a product like yi1 . . . yik evaluates to unity by default, and a holor likeui1...ik is to be understood as an index-free quantity, u [35, 42].

By yi and yi = ηij yj we denote the components of vec-

tors (e.g. spatial directions), respectively dual covectors (e.g.normalized diffusion sensitizing gradients), confined to theunit sphere, and by ηij (respectively ηij ) the components ofthe Euclidean metric tensor (respectively its inverse) in theembedding 3-space. In Cartesian coordinates ηij = ηij = 1iff i = j , otherwise 0. As outlined in the previous sectionu(y) generically represents any HARDI-related scalar ob-servable that can be represented as a function on the unitsphere. In typical cases in practice it is stipulated that

u(y) = u(−y), (2)

in which case only even orders will be of interest. This sym-metry property, or equivalently the identification of antipo-dal points, −y ∼ y, implies that y actually represents ori-entation, not direction. We will only occasionally need thissymmetry, in which case it will be explicitly indicated. Notethat u(y) represents a single codomain sample (e.g. signalstrength as a function of direction/orientation at a fiducialpoint in space), its dependence on the spatial domain vari-able x ∈ R

3 is suppressed in the notation.The collection of polynomials on the sphere,

B =⋃

k∈Z+0

Bk with Bk = {yi1 . . . yik | k ∈ Z

+0

}, (3)

is complete, but redundant. Apart from the fact that odd or-ders might be of no interest, redundancy is evident fromthe fact that lower order even/odd monomials can be repro-duced from higher order ones of equal parity through con-tractions as a result of ηij yiyj = 1 (unit sphere confinementor “single-shell” restriction). Thus any monomial of orderk ≤ N ∈ Z

+0 is linearly dependent on the set of N -th or-

der monomials of equal parity. This, of course, justifies theapproach by Özarslan and Mareci, in which data are fittedagainst linear combinations of N -th order monomials, dis-carding all lower order terms.

However, exploiting the redundancy of B, (3), we maychoose to encode residual information in higher order tensorcoefficients. As N → ∞ this residual tends to zero, whileall established tensor coefficients of ranks lower than N re-main fixed in the process of incrementing N . The coeffi-cients are constructed as follows. Suppose we are in posses-sion of ui1...ik for all k = 0, . . . ,N − 1, then we consider thefunction

EN(uj1...jN ) =∫ (

u(y) −N∑

k=0

ui1...ik yi1 . . . yik

)2

dy (4)

to find the N -th order coefficients by minimization (integra-tion here and henceforth takes place over the unit sphere).Setting

∂EN(uj1...jN )

∂ui1...iN= 0, (5)

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one obtains the following linear system for the contravarianttensor coefficients ui1...iN :

Ai1...iN j1...jNuj1...jN

=∫

u(y)yi1 . . . yiN dy −N−1∑

k=0

Ai1...iN j1...jkuj1...jk , (6)

with symmetric covariant tensor coefficients

Ai1...ik =∫

yi1 . . . yik dy.

Note that the second inhomogeneous term on the r.h.s. of (6)is absent in the scheme proposed by Özarslan and Mareci.By symmetry considerations constant odd-rank tensors van-ish:

Ai1...i2k+1 = 0 (k ∈ Z+0 ). (7)

All even-rank constant tensors must be products of theEuclidean metric tensor, so we stipulate

Ai1...i2k= γkη(i1i2 . . . ηi2k−1i2k), (8)

for some constant γk . Parentheses symbolize index sym-metrization, i.e. if Ti1...ip is a rank-p tensor, then

T(i1...ip) = 1

p!∑

π

Tπ(i1...ip), (9)

in which the summation runs over all p! permutations π

of index positions. The constant γk needs to be determinedfor each k ∈ Z

+0 . One way to find γk is to evaluate (8) for

i1 = · · · = i2k = 1 in a Cartesian coordinate system, sincethe symmetric product of metric tensors on the r.h.s. evalu-ates to 1 for this case:

γk = A1...←2k indices→...1 =∫

y2k1 dy. (10)

This is a special case of the closed-form multi-index repre-sentation by Folland [21] and Johnston [28]:

∫y

α11 . . . yαn

n dy = 2

Γ ( 12 |α| + n

2 )

n∏

i=1

Γ

(1

2αi + 1

2

), (11)

if all αj are even (otherwise the integral vanishes). Here|α| = α1 + · · · + αn = 2k denotes the norm of the multi-index, and

Γ (t) =∫ ∞

0st−1e−sds (12)

is the gamma function. Relevant properties: Γ (�) = (�− 1)!and Γ (� + 1

2 ) = (� − 12 ) . . . 1

2

√π = (2�)!√π/(4��!) for

� ∈ Z+0 .

From (6) and (7–8) it follows that, for general spatial di-mension n,

Ai1...i2k= 2Γ (k + 1

2 )Γ ( 12 )n−1

Γ (k + n2 )

η(i1i2 . . . ηi2k−1i2k). (13)

Equations (7) and (13) form the tensorial counterpart of (11).Note that the coefficients Ai1...ik are independent of thephysical interpretation of the expansion on the r.h.s. of (1)as long as its construction is based on an energy minimiza-tion principle of the form (4) for the observable of interest.

The interesting property of our inhomogeneous expan-sion, viz. that it realizes a (partial) hierarchical ordering ofdegrees of freedom akin to the one induced by spherical har-monic decomposition, is manifest in the following observa-tion. Consider the Laplace-Beltrami operator Δ on the unitsphere, then for any N ∈ Z

+0 ∪ {∞} and t > 0,

uN(y, t) ≡ etΔuN(y) =N∑

k=0

ui1...ik (t)yi1 . . . yik , (14)

with

ui1...ik (t) = e−k(k+1)tui1...ik . (15)

A proof of (14–15) is provided elsewhere [17], where itis shown that the span of the homogeneous polynomi-als ui1...ik yi1 . . . yik for fixed k coincides with that of thespherical harmonics Ykm(y) (with m = −k,−k + 1, . . . ,

k − 1, k) of the same order, and hence constitutes a de-generate eigenspace of the Laplace-Beltrami operator witheigenvalue −k(k + 1). (It is understood, in this case, thatn = 3, and that the coordinates y ∈ R

3 of the embeddedunit sphere are parameterized in terms of the usual sphericalcoordinates θ,φ.) This is nontrivial, since the monomialsyi1 . . . yik themselves are not eigenfunctions of Δ. Indeed,(14–15) show that the degrees of freedom in the polynomialexpansion are segregated in such a way that we may inter-pret each homogeneous term as an incremental refinementof detail relative to that of the lower order expansion. To ap-preciate the significance of (14), note that uN(y, t) satisfiesthe heat equation on the unit sphere:

∂u

∂t= 1√|g|∂μ

(gμν

√|g|∂νu)

= Δu, (16)

in which the initial condition corresponds to the N -th or-der expansion of the raw data, uN(y,0) = uN(y), and |g| =detgμν . Here the Riemannian metric of the embedded unitsphere is given by

gμν = ∂yi

∂ξμηij

∂yj

∂ξν, (17)

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in which ξμ (μ = 1, . . . , n− 1) parameterize the unit spherein R

n. It is evident from the above that the linear com-binations ui1...ik yi1 . . . yik , unlike the monomials yi1 . . . yik

themselves, are eigenfunctions of the heat operator exp(tΔ),i.e. self-similar polynomials on the unit sphere, with eigen-values e−k(k+1)t . Recall that the heat operator can be seenas the canonical resolution degrading2 semigroup opera-tor [29]. The parameter t captures inverse angular resolu-tion at which the raw HARDI data are resolved (at a fixedpoint in space). Indeed, (14) shows that each homogeneouspolynomial (k fixed) retains its form upon blurring (increas-ing t), up to a t-dependent attenuation factor e−k(k+1)t . Thischaracteristic decay is analogous to the e−t‖ω‖2

-attenuationof a Gaussian blurred image in the Euclidean frequency do-main, with frequency coordinate ω ∈ R

n. Cf. Descoteauxet al. [14] and Hess et al. [25] for similar regularizationschemes expressed relative to a spherical harmonic basis,and to Florack et al. [18] for a comparison between thespherical harmonic and tensorial approaches towards regu-larization.

