christopher p. hess et al- q-ball reconstruction of multimodal fiber orientations using the...

8/3/2019 Christopher P. Hess et al- Q-Ball Reconstruction of Multimodal Fiber Orientations Using The Spherical Harmonic Basis

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Q-Ball Reconstruction of Multimodal Fiber OrientationsUsing The Spherical Harmonic Basis

Christopher P. Hess,1 Pratik Mukherjee,1* Eric T. Han,2 Duan Xu,1 andDaniel B. Vigneron1

Diffusion tensor imaging (DTI) accurately delineates white mat-

ter pathways when the Gaussian model of diffusion is valid.

However, DTI yields erroneous results when diffusion takes on

a more complex distribution, as is the case in the brain when

fiber tracts cross. High angular resolution diffusion imaging

(HARDI) overcomes this limitation of DTI by more fully charac-

terizing the angular dependence of intravoxel diffusion. Among

the various HARDI methods that have been proposed, QBI of-

fers advantages such as linearity, model independence, and

relatively easy implementation. In this work, reconstruction of

the q-ball orientation distribution function (ODF) is reformulated

in terms of spherical harmonic basis functions, yielding an

analytic solution with useful properties of a frequency domain

representation. The harmonic basis is parsimonious for typical

b-values, which enables the ODF to be synthesized from a

relatively small number of noisy measurements and thus brings

the technique closer to clinical feasibility from the standpoint of

total imaging time. The proposed method is assessed using

Monte Carlo computer simulations and compared with conven-

tional q-ball reconstruction using spherical RBFs. In vivo results

from 3T whole-brain HARDI of adult volunteers are also pro-

vided to verify the underlying mathematical theory. Magn Re-

son Med 56:104117, 2006. 2006 Wiley-Liss, Inc.

Key words: diffusion tensor imaging (DTI); high angular resolu-

tion diffusion imaging (HARDI); q-ball imaging; fiber tractogra-

phy; white matter

The past decade has seen rapid advances in the develop-ment of diffusion MRI techniques that permit for the firsttime the noninvasive characterization of neural architec-ture. While the exact biophysical determinants of the dif-fusion signal have yet to be completely elucidated, it isnow generally accepted that microscopic boundaries todiffusion in the brain coincide with the local orientationsof white matter (WM) fiber tracts (1). The advent of diffu-sion tensor imaging (DTI) as a tool for modeling intravoxeldiffusion has inspired a number of promising applicationsin which WM connectivity can be evaluated in both healthand disease (2,3). Measures derived from the tensor arenow widely used to characterize regional anisotropy and

orientation of WM throughout the brain, and more recentlyhave been incorporated into tractography algorithms toallow 3D delineation of fiber pathways (46).

A major shortcoming of DTI lies in its inability to accu-rately characterize diffusion in complex WM, where fibertracts with different orientations intersect or are otherwisepartial volume averaged within a voxel (79). This limita-tion presents a significant obstacle to routine clinical ap-plication of DTI. For example, in brain tumor patients,presurgical DTI tractography fails to delineate the courseof the pyramidal tract in regions of fiber crossing (10).Although intravoxel diffusion in these areas of complexWM can be more accurately characterized using the q-space formalism (11), long experiment times and heavygradient demands preclude routine in vivo application ofthis technique in humans. High angular resolution diffu-sion imaging (HARDI) represents an alternative approachthat strives to improve imaging efficiency by recoveringthe angular structure of diffusion in lieu of the 3D spin-displacement probability function. Shorter imaging timesmake HARDI particularly promising for clinical applica-tions, especially when combined with the greater signal-to-noise ratios (SNRs) afforded by high-field MR systems(12) and the reduction in single-shot echo-planar imageartifacts provided by parallel imaging (1214).

Several approaches for reconstructing fiber orientationsfrom HARDI data have been proposed, including sphericalharmonic modeling of the apparent diffusion coefficient(ADC) profile (15,16), multitensor modeling (17), generalizedtensor representations (18,19), circular spectrum mapping(20), q- ball imaging (QBI) (2124), persistent angular struc-ture (25), and spherical deconvolution (26). Among thesetechniques, QBI and spherical deconvolution have generatedconsiderable interest due to their linearity and sensitivity tomultimodal diffusion. QBI has the additional desirable prop-erty of model independence because it does not require adhoc measurement of a response function in order to recon-struct fiber orientations. While the two methods differ signif-

icantly in their approach to recovering angular informationfrom the measured diffusion data, both define a continuousfunction over the sphere that encodes the anisotropy of dif-fusion within each voxel. The correspondence betweenpeaks of this function and multimodal fiber orientations has

been established experimentally for QBI using phantommodels (27,28).

