Simultaneous ODF Estimation and RobustProbabilistic Tractography from HARDI

Hasan Ertan Çetingül1, Laura Dumont2, Mariappan S. Nadar1,Paul M. Thompson3, Guillermo Sapiro4, and Christophe Lenglet4,5

1Siemens Corporation, Corporate Research and Technology, Princeton, NJ, USA2Engineering School of Chemistry, Physics, and Electronics (CPE Lyon), Lyon, France

3Laboratory of Neuro Imaging, UCLA School of Medicine, Los Angeles, CA, USA4Dept. of Electrical & Computer Engineering, U of Minnesota, Minneapolis, MN, USA5Center for Magnetic Resonance Research, U of Minnesota, Minneapolis, MN, USA{hasan.cetingul,mariappan.nadar}@siemens.com; laura.dumont@cpe.fr;

thompson@loni.ucla.edu; {guille,clenglet}@umn.edu

Abstract. We consider the problem of extracting white matter fibertracts from high angular resolution diffusion imaging (HARDI) data whilesimultaneously estimating the local fiber orientation profile. Prior workshowed that if parametric mixture models such as mixtures of tensors areused to characterize diffusion, nonlinear state-space filtering techniques(e.g., the unscented Kalman filter (UKF)) can provide a solution to thisproblem. In this work, we describe how to use the UKF on HARDI datamodeled by orientation distribution functions (ODFs), a more genericand versatile diffusion model, by considering the spherical harmonicrepresentation of the HARDI signal as the state. We apply the residualbootstrap method to develop a UKF-based probabilistic tractographymethod, which has the capability of rejecting outliers, i.e., erroneoustrajectories, by measuring the abrupt changes in the state covariance.We demonstrate the robustness of the proposed method by means ofexperiments on synthetic, phantom, and real HARDI data.

1 Introduction

Quantitative characterization of the white matter (WM) circuitry of the humanbrain is an important problem in neuroradiology, with applications in detecting,diagnosing, and tracking the progression of neurological diseases. Diffusion mag-netic resonance imaging (DMRI) can describe the WM circuitry by capturingvariations in water diffusion patterns. More specifically, DMRI quantifies Brown-ian motion of water molecules in biological tissues and produces images that areweighted according to the amount (i.e., anisotropy) and direction of diffusion.The anisotropy arises from the presence of axonal membranes and myelin, therebyallowing the mapping of neural fiber tracts in WM. State-of-the-art DMRI tech-niques such as high angular resolution diffusion imaging (HARDI) enables thereconstruction of the orientation distribution function (ODF) [1–3], which offersimproved accuracy in resolving intra-voxel complexities over the diffusion tensor(DT) model, currently the de facto standard for clinical applications.

2 H.E. Çetingül et al.

Conventional WM tractography methods consider a volume of diffusion models(DTs, ODFs, etc.) estimated from the HARDI signal as input, and extracttrajectories of the underlying fiber tracts using deterministic or probabilisticapproaches (see [4–8] and references therein). Contrary to these techniques, whichassume estimation of diffusion models and tractography to be two problemsindependent of each other, Malcolm et al. [9] considered tracking as a causalprocess (a common assumption in deterministic approaches) and proposed tosimultaneously solve the aforementioned problems using an unscented Kalmanfilter (UKF). This method assumes a diffusion model with a fixed number ofmixtures of diffusion tensors. In other words, the parameters of a multi-tensormodel form the state. The UKF recursively updates the state, provides thecovariance of the estimate, and determines the most appropriate direction tofollow. In theory, this strategy is fast (since it avoids estimation at every voxel)and robust to noise (due to the nature of filtering). However, since the state vectoris the concatenation of several elements (e.g., principal eigenvectors/eigenvaluesand mixture weights) with different statistical properties, parameter selectionand preservation of physically meaningful states during the evolution are vitalfor accuracy and robustness in tracking. This problem is addressed in [10] usingan intrinsic UKF framework in which the states (based on the two-tensor model)are guaranteed to be in the space of DTs, i.e., 3×3 symmetric positive definitematrices. Nevertheless, to the authors’ knowledge, mixture models are an integralpart of these methods and reproducibility of tracking results remains unanswered.