3 Examples

We consider some examples of possible application contextsfor the generic theory presented in the previous section.

3.1 Relation to Higher Order DTI and Regularization

In this example we consider the Stejskal-Tanner equa-tion [43],

S(y) = S0 exp (−bD(y)) , (18)

and identify the generic function u(y) from the previous the-ory with the ADC D(y), considered as a function of orien-tation (in particular, D(y) = D(−y)).

Here are some examples of (13) for the relevant case,n = 3:

k = 0: A = 4π,

k = 1: Aij = 4π

3ηij ,

k = 2: Aijk� = 4π

15(ηij ηk� + ηikηj� + ηi�ηjk).

The corresponding linear systems, (6), are as follows:

AD =∫

D(y)dy,

AijDj =

∫D(y)yidy − AiD,

Aijk�Dk� =

∫D(y)yiyj dy − AijD − AijkD

k.

2“Degrading” is meant in a sense akin to cartographic generalization,i.e. without negative connotation.

It follows that the scalar constant D is just the average dif-fusivity:

D =∫

D(y)dy∫dy

. (19)

The constant vector Di vanishes identically, as it should(since no basis independent constant vectors exist). For therank-2 tensor coefficients we find the traceless matrix

Dij = 15∫

D(y)yiyj dy − 5∫

D(y)dyηij

2∫

dy, (20)

and so forth. If, instead, we fit a homogeneous second orderpolynomial to the data (by formally omitting the

∑-terms

on the r.h.s. of (6)), as proposed by Özarslan and Mareci, weobtain the following rank-2 tensor coefficients:

DÖ.M.

ij = 15∫

D(y)yiyj dy − 3∫

D(y)dyηij

2∫

dy, (21)

which are clearly different. However, one may observe thatthe full second order expansions coincide. The difference incoefficients, in this example, is explained by the contributionalready contained in the lowest order term of our polyno-mial, which in Özarslan and Mareci’s scheme has migratedto the second order tensor.

In fact, we conjecture that equality holds to any order N .More precisely, if DN(y) denotes the truncated expansionof (1) including monomials of orders k ≤ N only, withcoefficients constructed according to (6–13), DÖ.M.

N (y) theN -th order homogeneous polynomial expansion proposedby Özarslan and Mareci, and DS.H.

N (y) the canonical spheri-cal harmonic decomposition, cf. Frank [22] and Alexanderet al. [1], then

DN(y) = DÖ.M.

N (y) = DS.H.

N (y). (22)

(The formal limit N → ∞ makes sense only for left andright hand sides.) This shows the equivalence of all threerepresentations.

In particular (14) reveals that the classical rank-2 DTIrepresentation, defined via the Stejskal-Tanner formula,(18), arises not merely as an approximation under the as-sumption that the apparent diffusion coefficient can be writ-ten as

D(y) ≈ DDTI(y) = DijDTIyiyj , (23)

but expresses the exact asymptotic behaviour of D(y, t) =D∞(y, t) as t → ∞, recall (14):

D(y, t) =(Dηij + e−6tDij

)yiyj

︸ ︷︷ ︸DDTI(y, t) = D

ijDTI(t)yiyj

+O(e−20t ). (24)

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Thus the DTI tensor is not self-similar, but has a bimodalresolution dependence. The actual limit of vanishing reso-lution is of course given by a complete averaging over thesphere:

limt→∞D(y, t) = lim

t→∞DDTI(y, t) = D, (25)

recall (19).The generalized counterpart of (24) suggests a Finsler [6,

7, 34] rather than Riemannian [5, 16, 31, 32, 38, 39] frame-work for tractography and connectivity analysis, wherebyone replaces the contravariant rank-2 (dual metric) tensorDDTI(y, t) by

DijHARDI(y, t) = Dij

DTI(t) +∞∑

k=1

Di1...ikij (t)yi1 . . . yik , (26)

with multimodal additional y-dependent terms as definedin (14). For details on Finsler geometry the books by Baoet al. [8] and Shen [41] are highly recommended. Litera-ture on Riemannian geometry is abundant. Spivak [42] andMisner et al. [35] are classics. Rund [40] provides a usefulvariational perspective.