Although it was originally formulated using sphericalradial basis functions (RBFs), the q- ball method can beimplemented using a number of well-characterized spher-ical basis sets, each of which yields a reconstruction withdifferent properties. In this paper we describe an imple-mentation of QBI using spherical harmonics, a basis setthat has been widely applied to various scientific applica-

1Department of Radiology, University of CaliforniaSan Francisco, San Fran-cisco, California, USA.2GE Healthcare Global Applied Sciences Laboratory, Menlo Park, California,USA.

Grant sponsor: NIH; Grant numbers: NS40117; 1 T32 EB001631-01A1; Grantsponsor: Department of Radiology, University of CaliforniaSan Francisco.

*Correspondence to: Pratik Mukherjee, M.D., Ph.D., Department of Radiol-ogy, Neuroradiology Section, University of CaliforniaSan Francisco, 505Parnassus Ave., Box 0628, San Francisco, CA 94143-0628. E-mail:[email protected]

Received 15 September 2005; revised 21 March 2006; accepted 22 March2006.

DOI 10.1002/mrm.20931Published online 5 June 2006 in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 56:104117 (2006)

2006 Wiley-Liss, Inc. 104


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tions in which a spherical geometry arises. This choice ofbasis leads to an analytic solution for the reconstruction,and allows frequency domain interpretation of the prop-erties of the resulting q-ball reconstruction. In this paperwe compare spherical harmonic q-ball reconstruction withRBF q-ball reconstruction, using both Monte-Carlo simu-lations and in vivo human brain experiments. Further-

more, we demonstrate how the choice of a spherical har-monic basis enables one to reliably determine multimodalfiber orientations from a relatively small number of noise-corrupted measurementsan important practical advan-tage for the potential clinical application of QBI.

THEORY

Linear HARDI and QBI

According to the q-space formalism, the wavevector thatdescribes diffusion encoding in a pulsed-gradient spin-echo experiment is defined as q (2)1g, where represents the gyromagnetic ratio, is the diffusion gradi-

ent duration, and g is the diffusion gradient vector. HARDIacquisition schemes measure samples from an underlyingdiffusion-attenuated signal E(q) at a finite set of points onthe sphere that are relatively uniform in their angulardistribution. The choice of the spherical sampling radiusin q-space, q0, depends on the desired angular resolution,the available SNR, and the gradient performance specifi-cations. In practice, b-values of 3000 s mm2 or greater aretypically required in order to distinguish among differentintravoxel fiber populations (9).

The goal of the HARDI reconstruction problem is to usethe available measurements to construct a spherical func-tion F(u) that characterizes the relative likelihood of waterdiffusion along any given angular direction u. For the

q-ball technique, this function is termed the fiber orienta-tion distribution function (ODF) (21). To simplify the nu-merical solution, both q and u are discretized to reflectfinite sampling of m measurements and reconstructionover a fixed number of points n. Because both the ODF andthe diffusion signal are defined on the domain of thesphere, it is convenient to normalize spherical points tounit magnitude and adopt a spherical coordinate systemq q(,) and u u(,), where and denote elevationand azimuth, respectively.

Linear HARDI reconstruction algorithms such as QBIand spherical deconvolution have in common that eachpoint of the reconstruction is computed as a linear combi-

nation of the diffusion measurements. Enumerating pointson the sphere to construct a vector representation, thisrelationship can be expressed as

fAe, [1]

where f and e denote n 1 and m 1 column vectorscomposed of the estimated values of the ODF and thediffusion measurements, respectively. Depending on thereconstruction algorithm employed, the n m reconstruc-tion matrix A is constructed using the spherical samplinggeometry and the assumed relationship between the diffu-sion space and the ODF. Linearity is desirable not only

from the standpoint of computational complexity, but also

because it facilitates the analysis of noise sensitivity andother properties of the reconstruction.

