Finsler Geometry on Higher Order Tensor Fields - Materials ...

Int J Comput Vis (2011) 92: 325–336DOI 10.1007/s11263-010-0377-z<strong>Finsler</strong> <strong>Geometry</strong> on Higher Order Tensor Fieldsand Applications to High Angular Resolution Diffusion ImagingLaura Astola · Luc FlorackPublished online: 31 August 2010© The Author(s) 2010. This article is published with open access at Springerlink.comAbstract We study 3D-multidirectional images, using<strong>Finsler</strong> geometry. The application considered here is in medicalimage analysis, specifically in High Angular ResolutionDiffusion Imaging (HARDI) (Tuch et al. in Magn. Reson.Med. 48(6):1358–1372, 2004) of the brain. The goal is toreveal the architecture of the neural fibers in brain whitematter. To the variety of existing techniques, we wish to addnovel approaches that exploit differential geometry and tensorcalculus.In Diffusion Tensor Imaging (DTI), the diffusion of wateris modeled by a symmetric positive definite second ordertensor, leading naturally to a Riemannian geometric framework.A limitation is that it is based on the assumption thatthere exists a single dominant direction of fibers restrictingthe thermal motion of water molecules. Using HARDIdata and higher order tensor models, we can extract multiplerelevant directions, and <strong>Finsler</strong> geometry provides thenatural geometric generalization appropriate for multi-fiberanalysis. In this paper we provide an exact criterion to determinewhether a spherical function satisfies the strong convexitycriterion essential for a <strong>Finsler</strong> norm. We also showa novel fiber tracking method in <strong>Finsler</strong> setting. Our modelincorporates a scale parameter, which can be beneficial inview of the noisy nature of the data. We demonstrate ourmethods on analytic as well as simulated and real HARDIdata.L. Astola () · L. FlorackDepartment of Mathematics and Computer Science, EindhovenUniversity of Technology, PO Box 513, 5600 MB Eindhoven,The Netherlandse-mail: l.j.astola@tue.nlL. Floracke-mail: l.m.j.florack@tue.nlKeywords <strong>Finsler</strong> geometry · Higher order tensors · HighAngular Resolution Diffusion Imaging (HARDI)1 IntroductionDiffusion Weighted Imaging (DWI) is a non-invasive medicalimaging modality that measures the attenuation of MagneticResonance Imaging (MRI) signal due to thermal motionof water molecules. High Angular Resolution DiffusionImaging (HARDI) is a collective name for techniques to analyzea dense sample of diffusion weighted measurements,typically ranging from 50 to 200 angular directions. It isassumed that the motion of water molecules reveals relevantinformation of the underlying tissue architecture. Theso-called apparent diffusion coefficient D(g), is computedfrom the signal S(g) using the Stejskal and Tanner (1965)formulaS(g)= exp(−bD(g)), (1)S 0where g is the gradient direction, S 0 the signal obtainedwhen no diffusion gradient is applied, and b is a parameterassociated with the imaging protocol (Mori 2007). Thisformula assumes that the diffusion is Gaussian distributed.This is indeed justified in view of the large number of hydrogennuclei (≈ 10 18 /voxel) and the central limit theorem(Rosner 2006).In the Diffusion Tensor Imaging framework, (1) is interpretedasS(g)S 0= exp(−bg T Dg), (2)with a direction independent 3 × 3 two-tensor D describingthe probability of directional diffusivity at each voxel.