3.2 Derivation of an ODF via the Funk-Radon Transform

In the second example we identify the function u(y) fromSect. 2 with a raw HARDI signal (again at an implicitly de-fined spatial locus x ∈ R

3). The precise way of identificationwill become clear soon.

To begin with, we consider the hypothetical signal func-tion S(q) defined for all Euclidean diffusion wave vectorsq ∈ R

3, and recall its relation to the diffusion probabilityfunction P(r), with Euclidean displacement vector r ∈ R

3,via Fourier transformation:

P(r) =∫

R3S(q)e2πir·qdq. (27)

Considerations of signal-to-noise ratio and acquisition timehave led Tuch [44] to introduce the single-shell orienta-tion diffusion function (ODF), defined on the unit sphere‖y‖ = 1,

Ψ (y) =∫ ∞

0P(λy)dλ, (28)

and to consider an approximation ΨQ(y) ≈ Ψ (y) in theform of the so-called extended Funk-Radon transform interms of a single-shell acquisition in q-space:

ΨQ(y) =∫

R3S(q)δ(q · y)δ(‖q‖ − Q)dq, (29)

in which Q > 0 denotes the radius of the acquisition shell inq-space. This “Q-ball” reconstruction has gained popular-ity because of the data acquisition efficiency of single-shellrecordings and for its robustness (for not too large Q).

There exist similar analytical approaches towards Q-ballimaging, cf. Descoteaux et al. [14], Anderson [2], and Hesset al. [25], all of which exploit spherical harmonic decompo-sition. We focus on the approach by Descoteaux et al. basedon the so-called Funk-Hecke theorem, which yields for anyspherical harmonic function Y�m(y) of capital order �

∫

‖η‖=1δ(y · η)Y�m(η)dη = 2πY�m(y)P�(0) (30)

in which P�(t) denotes the Legendre polynomial of de-gree �. Note that for even � = 2k we have

P2k(0) = (−1)k(2k)!22k(k!)2

= (−1)k√π

Γ (k + 12 )

Γ (k + 1). (31)

Taking into account that the homogeneous polynomials, i.e.the terms in (1) and (14) with k fixed, belong to the span ofthe spherical harmonics of the same order k, (29) and (30)allow us to write the ODF in tensor representation with ten-sorial coefficients Si1...ik induced by the signal S = S∞,

S(y) =∞∑

k=0

Si1...ik yi1 . . . yik , (32)

according to the general recursive scheme given by (6):

Ψ (y) = 2π

∞∑

k=0

Pk(0)Si1...ik yi1 . . . yik . (33)

Analogous to (14) we may incorporate a regularization para-meter via a suitable generator based on the Laplace-Beltramioperator on the unit sphere, e.g.

Ψ (y, t) = 2π

∞∑

k=0

Pk(0)Si1...ik (t)yi1 . . . yik , (34)

with t-dependent attenuation of coefficients, recall (15).(Truncation at finite order N results in an additional regu-larization effect.)

In particular, the example shows that no explicit detourvia a spherical harmonic representation is needed to achievea closed-form analytical Q-ball representation within a ten-sorial framework. The resulting diffusion ODF is equivalentto the one obtained from the established analytical Q-ballalgorithm in the spherical harmonic basis. This is of inter-est in Q-ball applications where the tensor formalism is thepreferred choice, such as in a differential geometric (notablyFinslerian) approach towards tractography and connectivityanalysis.