In this work, we present an UKF-based probabilistic tractography frameworkfor HARDI data modeled by ODFs. Our approach considers the spherical har-monic representation of the HARDI signal as the state and enforces nonnegativityof the reconstructed ODFs during the state transition [11]. The mode detectionscheme proposed in [12] is used to identify the directions at which the ODFsattain their peaks. We apply the residual bootstrap method [8] to develop theprobabilistic framework, which computes a large number of trajectories from eachseed point by propagating streamlines along the modes that are most consistentwith the previous direction. Our method produces likelihood maps that can befurther improved by rejecting outliers, i.e., erroneous trajectories, by measuringchanges in the state covariance. We quantify and depict the uncertainty (equiva-lently the robustness or reproducibility) in tracking through crossing fibers insynthetic and phantom data, as well as WM fiber tracts in a real HARDI dataset.

2 Diffusion Representation using Spherical Harmonics

In a typical DMRI protocol, one acquires images of the MR signal attenua-tion {Sn

.= S(θn, φn)} along N different gradient directions {(θn, φn)}Nn=1 on the

2-sphere S2, as well as baseline images S0 with no diffusion sensitization.1 To repre-sent diffusion at a specific spatial location, we consider the single shell q-ball imag-ing formulation within constant solid angle [3]. According to this formulation, the

1 θn ∈ [0, π] and φn ∈ [0, 2π) are the polar and azimuthal angles, respectively, repre-senting the direction (θn, φn) in spherical coordinates.

Simultaneous ODF Estimation and Robust Probabilistic Tractography 3

value of the ODF in the direction (ϑ, ϕ) (equivalent to (θn, φn) in the transformedspace) is given by p(ϑ, ϕ) = 14π +

116π2FRT{∇

2b ln(−ln(Sn/S0))}, where ∇2b is the

Laplace-Beltrami operator on S2 and FRT is the Funk-Radon transform. Since Snis in practice real and symmetric, the real spherical harmonics (SHs) Yj(θn, φn)

[13] are used to approximate the signal as ln(− ln(Sn/S0)) ≈∑Jj=1 cjYj(θn, φn),

where cj ∈ R is the SH coefficient associated with the j-th basis function.2 TheSH expansion of the signals {ln(−ln(Sn/S0))}Nn=1 can also be written in vectorform as s ≈ Bc, where s =

[ln(− ln(S1/S0)), . . . , ln(− ln(SN/S0))

]> ∈ RN and[Y1(θn, φn), . . . , YJ (θn, φn)

]is the n-th row of B ∈ RN×J . Once the least-squares

solution of s ≈ Bc for c∈RJ is found, a discrete representation of the ODF canbe reconstructed using the fact that SHs are eigenfunctions of ∇2b and FRT [3,14].The reconstruction method in [14] also enforces the ODFs p to be nonnegativeand it is used, in this work, to initialize the UKF.

3 The UKF for Simultaneous Estimation and Tracking

The main idea in the UKF-based tractography methods [9, 10,15] is to estimateand recursively update the diffusion model given the measured signal at a spatiallocation and trace the fiber by propagating in the direction consistent with theprevious directions of propagation. The four components of the UKF are thestate x (the diffusion model), state transition function f (to predict how thediffusion model changes), observation function h (to predict the observed signalfor a particular state), and measurement y (the acquired HARDI signal). In ourframework, we consider the SH coefficient vector c =

[c1, . . . , cJ ]

> as the statex and identity dynamics for the state transition f assuming that the diffusionprofile does not change drastically in the vicinity of the location of interest. Themeasurement y is the signals {Sn/S0} at that location, and the observation h isthe reconstruction of the signals as {exp(− exp(

∑j cjYj(θn, φn)))}.

3.1 Review of the UKF Formulation and the UKF-ODF Algorithm

The UKF uses a sampling technique known as the unscented transform (UT),which calculates the statistics of a random variable undergoing a nonlineartransformation. The UKF uses the state transition model to predict the nextstate (and observation) via the UT and the measurement to update the stateestimate. One iteration of this recursive algorithm can be outlined as follows.

Let xk ∈ RJ and Pk ∈ RJ×J be the estimated mean and covariance matrixof the current state at discrete time k. In the prediction stage, a set Xk ={χi}2J+1i=1 ⊂ RJ of 2J + 1 sigma points with associated weights {wi} ⊂ R+ isgenerated by perturbing xk such that χ0 = xk with weight w0 = κ/(J+κ),and χi = xk + [

√(J+κ)Pk]i, χi+J = xk− [

√(J+κ)Pk]i with weights wi =

wi+J = 1/(2J+2κ) for 1 ≤ i ≤ J .3 Next, the predicted sample set of states2 We consider the SH basis of degree L=4 and hence J=(L+1)(L+2)/2 = 15.3 [A]i denotes the i-th column of the matrix A and κ is a scaling parameter set to 0.01.