QBI assumes that the orientational structure of diffusionwithin each voxel can be accurately characterized by ra-dial projection of the q-space diffusion probability distri-

bution function (2124). This projection transforms the 3Dq-space probability distribution function into the 2D ODF

surface. Invoking the Funk transform, this idealized ODFis elegantly approximated by a great circle integration onthe sphere, i.e.,

F(u) qu

E(q)dq. [2]

When the method is implemented directly, it requiresinterpolation of the measurements in order to estimate thenecessary great circle integrals. Selecting a suitable linearrepresentation for the diffusion signal facilitates numericalcalculation of the necessary great circle integrals, so that

efficient reconstruction can be performed according to Eq.[1]. As noted by Tuch (23), who originally proposed re-construction in terms of spherical RBFs, there exist manyspherical basis sets that can be used to recover the ODFfrom the measurements. In what follows, we describe ef-ficient q-ball reconstruction on the spherical harmonic

basis. This form for the reconstruction is characterized bydifferent resolution and noise properties than RBF recon-struction, and allows efficient recovery of the ODF from asmaller number of measurements.

Spherical Harmonic Representation

With the proposed approach, the q-ball ODF is represented

as a linear combination of spherical harmonics Ylm(u) withorder l and phase factor m:

Ful0

L

ml

l

slmYl

mu, [3]

where slm denote the harmonic series coefficients, and L is

the harmonic series order. In spherical coordinates, thebasis elements can be written

Ylmu Yl

m,

1m2l

14

1/2

l

m!lm!

1/2

Plm(cos)eim, [4]

where Plm(x) are the associated Legendre polynomials:

Plmx

1 x2m/2

2ll!

dlm

dlmxx2 1 l. [5]

Spherical harmonic expansion of a function represents thegeneralization of the Fourier transform to the domain ofthe sphere, motivating its use for frequency-domain repre-sentation of spherical functions in a variety of scientificapplications (29). For example, representation of the ODF

using the spherical harmonic basis is an established

q-Ball Reconstruction 105


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method in materials science, and there is a considerablebody of literature in this field describing its use for char-acterizing crystal structure (30). Of note, the harmonicseries has also been applied to the HARDI reconstructionproblem by spherical deconvolution (26) and in analysis ofmultimodal diffusion using ADC profiles (15,16).

Because the spherical harmonics form a complete basis

set for the sphere, the representation is capable of describ-ing any bounded, finite-energy ODF given a sufficientlylarge order L. In practice, the number of diffusion measure-ments limits the maximum order that can be used, and theharmonic series must be truncated. If the convergence ofthe harmonic series is sufficiently rapid, the ODF can beaccurately represented using a small order. If the coeffi-cients do not converge or converge too slowly, truncationartifacts will distort the reconstruction. The rate of seriesconvergence determines the efficiency of any linear repre-sentation for the ODF, whether in terms of spherical har-monics, RBFs, or any other spherical basis elements.

Truncation of the spherical harmonic expansion is con-veniently studied using the concept of convolution on the

sphere. Whereas linear convolution is defined by integra-tion over the space of Cartesian translations, sphericalconvolution is performed by integration over the space of3D rotations. Driscoll and Healy (31) and Healy et al. (32)have shown that convolution in the angular domain ismathematically equivalent to multiplication in the angularfrequency domain. Restricting the harmonic representa-tion of the q- ball ODF to a finite model order L can beequated to apodizing the ideal, infinite-length harmonicseries with a uniform windowing function that is zero forall l L. Making use of spherical convolution, the result-ing ODF can be expressed as the convolution of the idealODF with the angular point spread function (PSF):

H l0

L

Pl0(cos). [6]