3.3 Spatial Regularization

Realizing that the unit-sphere function and its coefficientsintroduced in (14–15) also depend on a spatial variable

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x ∈ R3, we may consider to combine regularization in the

spatial domain (with a control parameter s > 0 for spatialcoherence, cf. Assemlal et al., Chen et al. and Goh et al.[3, 12, 24]) with codomain regularization (with the indepen-dent parameter t > 0 for signal smoothing at fixed position),

uN(x, y, s, t) = esΔx etΔy uN(x, y)

=N∑

k=0

ui1...ik (x, s, t)yi1 . . . yik , (35)

in which Δx is the standard Laplacian for Euclidean 3-space, and Δy the one appropriate for the unit sphere (Δ inthe foregoing). For example, on an unbounded domain theoperator esΔx can be identified (in spatial representation) asGaussian convolution:

ui1...ik (x, s, t) = e−k(k+1)t (ui1...ik ∗ φs)(x), (36)

in which

φs(x) = 1√

4πs3

exp

(−‖x‖2

4s

). (37)

Other types of regularization and domain boundary restric-tions can be straightforwardly accounted for. In particularthis combined domain–codomain regularization can be ap-plied to the polynomial representations of the spatially ex-tended ADC function D(x,y) from Sect. 3.1 and signalfunction S(x, y) from Sect. 3.2.

As a special case, recall the Q-Ball representation of (34)and consider it as a function of position in Euclidean spaceR

n, then we may write the doubly-regularized ODF as

Ψ (x, y, s, t) = 1

(2π)n

∞∑

k=0

∫

Rn

eiω·xe−s‖ω‖2e−k(k+1)t

× Ψ̂ i1...ik (ω)yi1 . . . yik dω, (38)

with spatial Fourier coefficients

Ψ̂ i1...ik (ω) =∫

Rn

e−iω·xΨ i1...ik (x)dx, (39)

and, in terms of the coefficients of the actual raw signal,

Ψ i1...ik (x) = 2πPk(0)Si1...ik (x). (40)

This double-frequency representation, (38–40), reveals theeffect of simultaneous regularization in the (infinite) spa-tial domain and on the (compact) unit sphere as a high-frequency attenuation process (with continuous, respec-tively discrete frequency spectrum for ω and k). Of course,an equivalent representation can be obtained using the spher-ical harmonic basis {Ykm|k = 0,1,2, . . . ;−k ≤ m ≤ k} in-stead of the redundant set B of (3).

In the Riemann geometric rationale the (pointwise) in-verse of the contravariant diffusion tensor image is identifiedwith the (covariant) metric tensor. In the Finsler geometricrationale one may instead stipulate a Finsler norm functionF(y) as a generalization of a Riemannian metric induced

norm√

gij yiyj , so that the corresponding (regularized) dual

Finsler metric is given by

gij (x, y, s, t) = ∂2Ψ (x, y, s, t)

∂yi∂yj

. (41)

By definition (ignoring our control parameters for regular-ization)

gij (x, y) = ∂2F 2(x, y)

∂yi∂yj, (42)

is called the (covariant) Finsler metric, and so we have thefollowing relation with the ODF, recall (38):

∂2Ψ (x, y, s, t)

∂yi∂yk

∂2F 2(x, y, s, t)

∂yk∂yj= δ

ji . (43)

We stipulate this as our conjecture for a rigorous Riemann-Finsler geometric approach towards tractography and con-nectivity analysis.

To actually prove this conjecture beyond heuristics amore thorough investigation of this “correspondence prin-ciple” will need to be conducted. In any case, the interestingfeature of any Finsler metric, such as (41), is its directionaldependence, i.e. its dependence on y, which distinguishesit from a Riemannian metric. The latter arises as a specialcase, either as the hypothetical case in which all coefficientsof orders k > 2 happen to vanish identically, or, in an oper-ational sense, in the asymptotic limit of large regularizationt � 0 in the signal codomain akin to (24). We refer to theliterature for details on Riemann-Finsler geometry [8].