4 H.E. Çetingül et al.

Xk+1|k is obtained by propagating Xk through the state transition function,i.e., Xk+1|k = {f(χi)} = {χ̂i}. These sample states are used to calculate themean x̂k+1|k of the predicted system state as well as its covariance Pxx, which iscorrupted by the process noise with covariance Qc ∈ RJ×J . Then the predictedset of observations is obtained as Yk+1|k = {h(χ̂i)} = {γi}. These observationsare used to calculate the mean of the predicted measurement ŷk+1|k as well asits covariance Pyy, which is corrupted by the measurement noise with covarianceRs ∈RN×N . The Kalman gain K = PxyP−1yy ∈ RJ×N is finally used to correctour prediction and obtain the mean and covariance of the updated state asxk+1 = x̂k+1|k +K(yt − ŷk+1|k) and Pk+1 = Pxx −KPyyK>, respectively. Thereader is referred to seminal papers such as [16] for further details on the UKF.

The Algorithm The UKF-ODF algorithm is initialized, at user-specified seedpoints, by estimating the SH representation of the HARDI signal and recon-structing the ODFs via the constrained optimization strategy in [14], whichenforces nonnegativity of the ODFs. For each seed point on the fiber of interest,the SH coefficient vector c is taken as the initial state x0 and the number andlocations of the mode(s) of the ODF are identified using the weighted sphericalmean shift clustering [12]. These modes represent the directions to be followed atthe current spatial location. At the k-th iteration, we predict the new state asx̂k+1|k = f(xk) = xk, use the set of predicted observations along with the mea-sured signal interpolated at the current spatial location, and find the new statexk+1.

4 This estimate is used to reconstruct the ODF whose modes (directions tobe followed at k + 1) are subsequently identified. The UKF-ODF algorithm pro-duces one streamline trajectory by propagating forward in the direction consistentwith the previous direction until a user-defined termination criterion (e.g., highcurvature, low generalized fractional anisotropy (GFA) [1], etc.) is met.

3.2 Probabilistic Tractography using the Residual Bootstrap

We develop an UFK-based probabilistic tractography framework, which we callas rbUKF-ODF, by integrating the UKF-ODF algorithm into the residual bootstrapmethod [8]. Through residual bootstrapping, one can consider a trajectory givenby UKF-ODF as an outcome of a probabilistic experiment and use it to constructthe likelihood maps, which represent the probability that a track (or trajectory)emanating from a seed point passes through other spatial locations.

Consider the SH expansion of the HARDI signal in §2, which can be writtenin vector form as s = Bc+ � with � being the noise vector. The signal ŝ predictedby the least-squares fit to the measured signal s is given as

ŝ = Bĉ = B(B>B)−1B>s.= Hs. (1)

Now consider the residual vector �̂ =[�̂1, . . . , �̂N

]>= s− ŝ and the raw residuals

corrected for leverage as�̂ci = �̂i/

√1−Hii, (2)

4 Qc and Rs are assumed diagonal with entries qc ∈ [0.05, 0.1] and rs ≈ 0.02, whichare selected according to the spectra of the SH coefficients and the HARDI signal.

Simultaneous ODF Estimation and Robust Probabilistic Tractography 5

where Hii is the i-th diagonal entry of H in (1). In the residual bootstrap method,the entries of the vector �̂c are randomly chosen with replacement to form anew bootstrapped residual �̂∗. Then, a synthetic bootstrap realization ŝ∗ of scan be generated as ŝ∗ = ŝ + �̂∗. By generating Nb bootstrap realizations fromthe measured signal s and running the UKF-ODF algorithm on each realizationfrom a seed point of interest, one can produce Nb trajectories emanating fromthat seed point. The resulting trajectories are used to obtain the likelihood mapof the region of interest by assigning to each voxel the ratio of the number oftrajectories passing through that voxel to Nb.