326 Int J Comput Vis (2011) 92: 325–336A natural way to do geometric analysis on the image, isto use the inverse of the diffusion tensor D as the Riemannmetric tensor. The heuristic for this is that large diffusionimplies short travel time (distance) for an ensembleof diffusing particles. This approach has been exploited tosome extent in the DTI literature (O’Donnell et al. 2002;Lenglet et al. 2004; Astolaetal.2007; Astola and Florack2008; Fuster et al. 2009). Since HARDI data typically containsmore directional measurements than DTI, it requiresa model that has richer directional information than a localposition dependent inner product i.e. Riemannian metric.Higher order tensor representations (Özarslan and Mareci2003; Barmpoutis et al. 2007; Florack and Balmashnova2008) form an interesting alternative to the popular sphericalharmonic representation of HARDI data and this is especiallysuited for a <strong>Finsler</strong> geometric approach. <strong>Finsler</strong> geometryhas already been considered for HARDI analysis inMelonakos et al. (2008), where the necessary homogeneitycondition is imposed on the <strong>Finsler</strong> norm by definition. InSMelonakos et al. (2008) a third power of a ratioFR(S/S 0 ) ofthe signal S and the Funk-Radon transform of the attenuationFR(S/S 0 ) is used as the cost function, and the optimalpaths that minimize geodesic distances w.r.t. this cost functionare computed using dynamic programming.In this paper a different approach is proposed, usinghigher order monomial tensors to approximate the sphericalfunctions that represent the complex diffusion/fiber orientationprofiles, such as the orientation distribution function(ODF), and constructing a homogeneous <strong>Finsler</strong> norm fromthese. Fibers are not modeled here as geodesics. For examplethe subcortical fibers, having high curvature but modestanisotropy, cannot be modeled as geodesics neither w.r.t.the Riemann metric (D −1 ) nor the <strong>Finsler</strong> norm we proposehere. Instead, the streamline tractography, commonly usedin DTI, is generalized here to the HARDI case.This paper is organized as follows. Section 2 contains avery short introduction to <strong>Finsler</strong> geometry and in Sect. 3 weverify that indeed HARDI measurements can be modeledwith a <strong>Finsler</strong>-structure and give a specific condition whichensures this. In Sect. 4 an explicit method to transforma polynomial tensor presentation, which allows Laplace-Beltrami smoothing, to a monomial one convenient for constructinga <strong>Finsler</strong>-norm is given. In Sect. 5 we show someresults of <strong>Finsler</strong> fiber-tracking on simulated data sets aswell as on two human brain HARDI data. In the appendixwe show the details of computing the new strong convexitycriterion.2 <strong>Finsler</strong> <strong>Geometry</strong>In a perfectly homogeneous and isotropic medium, geometryis Euclidean, and shortest paths are straight lines.In an inhomogeneous space, geometry is Riemannian andthe shortest paths are geodesics induced by the Levi-Civitaconnection (Carmo 1993). The Levi-Civita connection is aset of rules that tells one how to take derivatives on a Riemannianmanifold. If the connection is linear, symmetricand satisfies the product (Leibniz) rule, then it is unique.If a medium is not only inhomogeneous, but also anisotropic,1 i.e. it has an innate directional structure, the appropriategeometry is <strong>Finsler</strong>ian (Bao et al. 2000; Shen2001) and the shortest paths are correspondingly <strong>Finsler</strong>geodesics.As a consequence of the anisotropy, the metrictensor (to be defined below) depends on both position anddirection. This is also a natural model for high angular resolutiondiffusion images.Another way to look at the difference of Riemann and<strong>Finsler</strong> geometry is that while Riemannian distances are definedby a position dependent inner product, <strong>Finsler</strong>ian distancesare defined by a direction dependent inner product,computed from a position dependent norm. Since the levelsets determined by norms can be more complex than those(ellipsoidal) given by an inner product, this approach is suitablefor modeling complex diffusion profiles.Definition 1 Let M be a differentiable manifold and T x Mbe a (Euclidean) tangent space (approximating the manifoldat each point) at x ∈ M. LetTM= ⋃ x∈M T xM be the collectionof tangent spaces. Take y to be a vector with positivelength in T x M. A <strong>Finsler</strong> norm is a function F : TM →[0, ∞) that satisfies each of the following criteria:1. Differentiability: F(x,y) is C ∞ on TM.2. Homogeneity: F(x,λy)= λF (x, y).3. Strong convexity: The <strong>Finsler</strong> metric tensor, derived fromthe <strong>Finsler</strong> norm, with componentsg ij (x, y) = 1 ∂ 2 F 2 (x, y)2 ∂y i ∂y j , (3)is positive definite at every point (x, y) of TM.√Note that if F(x,y) = g ij (x)y i y j , then g ij (x, y) = g ij (x)i.e. it reduces to a Riemannian metric.3 <strong>Finsler</strong> Norm on HARDI Higher Order Tensor FieldsWe want to show that higher order tensors, such as thosefitted to HARDI data, do define a <strong>Finsler</strong> norm, which can1 We will call a medium isotropic if it is endowed with a direction independentinner product, or Riemannian metric. In the literature sucha medium is also often referred to as anisotropic due to the directionalbias of the metric itself.