Figure 1 shows a synthetic example of a single-shellsignal, illustrating the effect of combined domain andcodomain regularizations. The example was created usinga Gaussian mixture model:

S = 1

2S0

(e−bgT D1g + e−bgT D2g

), (44)

in which D1 and D2 are linear tensors with principal eigen-vectors oriented along stipulated fibers, and eigenvalues(3,3,17) × 10−4 mm2/s. The b-value is 1000 s/mm2 andthe number of gradient directions 80.

The glyphs in the first column of Fig. 1 vary from topto bottom due to spatial averaging only (a Gaussian scalespace process), i.e. no regularization is applied to their lo-cal profiles. On the other hand, going from left to right, weobserve a simplification of these profiles that is exclusivelydriven by pointwise regularization in the signal domain, i.e.

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Fig. 1 Artificial data ofcrossing fibers built with aGaussian mixture model, andperturbed by Rician noise withSNR = 15.3. From left to right:signal (codomain)regularization, witht = 0.00,0.018,0.050,0.14.From top to bottom: spatial(domain) regularization, withs = 0.00,0.22,0.37,0.61.Recall (35). Cf. text forexplanations

a parallel process involving no spatial neighbourhood inter-actions. In general one will need to incorporate both typesof regularization for reasons of robustness and dependingon data and objective. It is not a priori self-evident that asingle pair of parameters (s, t) (“scale selection”) may workin practical applications, although it is an obvious first steptowards more sophisticated attempts to solve the notorious

problem of scale (“deep structure”), e.g. through a coarse-

to-fine approach. Even in the one-parameter case this is still

an outstanding problem [20, 29].

Figure 2 shows an RGB map representing the princi-

pal diffusion directions (main eigenvector of DTI tensor) in

a Wistar rat brain (red: left-right, green: anterior-posterior,

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Fig. 2 Brain of 17-week oldmale Wistar rat, and region ofinterest

blue: superior-inferior). The luminance is weighted by thefractional anisotropy value.

The rat brain data were measured on a 9.4T BrukerBiospec AVANCE-III system. HARDI was acquired usinga diffusion-weighted spin-echo sequence with two unipo-lar pulsed field gradients placed symmetrically around the180 degree pulse (TE 27 ms, TR 4000 ms, NA 1, total time19 hours). Fifteen coronal slices with slice thickness 500 µmand interslice gap 50 µm were measured. The matrix sizewas 128 × 128, zero-filled to 256 × 256 pixels. The FOVwas 25.6 × 25.6 mm2, leading to an in-plane pixel dimen-sion of 100 µm. A series of 54 images with different gra-dient directions and b-value 3000 s/mm2, together with anunweighted image, was measured.

The enlarged ROI3 on the right-hand side in Fig. 2 fo-cuses on the junction of the internal capsule (large whitematter bundle, represented as a bright green structure, partlyseen in the lower left corner of the ROI) and the zona in-certa (gray matter, lying just above the internal capsule). Inthe zona incerta, many small fiber bundles are intermingled,causing crossings to occur in the HARDI data. These fiberbundles include tracts from the subthalamic nucleus to morelateral nuclei such as the globus pallidus.

Figure 3 shows the combined (three-dimensional) regu-larization in domain and codomain for the ROI. Again, no-tice that the apparent smoothing effects visible in the glyphrepresentations have various causes. On the top row smooth-ing is entirely due to individual glyph regularization (low-ering of angular resolution). In the left column smoothingof glyphs is caused by Gaussian weighted spatial averag-ing only (lowering of spatial resolution). For tractographypurposes one may want to control both spatial and angularresolution depending on the size of the structures of interest,signal-to-noise ratio, robustness, et cetera.

3In human subjects, proper fiber tracking in a similar region might beimportant for neurosurgical purposes such as the planning of electrodeimplantation for deep brain stimulation of the subthalamic nucleus.