This is the inverse spherical Fourier transform of the uni-form windowing function. This PSF, which is illustratedin profile for several harmonic model orders in Fig. 1a, iscomprised of a single main lobe and L/2-1 sidelobes. Notethat the width of the main lobe varies inversely with theharmonic order L. Note also that as L increases, the numberof sidelobes increases and the width of each sidelobe de-creases. For all values of L, the PSF shows axial and

antipodal symmetry.By analogy with the convolution property of a linear

Fourier series, the relationship between the true ODF andthe reconstructed ODF using a truncated spherical har-monic series follows from the properties of the angularPSF. Convolution with the main lobe of the PSF smoothsthe ODF in the angular domain, and thus determines theeffective angular resolution. Consider as an example theharmonic representation of two delta functions on thesphere, each of which describes the orientation of a dis-tinct intravoxel fiber population. When they are separated

by sufficient angular distance, the delta functions can beindividually resolved. Below some minimum angular sep-

aration, the fibers are blurred together by the convolution

and can no longer be separately resolved. The width of themain lobe of the PSF, and thus the harmonic series order L,determines this minimum angular separation. The side-lobes of the PSF give rise to spurious oscillation in thereconstruction, the magnitude of which depends on thesmoothness of the true ODF.

In practice, the q-ball ODF is not comprised of delta

functions. Instead, measurement at a fixed b-value boundsthe rate at which angular features in the ODF may vary.The theoretically achievable angular resolution of the q-

ball technique depends on q0, the radius at which samplesare obtained in q-space. It has been shown that the linearspatial resolution of the radial projection operation variesaccording to 0/q0, where 0 denotes the first zero of azero-order Bessel function (23). This radial uncertaintyalso bounds the angular uncertainty of the ODF. However,since only a finite number of samples on the sphere areavailable for reconstruction, the angular resolution that isactually achieved is also subject to how the ODF is com-puted numerically. This important difference between theideal analog angular resolution and the actual digital

angular resolution of the reconstruction is determined bythe properties of the spherical basis functions that are usedto represent the ODF.

Resolution in traditional linear systems analysis is de-termined by the Rayleigh limit. Subject to finite sampling,this value is proportional to the full-width at half maxi-mum (FWHM) of the main lobe of the PSF of the recon-struction (33). Convolution on the sphere allows a similaranalysis to be undertaken when the ODF is representedusing spherical harmonics. Because angular resolution be-comes an independent function of azimuth and elevation,it is governed by the width of the main lobe of the spher-ical PSF. Calculated using Eq. [5], the width of the angular

PSF is plotted as a function of harmonic series order in Fig.1b. As may be expected by analogy with a Cartesian Fou-rier series, angular resolution is proportional to the order Lof the highest order harmonic in the series.

Parameter Estimation

The spherical harmonic basis also enables an analyticsolution for the q-ball reconstruction. To obtain the valuesof the q-ball ODF at the desired points, we first use theacquired data to expand the diffusion signal over an or-der-L harmonic representation

Eq

l0L

mll

clm

Ylm

q, [7]

where clm denote the harmonic series coefficients. This

step is equivalent to discrete spherical Fourier transforma-tion of the measurements. By substituting this representa-tion for E(q) into Eq. [2] and exchanging the order ofsummation and integration, any point ui of the resultingODF can be expressed as a linear combination of orthogo-nal great integrals of spherical harmonic basis elements:

Fui l0

L

ml

l

clm

q

u

Ylmqdq. [8]

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The necessary integrals can be computed numerically toresult in a linear solution, as done in the spherical RBFimplementation. Unlike RBFs, however, spherical har-monics are analytic with respect to great circle integrals.Specifically,

1

2qu

Ylmqdq Pl0Yl

mu, [9]

where Pl(x) denotes the unassociated Legendre polynomialof order l, and u is the unit vector orthogonal to the greatcircle over which the integration is performed. A proof ofthis useful property of the spherical harmonics is providedin the Appendix. Exploiting this analytic form for spheri-

cal harmonic great-circle integrals, the ODF is obtained bysimple scalar multiplication with the harmonic coeffi-cients of the diffusion signal, i.e.,

Eu l0

L

ml

l

2Pl0clmYl

mu, [10]

To summarize, reconstruction of the ODF is performed inthree steps: 1) spherical harmonic decomposition of themeasurements, 2) multiplication of the harmonic coeffi-cients of the data by Legendre polynomials Pl(0), and 3)spherical harmonic synthesis from the resulting ODF har-

monic coefficients.