Int J Comput Vis (2011) 92: 325–336 327be used in the geometric analysis of this data. We take as apoint of departure a given orientation distribution function(ODF), which if normalized, is a probability density functionon the sphere. Such a spherical function, assigning unitvectors a probability that it coincides with fiber- or diffusionorientations, can be computed from HARDI data byusing one of the methods described in the literature (Tuch2004; Jansons and Alexander 2003; Özarslan et al. 2006;Jian et al. 2007; Descoteaux et al. 2006). The ODF is assumedto be symmetric w.r.t. the origin i.e. an even orderfunction. A higher order symmetric tensor restricted tothe sphere, can be defined as follows. Let y be a unit vectory(θ, ϕ) = (u(θ, ϕ), v(θ, ϕ), w(θ, ϕ))= (sin θ cos ϕ,sin θ sin ϕ,cos θ). (4)It is well known that every even order real spherical polynomialup to order l can be represented as a linear combinationof the spherical harmonics basisY0 0 −2(θ, ϕ), Y2 (θ,ϕ),...,Yl l−1 (θ, ϕ), Yl l (θ, ϕ). (5)In tensor formulation we use the following basis (omittingthe arguments (θ, ϕ))u l ,u l−1 v 1 ,u l−1 w 1 ,...,u p v q w r ,...,w l (6)where p + q + r = l. For simplicity we consider fourthorder tensors (l = 4) in what follows, but any even ordertensor can be treated in similar way.We denote a fourth order symmetric tensor (using theEinstein summation convention, i.e. a i b i = ∑ i a ib i )as,T(x,y) = T ij kl (x)y i y j y k y l . (7)We fit such a tensor to the ODF-data, using linear leastsquares method, following Özarslan and Mareci (2003).Suppose we obtain measurements m 1 ,m 2 ,...,m N in N differentdirections. First we recall that the number of distinctnth order monomials of 3 variables (u, v and w)is(3 + n − 1)!. (8)(3 − 1)!n!Thus a fourth order symmetric tensor is completely determinedby 15 coefficients. We denote the multinomial coefficientasμ(û, ˆv,ŵ) =(û +ˆv +ŵ)!, (9)(û!ˆv!ŵ!)where û +ˆv + ŵ = 4 and û denotes the multiplicity ofcomponent u etc. We omit zero multiplicities from notationμ(0, ˆv,0) = μ(ˆv). To fit tensor (7) to data, first the following(N × 15) matrix M is constructed:⎛⎞μ(4)u 4 1 μ(3, 1)u 3 1 v 1 ... μ(2, 2)v1 2w2 1 ... μ(4)w14 μ(4)u 4 2 μ(3, 1)u 3 2 v 2 ... μ(2, 2)v2 2w2 2 ... μ(4)w 4 2⎜⎝ ..,.. ⎟⎠μ(4)u 4 N μ(3, 1)u 3 N v N ... μ(2, 2)vN 2 w2 N ... μ(4)wN4(10)where u i = u(θ i ,ϕ i ), v i = v(θ i ,ϕ i ), etc. corresponding tothe ith measurement. Setting the unknown tensor coefficientsto be d 1 ,d 2 ,d 3 ,...,d 15 and defining vectors⎛⎛d 1m 1⎜ ⎟⎜ ⎟⎞d 2d =⎜ .⎝ .⎟⎠d 15⎞m 2and m =⎜ ., (11)⎝ .⎟⎠m Nwe can solve d from the equationM · d = m, (12)using the pseudo-inverse.Next we define a <strong>Finsler</strong> norm F(x,y) corresponding tofourth order ODF-tensor asF(x,y) = (T ij kl (x)y i y j y k y l ) 1/4 . (13)A general form for any even order n would be thenF(x,y) = (T i1 ...i n(x)y i1 ···y i n) 1/n . (14)In the following, we verify the necessary criteria for this(13) to be a <strong>Finsler</strong> norm in Definition 1.1. Differentiability: The tensor field T (x,y) is continuousin x by linear interpolation between the sample pointsand differentiable w.r.t. x using Gaussian derivatives.As a probability T(x,y) is assumed to be non-negative,and the differentiability of F(x,y) w.r.t. x follows. Thedifferentiability of F(x,y) in y is obvious from the formula(13).2. Homogeneity: Indeed for any α ∈ R + , x ∈ MF(x,αy) = (T ij kl (x)αy i αy j αy k αy l ) 1/4= αF(x,y). (15)3a. Strong convexity for a tensor based norm: Since this isa pointwise criterion, we omit here the argument x. Wecompute explicitly the metric tensor from norm (13)g ij (y) = 1 ∂ 2 F 2 (y)2 ∂y i ∂y j