Inferring Track Uncertainty from the UKF It is worth noting that the co-variance Pk provides a measure of confidence in the k-th estimation. In particular,the Frobenius norm of Pk can be considered as a scalar measure of uncertaintyat k [17]. This information can be used to eliminate outliers, i.e., erroneoustrajectories, and improve the quality of the likelihood maps. More specifically,one can eliminate a trajectory (or portion of a trajectory) before computingthe likelihood map if there is a drastic increase in the Frobenius norm of thecovariance {Pk}, i.e., if

‖Pk+1‖F − ‖Pk‖F > τ, (3)

for some empirically-tuned threshold τ >0. [17] uses the moving average of theFrobenius norm of the covariance as a measure of track uncertainty to detecterroneous tracks, whereas, in this work, we propose to analyze the sequence(‖P0‖F , ‖P1‖F , ‖P2‖F , . . . ) to eliminate the erroneous portions of the trajectories.

4 Method Validation

4.1 Experiments on Synthetic Data

Experiments on synthetic data evaluate the sensitivity of the rbUKF-ODF methodto noise (without outlier rejection) and compare its performance to that of thefiltered 2-tensor tractography method (with parameters given in [15]) extendedinto a probabilistic one (rbUKF-2T) using the residual bootstrapping method. Wesimulate diffusion weighted images of two randomly generated configurations,each of which contains two intersecting fiber bundles (see Figs. 1(a)-1(b)). Thecenterline of a fiber bundle is formed by fitting cubic splines through threerandomly selected points in a 30×30 lattice. We use the two-tensor model in [13],where the HARDI signals {Sn}Nn=1 at N = 81 gradient directions, with S0 = 1and b= 2,000 s/mm2, are simulated to represent an isotropic background and1- or 2-fiber ODFs according to the shape of a fiber centerline. Noisy signalsare generated by adding Rician (complex Gaussian) noise with zero mean andstandard deviation σ = S0/ζ, where ζ is the signal-to-noise ratio (SNR).

Figs. 2(a) and 2(b) show the likelihood maps of both configurations, obtainedusing rbUKF-2T (top rows) and rbUKF-ODF (bottom rows) at different levels ofSNR (from left to right: 30, 20, 10, 5 dB). Initiated at two different seed points(shown in yellow), the UKF-based tractography is performed on Nb = 1,000bootstrapped realizations of the signal. Superimposed on the GFA maps of the

6 H.E. Çetingül et al.

(a) Configuration 1 with the ODFs (b) Configuration 2 with the ODFs

Fig. 1. Two synthetic configurations with fiber centerlines shown in red and the corre-sponding ODFs reconstructed for visualization purposes using the method in [14].

synthetic configurations at the corresponding SNR level, the probability of con-nectivity is depicted using the colors red and blue for the two fiber bundles in theconfigurations (i.e., dark red/blue∼high probability, light red/blue∼low probabil-ity). It is observed that rbUKF-ODF produces a higher likelihood of connectivityalong the true fiber centerlines and less dispersion than rbUKF-2T does. Morespecifically, for configuration 1, rbUKF-ODF produces considerably less dispersion(i.e., less false negatives) when SNR = 30, 20 dB. Yet the performance of themethods is comparable when SNR = 5 dB, and rbUKF-2T slightly outperformsrbUKF-ODF when SNR = 10 dB. For configuration 2, rbUKF-ODF greatly outper-forms rbUKF-2T at all levels of SNR, with trajectories almost flawlessly followingthe fiber centerlines when SNR = 30, 20, 10 dB. We observe that for a consider-able number of realizations, the two-tensor model does not accurately resolvethe crossing configuration (especially when SNR = 30, 20 dB), and hence UKF-2Tproduces erroneous trajectories. These results demonstrate that by removing thedependency of the UKF on a more restrictive diffusion model, it is possible toresolve higher degrees of complexity and obtain more accurate likelihood maps.

4.2 Experiments on Phantom Data

Next, we test the proposed method on the biological phantom in [18], constructedfrom excised rat spinal cords and designed to have crossing fiber bundles (seeFig. 3(a)). The diffusion weighted images of the phantom were acquired with a40×9 image matrix (40 slices with an isotropic spatial resolution of 2.5 mm) anda diffusion sensitization at b = 1,300 s/mm2 applied along a set of 90 gradientdirections with 10 baseline images. For visualization purposes, we reconstruct theODFs at y=4 slice (see Fig. 3(b)) and show the underlying orientation profiles.