Imposing two additional constraints on the harmonicseries further reduces the number of parameters that must

be computed. First, since diffusion encoding along anydirection is incapable of distinguishing between diffusionalong that direction and diffusion opposite that direction,the ODF exhibits antipodal symmetry. Because only even-order spherical harmonics define symmetric functions,

odd-order harmonics are not included in the representa-tion. Second, while in general the harmonic series is ca-pable of representing any complex-valued function, boththe ODF and the diffusion-weighted (DW) measurementsare real-valued functions on the sphere. As a consequence,the m 0 component for all orders is real and the har-monic coefficients at each order exhibit conjugate symme-try. For a spherical harmonic representation with order L,these two constraints reduce the total number of harmoniccoefficients from (L1)2 to (L1)(L2)/2. An alternative toforcing conjugate symmetry to ensure realness of the re-construction is to use real spherical harmonics (29).

Table 1 summarizes the proposed reconstruction algo-rithm. We use a least-squares approach to obtain the spher-

ical harmonic decomposition of E(q) from the raw diffu-sion measurements, similarly to the technique detailed byAlexander et al. (16) in their analysis of the ADC profile.Specifically, we first construct matrices ZQ and ZU fromthe values of the spherical harmonics at the measurementand reconstruction points. We denote the matrices con-taining the coordinates of the locations of the measure-ment points and reconstruction points on the unit sphere

by Q [q1 q2 qm]T and U [u1 u2 un]

T, respectively.

FIG. 1. a: Angular PSFs for harmonic model orders L 2, 4, 6, and

8. b: Angular resolution of the harmonic representation in degrees

as a function of the model order L, as calculated from the FWHM of

the main lobe of the spherical PSF.

FIG. 2. a: ODF reconstructions for simulated fiber populations with

equal FA values of 0.7 and volume fractions of 0.5, crossing at 90 with

b 3000 s mm2. The data are corrupted by complex Gaussian noise

with SNR 10, and reconstructions were generated using harmonic

orders L 4 (left) up to L 10 (right). b: The RMSE as calculated from

the absolute difference between the noisy and noise-free reconstruc-

tions is plotted as a function of model order, using b-values ranging

from1000s mm2 to4000 s mm2, averaged over 1000 trials of noise

at each order. Reconstructions for both a and b were generated from

m 131 diffusion measurements.



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Harmonic decomposition and synthesis are then per-formed by multiplication with matrices derived from thespherical harmonic functions and the points in Q and U.For example, ZQ is formed from the matrix concatenationZQ [Z Q

0 ZQ0

ZQL ], where

ZQl Y

ll(q1) Yl

0q1 Yllq1

Yllqm Yl

0qm Yllqm)

[11]contains the values of the spherical harmonics for order lat each point qi. Note here that Y l

l(q1) denotes the complex

conjugate of Yll

(qi). ZU is generated in the same mannerusing the reconstruction points U.Direct harmonic synthesis can be performed by multi-

plication by ZU, and harmonic decomposition is achievedby multiplication with the matrix ZQ

, the Moore-Penrosepseudoinverse ofZQ. Rather than directly undertaking har-monic decomposition and synthesis, however, it is conve-nient to generate a single reconstruction matrix so that Eq.[1] can be used to obtain the values of the ODF at thedesired points. Combining the individual reconstructionsteps, the final reconstruction matrix can be computed asA ZUPZQ

, where P is a diagonal matrix containing thenecessary values of the Legendre polynomial at each har-monic order. This (L1)(L2)/2 (L1)(L2)/2 square

matrix is formed from Pl(0) according to Eq. [10]. Thematrix can be readily generated as P diag[p0, p2, , pL]from vectors pl Pl(0) 12l1, where we have denoted by12l1 the (2l1) row vector of ones.

Unique reconstruction with the proposed method re-quires selection of the harmonic model order L, up to amaximum determined by the number of measured datapoints. While it is desirable to use a larger series order toachieve the highest possible angular resolution, higher-frequency harmonics are less reliably estimated from themeasurements and can give rise to spurious peaks in thereconstruction. These artifactual peaks are the result ofnumerical instability in the least-squares solution for the

harmonic series coefficients of the diffusion signal. Spe-

cifically, noise in the measurements is amplified in thereconstruction according to the numerical conditioning ofthe matrix ZQ. The condition number of this matrix in-creases with L, such that higher SNR is required in order togenerate reconstructions with larger harmonic model or-ders. This relationship mirrors the magnitude of the har-monic series coefficients, which decay rapidly with har-monic order.