328 Int J Comput Vis (2011) 92: 325–336= 3T −1/2 T ij kl y k y l− 2T −3/2 T iabc y a y b y c T jpqr y p y q y r , (16)where we have abbreviated the scalar sum asT ij kl y i y j y k y l := T . To simplify the expression, wemultiply it with a positive scalar c = T 3/2 , since it doesnot affect the positive definiteness. We obtainc · g ij (y) = 3TT ij kl y k y l− 2T iabc y a y b y c T jpqr y p y q y r . (17)We define a special product inner product candidate〈v,v〉 T := T ij kl y k y l v i v j . (18)Then from (17) we have for an arbitrary vector v:c · g ij (y)v i v j =〈y,y〉 T 〈v,v〉 T + 2 ( 〈y,y〉 T 〈v,v〉 T−〈y,v〉 T 〈y,v〉 T). (19)In case (18) does define an inner product, the Cauchy-Schwarz inequality says that the last term must be nonnegative.Then we have indeed thatg ij (y)v i v j > 0. (20)Thus in case the norm function is a power of an even ordertensor, the strong convexity is satisfied if the 〈,〉 T ispositive, since it meets all other conditions for an innerproduct. As a result the original condition for positivedefinitenessg ij (y)v i v j = (T −1/2 T ij kl y k y l− 2T −3/2 T iabc y a y b y c T jpqr y p y q y r )v i v j> 0 (21)We denote v θ :=∂θ ∂ (v), v θθ := ∂2 (v) and similarly∂θ 2for ϕ. We define the following three matrices:⎛v 1 v 2 v 3 ⎞m = ⎜ v⎝1 θv 2 θv 3 θ⎟⎠ ,v 1 ϕ v 2 ϕ v 3 ϕ⎛v 1 θθv 2 θθv 3 ⎞θθm θ = ⎜ v⎝θv 2 θv 3 ⎟θ ⎠ ,(25)⎛v 1 ϕϕ v 2 ϕϕ v 3 ⎞ϕϕm ϕ = ⎜ v⎝θv 2 θv 3 θ⎟⎠ .v 1 ϕ v 2 ϕ v 3 ϕv 1 ϕ v 2 ϕ v 3 ϕThen the strong convexity requires:det(m θ )det(m) < 0 and det(m ϕ )det(m) < −(g ij v i θ vj ϕ) 2, (26)g ij v i θ vj θa proof of which is provided in Appendix.The goal of this section was to define <strong>Finsler</strong> metric tensorsg ij (x, y) corresponding to a given tensorial ODF-field.Following a <strong>Finsler</strong> approach, instead of one metric tensorper voxel one obtains a collection of metric tensors at any x.For an illustration, see Fig. 1.is reduced to a simpler conditionT ij kl y k y l v i v j > 0. (22)3b. Strong convexity for a general norm F :Wenowstatea strong convexity criterion for a general <strong>Finsler</strong> normF(x,y) in R 3 , by analogy to the R 2 -criterion by Baoet al. (2000). We have put the derivation of the conditioninto Appendix, and merely state the result here.We consider the so-called indicatrix of the norm functionF(x,y) at any fixed x, which is the set of vectorsv(θ, ϕ){v ∈ R 3 | F(v) = 1}. (23)The indicatrix forms a closed surface which is a unitsphere w.r.t. norm function F . When F is the Euclideanlength the indicatrix becomes the regular sphere.For brevity, we writev := v(θ, ϕ). (24)Fig. 1 (Color online) A fourth order spherical tensor (the light purplecolored blob), representing an ODF and 2 ellipsoids illustrating thelocal second order diffusion tensors corresponding to the 2 vectors.The red ellipsoid is determined by the red vector and similarly for thegreen ellipsoid