4 Conclusion and Recommendations for Future Work

We have posited a tensorial decomposition of HARDI-related functions on the unit sphere in terms of inhomo-geneous polynomials. The resulting polynomial represen-tation may be regarded as the tensorial counterpart of thecanonical spherical harmonic decomposition. Although for-mally equivalent to the homogeneous polynomial expan-sion proposed by Özarslan and Mareci [36], our inhomo-geneous expansion differs in an essential way. It segregatesthe HARDI degrees of freedom into a hierarchy of homoge-neous polynomials that are self-similar under the act of reso-lution degradation induced by the heat operator, exp(tΔ), orsimilar Tikhonov regularization operators of the type f (tΔ),each with a characteristic decay that depends on order (forfixed t ∈ R

+).The polynomial framework is generically applicable to

single-shell representations. In the context of HARDI wehave illustrated its potential use by several examples. Oneis the well-posed extension of rank-2 DTI to arbitrary rankswhile simultaneously controlling regularity via a control pa-rameter t > 0, exploiting the special properties of the ho-mogeneous terms. The asymptotic case of almost vanish-ing resolution (t → ∞) reproduces the diffusion tensor ofclassical diffusion tensor imaging (DTI), with one constantand one resolution-dependent mode. Whereas this exampleis based on the Stejksal-Tanner equation for the ADC, a sec-ond example involves only the HARDI signal itself, andmimics the popular analytical Q-ball rationale within thetensorial framework, again by virtue of the special proper-ties of the homogeneous terms in the polynomial expansion.A third example shows how the representations can be si-multaneously regularized in spatial and signal domains, in-troducing a second regularization parameter s > 0 (scale)for spatial coherence. A suggestion for future research isto consider regularization in domain and/or codomain, (35),for sufficiently small negative values of s and t , in com-bination with truncation at some optimal order N(s, t) forthe purpose of enhancement (inverse diffusion). Truncationmakes this scenario well-posed, but amplification of noise

J Math Imaging Vis

Fig. 3 Recall Fig. 2, and (35).From left to right: signal(codomain) regularization, witht = 0.00,0.0025,0.018,0.14.From top to bottom: spatial(domain) regularization, withs = 0.00,0.10,0.15,0.22. Cf.text for explanations

requires a careful balance of order versus enhancement pa-rameter(s). A similar case has been studied in the context ofgrey-value images (trivial codomain) and spatial deblurring[19] as an extension of traditional image sharpening. ODFenhancement in the HARDI codomain may facilitate fibertracking.

In all cases, and unlike in previous work, analyticalmanipulations are carried out directly within the tensorialframework, i.e. without the need to map terms to a sphericalharmonic basis. This is especially attractive in the contextof a differential geometric rationale. The theory in this pa-per naturally suggests a shift of paradigm for tractographyand connectivity analysis based on a Finslerian rather thanRiemannian geometry combined with Tikhonov-like regu-larization, and provides the basic operational concepts forthis.

Several major conceptual issues will need to be addressedin the future in order to apply Finslerian geometry to HARDIproblems. One concerns the requirement of positivity, bothdictated by the physical nature of diffusion weighted imag-ing as well as by the axiomatic constraint imposed on theFinsler function [8]. Positivity has been addressed in the

case of fourth order representations, cf. Barmpoutis et al.and Ghosh et al. [9, 23]. A second problem concerns therigorous derivation of the Finsler function itself in termsof HARDI measurement data and the underlying physics ofwater diffusion. Our suggestion in this paper is of a heuristi-cal nature. A third problem concerns the fact that choiceswill need to be made when extending methods for trac-tography and connectivity analysis from the Riemannian tothe more general Finslerian framework, for these extensionsare not necessarily unique. For instance, the uniqueness ofan a priori reasonable (torsion-free) connection compati-ble with the metric in the Riemannian framework is lost inthe Finslerian case, and ambiguities will need to be takenaway by physical considerations and/or experimental vali-dation.

Acknowledgements The Netherlands Organisation for ScientificResearch (NWO) is gratefully acknowledged for financial support. Wethank Erwin Vondenhoff and Mark Peletier for fruitful discussions, andVesna Prckovska and Paulo Rodrigues for software implementations.