Fig. 3(c) shows the color-coded trajectories obtained using UKF-ODF (initiatedat two different seeding regions marked in yellow) from the bootstrapped realiza-tions of the HARDI signal. It is observed that UKF-ODF is successful in resolvingthe crossing configuration and delineating the centerlines of the fibers. To depictthe track uncertainties, we plot the same trajectories in Fig. 3(d) using {‖Pk‖F }as the colormap. One can clearly see in the zoomed regions that when a trajectory

Simultaneous ODF Estimation and Robust Probabilistic Tractography 7

(a) Likelihood maps for synthetic configuration 1

(b) Likelihood maps for synthetic configuration 2

Fig. 2. Likelihood maps (superimposed on the GFA maps) of the configurations com-puted using rbUKF-2T (top two rows) and rbUKF-ODF (bottom two rows), without thecovariance-based outlier rejection, at different levels of SNR. The probability of connec-tivity is depicted using the colors blue and red for the fibers (dark∼high, light∼low).

deviates from the fiber centerline, the norm of the covariance increases suggestingthat confidence to that portion of the trajectory decreases. Superimposed onthe GFA maps, the likelihood maps of fiber 1 (resp. fiber 2) are obtained usingrbUKF-ODF (with Nb = 1,000) and shown in blue (resp. red) in Fig. 3(e) follow-ing the same visualization rule in §4.1. We observe that rbUKF-ODF produceslikelihood maps that are in accordance with the fiber centerlines. In particular,the small thickness of the fibers does not adversely affect the performance ofUKF-ODF and the intra-voxel complexity in the crossing region is resolved by theODF model. Thus, the resulting likelihood maps are more reliable than the mapsproduced by rbUKF-2T, which do not fully delineate fiber 1 (see Fig. 3(f)).

8 H.E. Çetingül et al.

(a) Image of the phantom (b) ODFs at y=4 (c) UKF-ODF trajectories

(d) UKF-ODF trajectories color-coded using {‖Pk‖F } with the zoomed views of 3 ROIs.

(e) Likelihood maps obtained from rbUKF-ODF at slices y = 3, 4, 5.

(f) Likelihood maps obtained from rbUKF-2T at slices y = 3, 4, 5.

Fig. 3. (a-b) Image of the phantom from [18] and the reconstructed ODFs at y=4 slice,(c) Trajectories obtained using UKF-ODF, (d) Trajectories color-coded using the valuesof ‖Pk‖F , (e-f) Likelihood maps (superimposed on the GFA maps) computed usingrbUKF-ODF and rbUKF-2T at y = 3, 4, 5, and depicted using the colormap in Fig. 2.

Simultaneous ODF Estimation and Robust Probabilistic Tractography 9

4.3 Experiments on Real Data

We finally test the rbUKF-ODF algorithm on a human brain dataset containingstructural, functional, and diffusion MR images provided for the Pittsburgh BrainConnectivity Challenge (PBCC Spring 2009).5 The diffusion weighted imageswere acquired with a 128×128 image matrix, (68 slices with an isotropic spatialresolution of 2 mm), and a diffusion sensitization at b = 1,500 s/mm2 appliedalong a set of 256 gradient directions with 29 baseline images. In the experiments,we consider a volume of size 40×71×30 containing the left hemisphere of thebrain. We apply rbUKF-ODF and rbUKF-2T (with Nb = 1,000) on several whitematter fiber tracts such as the body, splenium (SP), and genu (GN) of the corpuscallosum (CC), the internal capsule (IC), the cingulum (CG), and the fornix(FX). One seed point per tract is selected in the CG, FX, GN, and SP, whereastwo seed points per tract are selected for the CC and IC to better illustratehow successfully the two diffusion models resolve intravoxel fiber complexities(i.e., partial volume effects) between these two fiber tracts.

Fig. 4 shows not only the trajectories computed from the bootstrappedrealizations of the HARDI signal using UKF-2T and UKF-ODF, but also the resultinglikelihood maps depicted on selected axial, coronal, and sagittal slices. First, weobserve that both the two-tensor and ODF models successfully resolve intravoxelcomplexities in the region (shown in the coronal slices at the top right of Figs. 4(a)-4(b)) where the CC and IC meet. The trajectories generated by UKF-2T andUKF-ODF) are dense and in accordance with the neuroanatomy. The dispersionin the bottom portion of the trajectories (generated by UKF-ODF) for the IC isdue to the slightly low GFA threshold we selected for stopping. At this point,it is worth noting that dispersion due to the residual bootstrap method is notto be confused with anatomical dispersion. Bootstrap dispersion is due to noise(refer to §3.2) and should be viewed as a measure of robustness or reproducibilityfor the tractography algorithm under study [8]. From this perspective, in theanalysis of the CG, the reproducibility of the results of rbUKF-ODF is higherthan that of rbUKF-2T as UKF-ODF produces trajectories almost solely in theCG bundle. More specifically, the trajectories produced by UKF-2T for the CGshow large bootstrap dispersion (i.e., bundles of trajectories that “bifurcate”around the region indicated by one of the white arrows in Fig. 4(b)) towardsthe corpus callosum. UKF-ODF, on the other hand, is able to recover the anteriorand posterior loops of the CG, which are usually difficult to extract due to highcurvature. The dispersion in the anterior portion of the trajectories (generatedby UKF-ODF) is due to low GFA threshold for stopping, similar to the case forthe internal capsule.