Int J Comput Vis (2011) 92: 325–336 3294 Transforming a Polynomial Tensor to a MonomialTensorSay we wish to regularize our spherical data, and that wewish to use a tensorial representation of the data instead ofspherical harmonics. In Florack (2008), a scale space extensionfor a spherical signal is considered which is a generalizationof the Tikhonov regularization. This scale spacetakes particularly simple form when applied to spherical harmonics(Bulow 2004). Conveniently, in Florack and Balmashnova(2008) it is shown that this regularization techniquecan also be applied to tensorial case requiring onlythat the tensor representation is constructed in a hierarchicalway, such that the higher order components contain only“residual” information, that can not be contained in lowerorder terms. In practice this means that to a given ODF, onefirst fits a zeroth order tensor, and to the residual one fits asecond order tensor. Further, one fits iteratively higher ordertensors to the residuals until the desired order is reached(for details see Florack and Balmashnova 2008). However,as can be seen from the formulae below, it is not immediatelyclear how to homogenize a hierarchically expressedtensor, where higher order terms can have negative values.Thus for <strong>Finsler</strong> analysis, it is favorable to work with atensor representation of monomial formT(x,y) = T i1···i n(x)y i1 ···y i n, (27)and not with an equivalent polynomial expressionT(x,y) =n∑˜T i1···i k(x)y i1 ···y i k, (28)k=0but to still benefit from the pointwise regularization schemeof the latter:T(x,y) τ =n∑e −τk(k+1) ˜T i1···i k(x)y i1 ···y i k, (29)k=0where τ is the regularization parameter.Indeed, we can transform a polynomial expression to amonomial one using the fact that our polynomials are restrictedto the sphere (4), thus we may expand a nth ordertensor to a n + 2 order one and symmetrize it. Symmetrizationof a tensor amounts to a projection to the subspace ofsymmetric tensors (Yokonuma 1992). For an even order (2n)symmetric tensor T the multilinear mappingT : S 2 ×···×S 2} {{ }2n times→ R, T(y) = T i1 ...i 2ny i1 ···y i 2n, (30)can be equivalently expressed as a quadratic form˜T αβ ỹ α ỹ β , (31)where α, β = 1,...,3 n .Here ˜T is a square matrix withcomponents˜T αβ = T σα (i 1 )···σ α (i n )σ β (i 1 )···σ β (i n ), (32)andỹ α = y σ α(i 1) ···y σ α(i n ) , (33)where σ 1 ,...,σ 3 n are the permutations of componentsof y corresponding to each term of the outer producty ⊗ 1 ···⊗ n y.As an example to clarify the equivalence of (30) and (31),let us consider a spherical fourth order tensor A ij kl in dimensiontwo, with input vectorsy = (y 1 , y 2 ) = (u(ϕ), v(ϕ)) = (cos ϕ,sin ϕ). (34)We observe that since the tensor is symmetric, a sumA ij kl y i y j y k y l (35)is equal to an inner product (omitting the argument ϕ here)⎛⎞A 1111 A 1112 A 1211 A 1212(u2uv vu v 2 ) ⎜A 1111 A 1112 A 1211 A 1212⎟⎝A 2111 A 2112 A 2211 A 2212⎠A 2121 A 2122 A 2221 A 2222⎛u 2 ⎞× ⎜uv⎟⎝vu⎠ . (36)v 2In this matrix-form, for example the expansion of a secondorder tensor with components T ij to a fourth order tensorwith components T ij kl can be illustrated in a very simpleway in Fig. 2. Such expansion is possible, because the inputvectors are unit vectors. For example we can immediatelyverify the following identity of a zeroth and a second ordertensorc = ( u(ϕ) v(ϕ) ) ( )( )c 0 u(ϕ)= c(sin 2 ϕ + cos 2 ϕ).0 c v(ϕ)(37)The same principle extends to dimension three and higher.A formula for obtaining a fourth order symmetric tensorequivalent to a given second order Cartesian tensor (i.e. thatT ij y i y j = T klmn y k y l y m y n ) is thusT ij kl = 1 ∑T σ(i)σ(j) I σ(k)σ(l) , (38)4!σ ∈S 4where I is the identity matrix and S 4 the symmetric group ofall permutations of a set of four elements.