Open Access This article is distributed under the terms of the Cre-ative Commons Attribution Noncommercial License which permits

J Math Imaging Vis

any noncommercial use, distribution, and reproduction in any medium,provided the original author(s) and source are credited.

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Luc Florack was born in 1964 in Maastricht, The Netherlands, wherehe attended primary and secondary school. He received his M.Sc. de-gree in theoretical physics in 1989, and his Ph.D. degree cum laudein 1993 with a thesis on image structure, both from Utrecht Uni-versity, The Netherlands. During the period 1994–1995 he was anERCIM/HCM research fellow at INRIA Sophia-Antipolis, France, and

INESC Aveiro, Portugal. In 1996 he was an assistant research profes-sor at DIKU, Copenhagen, Denmark, on a grant from the Danish Re-search Council. In 1997 he returned to Utrecht University, were he be-came an assistant research professor at the Department of Mathematicsand Computer Science. In 2001 he moved to Eindhoven University ofTechnology, Department of Biomedical Engineering, were he becamean associate professor in 2002. In 2007 he was appointed full professorat the Department of Mathematics and Computer Science, retaining aparttime professor position at the former department. His research cov-ers mathematical models of structural aspects of signals, images, andmovies, particularly multi-scale and differential geometric representa-tions, and their applications to imaging and vision, with a focus on theanalysis of tagging magnetic resonance imaging for the heart and dif-fusion magnetic resonance imaging for the brain, and on biologicallymotivated models of “early vision”.

Evgeniya Balmashnova was born in 1976 in Ust-Ilimsk, Russia. Shereceived her M.Sc. degree in mathematics at Novosibirsk State Uni-versity, Russia, in 2000. In 2001–2003 she followed the postgraduateprogram Mathematics for Industry at the Stan Ackermans Institute ofEindhoven University of Technology, The Netherlands. She receivedher Ph.D. degree in 2007 from Eindhoven University of Technologyunder supervision of professor Bart ter Haar Romeny and associateprofessor Luc Florack. She is currently a postdoctoral fellow at theDepartment of Mathematics and Computer Science at the same univer-sity. Her research is focused on medical image analysis, high angularresolution diffusion imaging and diffusion tensor imaging, and in par-ticular, on multi-scale approaches in image analysis.

Laura Astola was born in 1967 in Turku, Finland. She got her highschool diploma from Tottori East prefectural high school in Japan.She began her studies in mathematics at Turku University, and sub-sequently moved to Helsinki University. In April 2000 she finishedher M.Sc. thesis on “Computer-Aided Visualization in Mathemat-ics Teaching” and began her Ph.D. studies on geometric analysis inHelsinki University of Technology. She completed her Licenciate’sthesis “Lattès Type Uniformly Quasi-Regular Mappings on CompactManifolds” in February 2009, obtaining the degree of Licenciate ofTechnology. In February 2006 she started as a Ph.D. student within theBiomedical Image Analysis group of Eindhoven University of Tech-nology. In 2007 she continued her Ph.D. research as a member ofCASA (Center for Analysis, Scientific Computing and Applications) atthe Department of Mathematics and Computer Science of EindhovenUniversity of Technology. In January 2010 she obtained her Ph.D. de-gree under the supervision of professor Luc Florack. The title of herthesis is “Multi-Scale Riemann-Finsler Geometry: Applications to Dif-fusion Tensor Imaging and High Angular Resolution Diffusion Imag-ing”.

Ellen Brunenberg was born in 1983 in Weert, The Netherlands. Shereceived her M.Sc. degree in Biomedical Engineering from EindhovenUniversity of Technology, The Netherlands, in 2007. She is currentlydoing her Ph.D. in Biomedical Engineering at the same university, sup-ported by a TopTalent grant from the Netherlands Organization forScientific Research, under the supervision of professor Bart ter HaarRomeny. Her research is focused on image analysis for deep brain stim-ulation in Parkinson’s patients.