We reach a similar conclusion for the other WM neural fiber tracts. Inparticular, since the trajectories (generated by UKF-ODF) emanating from themiddle portion of the FX show no severe bootstrap dispersion, rbUKF-ODF canbe viewed as an algorithm that is more robust than rbUKF-2T, which produceslikelihood maps with larger dispersion (due to the region indicated by the

5 pbc.lrdc.pitt.edu/?q=2009a-home

10 H.E. Çetingül et al.

corresponding white arrow). For the fourth and fifth WM fiber tracts of intereset(i.e., the GN and SP, respectively), rbUKF-2T presents large dispersion in theforceps minor (resp. forceps major) and some spurious connections, whereasrbUKF-ODF generates considerably less bootstrap dispersion and hence it is morerobust in delineating these fiber tracts.

(a) Delineating WM neural fiber tracts using rbUKF-ODF.

(b) Delineating WM neural fiber tracts using rbUKF-2T.

Fig. 4. Trajectories emanating from the seed points shown in yellow (larger images onthe left) and the corresponding likelihood maps superimposed on the axial, coronal,and sagittal GFA maps (five images in the right columns). The regions with “large”bootstrap dispersion (e.g., bundles of trajectories that “bifurcate”) are indicated by thewhite arrows. In the likelihood maps, the probability of connectivity is depicted usingthe colors blue and red for rbUKF-2T and rbUKF-ODF, respectively.

Simultaneous ODF Estimation and Robust Probabilistic Tractography 11

5 Discussions and Conclusions

In this work, we first explain how to use the UKF-based tractography techniques,which were shown to be successful for simultaneous estimation (of diffusionmodels) and tractography, for HARDI data characterized by ODFs. The resultingalgorithm, which we refer to as UKF-ODF, uses the SH representation of the HARDIsignal as the state, guarantees that the reconstructed ODFs are nonnegative,and produces a trajectory by considering the modes of the ODFs as candidatedirections to follow [11]. We propose to apply the residual bootstrap methodto develop a UKF-based probabilistic tractography algorithm called rbUKF-ODF,which has the capability of rejecting erroneous trajectories by measuring changesin the state covariance. Experiments on synthetic, phantom, and real datademonstrate the advantages of our method over its two-tensor based version(rbUKF-2T) in terms of accuracy in tract delineation and dispersion in likelihoodmaps under noisy conditions. In addition, even though UKF-ODF uses a higher-dimensional state vector and an iterative ODF mode detection scheme, we observethat UKF-ODF and UKF-2T have comparable computation times.

However, we should also mention that, in some cases, increased accuracyand reliability in tracking may come with an undesirable outcome as UKF-ODF isprone to producing slightly less smooth streamline trajectories. We believe thatgeneration of sigma points as well as perturbation of the predicted states andmeasurements play a critical role in obtaining smooth trajectories. Accordingly,future work includes investigating other state space representations that willformalize the UKF into an intrinsic (manifold-constrained) formulation [19] onthe Riemannian manifold of ODFs, as discussed in [10] for the space of diffusiontensors. In particular, we will investigate ideas from [20,21] to perturb the states(i.e., spherical harmonic coefficients) in an intrinsic manner. We will also considerusing a faster mode detection strategy that directly operates on the sphericalharmonic coefficients of the ODFs to further reduce computation time.

Acknowledgments: This work was supported in part by NIH grants R01EB008432, P41 RR008079 (NCRR), P41 EB015894 (NIBIB), P30 NS057091,P30 NS5076408, and the Human Connectome Project (U54 MH091657) fromthe 16 NIH Institutes and Centers that support the NIH Blueprint for Neuro-science Research. L. Dumont contributed to this work while she was at SiemensCorporation, Corporate Research and Technology, Princeton, NJ, USA.

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