330 Int J Comput Vis (2011) 92: 325–336Similar to the Riemannian case, in <strong>Finsler</strong> setting a geodesicsγ is an extremal path that minimizes the variationallength of a curve between fixed endpoints. The Euler-Lagrange equations of the length integral then give us a localcondition according to which an extremal curve has vanishinggeodesic curvature (Shen 2001). This amounts to equation¨γ i (t) + 2G i (γ (t), ˙γ(t))= 0, (39)whereG i (x, y) = 1 ( ∂F 2 )(x, y)4 gil (x, y)∂x k ∂y l y k − ∂F2 (x, y)∂x l , (40)is the so-called geodesic coefficient. Using the relationshipof the norm and the bilinear formg ij (x, y) = 1 ∂ 2 F 2 (x, y)2 ∂y i ∂y j , (41)Fig. 2 Top left: A second order symmetric tensor T in 2D. Top right:The Kronecker product of T and the identity matrix I. Here elementswith same color have same value (white = 0). Bottom left: The equivalenceclasses of permutation group S 4 on sets with elements {1, 2}.Bottom right: To symmetrize the matrix, each element is replaced bythe average of all elements in the equivalence class5 Fiber Tracking in HARDI Data Using <strong>Finsler</strong><strong>Geometry</strong>In DTI setting the most straightforward way of trackingfibers is to follow the principal eigenvector correspondingto the largest eigenvalue of the diffusion tensor until somestopping criterion is reached. When the ODF is modeledas a second order tensor, this straightforward method cannotreveal crossings, since a second order tensor typicallyhas only one principal eigenvector. With higher order ODFmodel, tracking along principal eigenvectors of <strong>Finsler</strong> metrictensors, we can do tractography even through crossings.We begin this section by briefly introducing geodesics in<strong>Finsler</strong> spaces followed by an analytical example, where theoptimal paths correspond to geodesics. This is to underlinea difference between <strong>Finsler</strong>ian and Riemannian approach.We stress that in general we do not assume fibers to be geodesicsin the norm field determined by ODF. In the remainingpart of the section some experimental results of <strong>Finsler</strong>tractographyis shown both in simulated crossing data aswell as in two real HARDI data sets.5.1 Geodesics in <strong>Finsler</strong> <strong>Geometry</strong>we see that (omitting arguments x,y for brevity)2G i = 1 ( [ ]∂ ∂2 gil ∂x k ∂y l g jny j y n − ∂ )∂x l [g jky j y k ]= 1 2 gil (2 ∂g jl∂x k yj y k − ∂g jk∂x l y j y k )= 1 2 gil (∂gjl∂x k yj y k + ∂g kl∂x j yk y j − ∂g jk∂x l y j y k ), (42)showing that the geodesic equation in the <strong>Finsler</strong>ian case isformally identical to that in the Riemann geometry¨γ i + Ɣ i jk ˙γ j ˙γ k = 0, (43)where Ɣjk i is the last term in (42). Note that, unlike in theRiemannian case, Ɣjk i depend on y.5.2 Analytic ExampleWe consider an analytic norm field in R 2 . From the applicationpoint of view we may consider this norm to be the inverseof an ODF. By the inverse we mean the Moebius inversionw.r.t. a sphere. The minima of the norm coincide withthe maxima of the ODF and the geodesics (shortest paths)would then coincide with paths of maximal diffusion. Let ustake as a convex norm function at each spatial pointF(ϕ)= (cos 4ϕ + 4) 1 4= ( 5 cos 4 ϕ + 2 cos 2 ϕ sin 2 ϕ + 5sin 4 ϕ ) 1 4. (44)This is an example of fourth order spherical tensor. Such atensor field could represent a structure/material which hastwo preferred orientations of diffusion, along which the distancemeasure is shorter (see Fig. 3). From the fact that F(ϕ)

Int J Comput Vis (2011) 92: 325–336 331the directional norm function is correspondingly small. In<strong>Finsler</strong> setting the connectivity strength m(γ ) is:∫ √ δ ij ˙γ i ˙γ j dtm(γ ) = ∫ √ , (45)g ij (γ, ˙γ)˙γ i ˙γ j dtwhere the δ ij (γ ) is the identity matrix, ˙γ(t) the tangentto the curve γ(t) and g ij (γ, ˙γ) the <strong>Finsler</strong>-metric tensor(which depends not only on the position on the curve butalso on the tangent of the curve). For illustration we computeexplicitly the metric tensors g, using Cartesian coordinates:( )1 g11 gg =122(4 + cos 4ϕ) 3/2 , (46)g 21 g 22whereg 11 = 5(6 + 3 cos 2ϕ + cos 6ϕ),g 12 = g 21 =−12 sin 2ϕ 3 ,g 22 =−5(−6 + 3 cos 2ϕ + cos 6ϕ).The strong convexity criterion in R 2 (Bao et al. 2000)ontheindicatrix {v(ϕ) | F(v(ϕ)) = 1}, for metric (46) is satisfiedfor every ϕ, since¨v 1 ˙v 2 − ˙v 1 ¨v 2 13 − 8 cos 4ϕ˙v 1 v 2 − v 1 = > 0. (47)˙v 2 (4 + cos 4ϕ) 2The connectivity measure for a (Euclidean) geodesic γ canbe computed analytically:m(γ ) =∫dt∫(4 + cos(4ϕ)) 1/4 dt , (48)which gives the maximal connectivities in directionsFig. 3 (Color online) Top: A field of fourth order spherical harmonicscos 4ϕ + 4 representing the norm. In the middle of the figure, the ODFprofiles are indicated and some best connected geodesics are drawn.Bottom: 50 ellipses representing metric tensors corresponding to directionsϕ =50 i 2π (i = 1,...,50) of the norm function F(ϕ),andanellipse with thick boundary corresponding to the metric tensor in directionϕ = π 4.Thethick red curve is the homogenized norm functionF(ϕ) 1 4π4 , 3π 4 , 5π 4 , and 7π 4 , (49)as expected. See Fig. 3 for an illustration. We remark thaton this norm field the Riemannian (DTI) framework wouldsimilarly result in Euclidean geodesics but the connectivitywill be constant over all geodesics, because the second ordertensor best approximating the fourth order tensor will be asphere, thus revealing no information at all of the preferreddirections.5.3 <strong>Finsler</strong> Fiber Trackinghas no x-dependence, we conclude that the geodesic coefficientsvanish and that the shortest paths connecting anytwo points coincide with the Euclidean geodesics γ(t) =(t · cos ϕ,t · sin ϕ), i.e. straight lines. However the so-calledconnectivity strength of a geodesic (Prados et al. 2006;Astola et al. 2007) is relatively large, only in cases, whereWe extend the standard streamline tractography⎧ċ(t) = arg max{D ij (c(t))h i h j },⎪⎨ |h|=1c(0) = p,⎪⎩ ċ(0) = arg max{D ij (c(0))h i h j },|h|=1(50)

332 Int J Comput Vis (2011) 92: 325–336which essentially follows the principal eigenvectors of thediffusion tensors to a <strong>Finsler</strong> tractography that solves thesystem⎧⎨ċ(t) = arg max{D ij (c(t), ċ(t))h i h j },|h|=1(51)⎩c(0) = p,where the second order tensor D ij (c(t), ċ(t)) is computedfrom the nth order tensor T n (c(t), ċ(t)) (approximatingODF) as followsD ij (c(t), ċ(t)) = 1 ∂ 2 ((T n (c(t), y)) 1/n ) 2 ∣ ∣∣∣y=ċ(t)2 ∂y i ∂y j . (52)The major difference between these two approaches is thatthe latter depends also on the tangent ċ(t) of the curve. Anotherdifference is that in (51) there is no single initial directionasin(50).Thus the tracking is initially done in manydirections, say in directions of all unit vectors of a nth ordertessellation of the sphere.The <strong>Finsler</strong>-fiber tracking can be summarized in the followingscheme.1. Fit a tensor T n (x, y) to the spherical function representingthe distribution of fiber orientations (e.g. the ODF).2. Compute the homogeneous higher order diffusion functionF(x,y) = (T n (x, y)) 1/n .3. Start tracking from point p by computing the N diffusiontensors D(k) k=1,...,N in directions y 1 ,...,y N , whereD(k) = D ij (p, y k ) = 1 ∂ 2 (F 2 ∣(p, y)) ∣∣∣y=yk2 ∂y i ∂y j . (53)4. Compute the fractional anisotropy (FA) and principaleigenvectors (PE) of diffusion tensors D(k). IftheFAislarger than required minimum and PE in the given angularcone of the tangent vector, take a step in the directionof the PE, otherwise stop.In the following experiments we have used as an angularcondition that the absolute value of the dot product of theprincipal eigenvector and the unit tangent of the streamline,is greater than 0.6, and that the FA (of D ij ) is greater than0.15, and stepsize is 0.2 voxel dimensions.5.4 Simulated Crossing DataWe show some results of <strong>Finsler</strong>-tracking in tensor data thatsimulate crossings. The fourth order tensors in crossingswere generated using a multi-tensor model (Frank 2002)S = S 02∑f k e −bD k, (54)k=1Fig. 4 (Color online) Top: A tensor field simulating fiber bundlescrossing at 25 degree angle. <strong>Finsler</strong>-tracks with initial points marked byred balls. Bottom: Same tensor field with Rician noise (SNR = 15.3)applied. <strong>Finsler</strong> tracks with same initial points as in the noise-free tensorfieldwhere f k is a weight associated to the kth tensor D k (cf. (1)).In Figs. 4, 6 and 8 we see that <strong>Finsler</strong>-streamline trackingcan indeed resolve crossings. The fibers are well recoveredalso when Rician noise (the characteristic noise in diffusionMRI) is applied. However, we observe that as the angle betweenthe bundles is narrow, this results in passing/kissingfibers instead of crossings.5.5 Real Human Brain DataWe have computed the <strong>Finsler</strong>-streamlines in two humanbrain HARDI data sets. These data sets were acquired withb-value 1000 and 132 gradient directions. The data has dimensions10 × 104 × 104 containing the corpus callosum,but does not contain data below the ventricles. We selectedinitial points on a horizontal plane, and tracked in 32 directions,which are the first order regular tessellations of a dodecahedron,projected on the circumscribed sphere. For anillustration of subsets of initial points and the correspondingtracks, see Figs. 5 and 7. To distinguish between thefiber orientations, the following color coding was used. If

Int J Comput Vis (2011) 92: 325–336 333Fig. 5 Left: <strong>Finsler</strong>-fibers on a subset of initial points. Right: A subsetof initial points on a horizontal planeFig. 7 As in Fig. 5, with another HARDI dataFig. 8 As in Fig. 4, but with 65 degree angle at the crossingFig. 6 As in Fig. 4, but with 45 degree angle at the crossingmost of the discretized fiber segments in a curve have orientationclosest to “anterior-posterior”, the curve is coloredyellow. If most of the discretized fiber segments in a curvehave orientation closest to “superior-inferior”, the curve iscolored blue. Similarly curves with major-orientation being“left-right” are colored red. The in-between colors are greenbetween yellow and blue, purple between red and blue andfinally orange, between red and yellow. For illustration seeFigs. 9 and 10.6 Conclusions and Future WorkWe have seen that it is indeed possible to analyze sphericaltensor fields using <strong>Finsler</strong> geometry. This gives newmethods to analyze the data and is likely to provide newinformation on the data. For example <strong>Finsler</strong> diffusion tensorsare geometrically well justified local approximationsof the more complex tensor describing ODF and <strong>Finsler</strong>streamlinescan easily propagate through crossings. Anothermerit of <strong>Finsler</strong>-streamline tracking is that it does not requireany solving of extrema of the ODF and is very simpleto implement.

336 Int J Comput Vis (2011) 92: 325–336Jian, B., Vemuri, B., Özarslan, E., Carney, P., & Mareci, T. (2007).A novel tensor distribution model for the diffusion-weighted MRsignal. NeuroImage, 37, 164–176.Lenglet, C., Deriche, R., & Faugeras, O. (2004). Inferring white mattergeometry from diffusion tensor MRI: application to connectivitymapping. In Proc. eighth European conference on computer vision(ECCV) (pp. 127–140). Prague, Czech Republic.Melonakos, J., Pichon, E., Angenent, S., & Tannenbaum, A. (2008).<strong>Finsler</strong> active contours. IEEE Transactions on Pattern Analysisand Machine Intelligence, 30(3), 412–423.Mori, S. (2007). Introduction to diffusion tensor imaging. Amsterdam:Elsevier.O’Donnell, L., Haker, S., & Westin, C. F. (2002). New approaches toestimation of white matter connectivity in diffusion tensor MRI:elliptic PDEs and geodesics in a tensor-warped space. In LNCS:Vol. 2488. Proc. medical imaging, computing and computer assistedintervention (MICCAI) (pp. 459–466). Berlin: Springer.Özarslan, E., & Mareci, T. (2003). Generalized diffusion tensor imagingand analytical relationships between diffusion tensor imagingand high angular resolution diffusion imaging. Magnetic Resonancein Medicine, 50, 955–965.Özarslan, E., Shepherd, T., Vemuri, B., Blackband, S., & Mareci, T.(2006). Resolution of complex tissue microarchitecture using thediffusion orientation transform. NeuroImage, 31, 1086–1103.Prados, E., Soatto, S., Lenglet, C., Pons, J. P., Wotawa, N., Deriche,R., & Faugeras, O. (2006). Control theory and fast marchingtechniques for brain connectivity mapping. In Conf. computer visionand pattern recognition (CVPR) (pp. 1076–1083). New York:IEEE Computer Society.Rosner, B. (2006). Fundamentals of biostatistics (6th ed.) London:Thomson Brooks/ColeShen, Z. (2001). Lectures on <strong>Finsler</strong> geometry. Singapore: World Scientific.Stejskal, E., & Tanner, J. (1965). Spin diffusion measurements: spinechoes ion the presence of a time-dependent field gradient. TheJournal of Chemical Physics, 42(1), 288–292.Tuch, D. (2004). Q-ball imaging. Magnetic Resonance in Medicine,52(4), 577–582.Tuch, D., Reese, T., Wiegell, M., Makris, N., Belliveau, J., & vanWedeen, J. (2004). High angular resolution diffusion imaging revealsintravoxel white matter fiber heterogeneity. Magnetic Resonancein Medicine, 48(6), 1358–1372.Yokonuma, T. (1992). Tensor spaces and exterior algebra (Vol. 108).Providence: American Mathematical Society.