Tournier et al. (26) address this problem in their imple-mentation of spherical deconvolution by attenuating high-er-frequency harmonic components. This apodization ofthe harmonic series coefficients is numerically equivalent

to applying a smoothing filter to the ODF in the angulardomain. A similar approach, used by Tuch (23) in the RBFimplementation of the q-ball method, is to directly apply aspherical smoothing filter to the reconstruction, the pa-rameters of which are selected empirically based on thevisual appearance of the calculated ODF. While both ofthese methods may be applied to the proposed sphericalharmonic q-ball reconstruction, we found that using acombination of two alternative techniques was more use-ful in our experiments, depending on the intrinsic resolu-tion of the ODF.

First, relatively small harmonic orders allow for accu-rate representation of the ODF at b-values typically used

for in vivo human brain imaging. When the SNR is low, asmay be the case in the clinical setting when examinationtime is limited, the choice of a small harmonic orderensures numerical stability in generating the reconstruc-tion. Heuristically, the choice of a lower model order en-ables a better fit to the measured data, since a smallernumber of parameters must be estimated from the avail-able measurements. Fluctuations in the data due to noiseare effectively removed by least-squares fitting because alower-order series is not capable of representing the high-frequency spikes introduced by measurement noise. Thepenalty of using this approach is to limit the angularresolution of the reconstruction. However, as demon-

strated in the following section, using L 4 permits accu-

Table 1

Summary of Q-Ball Reconstruction Algorithm Using Spherical Harmonics

Input

L Harmonic model order for q-ball reconstruction

e m 1 column vector of diffusion signal measurements

Q m 3 matrix of normalized diffusion gradient orientations

U n 3 matrix of desired reconstruction points on the unit sphere

Output

f n 1 column vector of ODF values at reconstruction points UAlgorithm

ZQ Matrix of spherical harmonics evaluated at measurement points Qa

ZU Matrix of spherical harmonics evaluated at reconstruction points Ua

pl Pl(0) 12l1 Form 2l 1 row vector of Legendre polynomials, l 0, 2, . . ., L

P diag[p0, p2, . . ., pL] Construct diagonal matrix from Legendre polynomial vectors

A ZUPZQ Construct n m reconstruction matrixb

fAe Compute ODFaThe matrices of spherical harmonics are constructed at the spherical points Q and U from the even-order spherical harmonic basis

elements up to order L (see text).bZQ

denotes the Moore-Penrose pseudoinverse of ZQ, i.e., ZQ (ZQ

HZQ)-1ZQ

H. As discussed in the text, matrix regularization may be

applied with larger values of L in order to ensure numerical stability. For Tikhonov regularization, the pseudoinverse is calculated as ZQ

(ZQHZQI)

-1ZQH where is the regularization parameter chosen on the basis of SNR (34).

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rate reconstruction at b 3000 s mm2 for both simulatedand in vivo experimental diffusion data.

When it is desirable to use a larger harmonic order, asecond approach for decreasing sensitivity to noise is touse matrix regularization to improve the numerical condi-tioning of harmonic decomposition. This approach per-mits a more balanced trade-off between angular resolution

and numerical stability than using a fixed harmonic order,and is useful with larger b-values in order to providehigher angular resolution. A simple regularization tech-nique, based on the method of Tikhonov, is to slightlyperturb the diagonal elements ofZQ

HZQ in computing thepseudoinverse of ZQ:

ZQHZQ 3ZQ

HZQ I, [12]

where is a regularization parameter, and I is the identitymatrix. The regularization parameter balances between thenumerical conditioning of the reconstruction and the de-gree of data misfit by imposing a uniform smoothness

constraint on the solution, and can be chosen on the basisof the data SNR (34). This form of regularization using thismethod permits the use of larger model orders while spu-rious peaks are still reduced in the ODF.

MATERIALS AND METHODS

Numerical Simulations

To investigate the accuracy of the proposed technique andcompare it with spherical RBF q-ball reconstruction, weperformed computer simulations using Monte Carlo sim-ulations similar to those described by Jones et al. (35) and

Jones (36) in their analyses of the tensor model. Two fiber

populations in slow Gaussian exchange, with orientationsseparated by a prescribed angle, were modeled as the sumof prolate Gaussian functions with equal volume fractionsof 0.5 and fraction anistropies (FAs) of 0.7. Samples of thecorresponding synthetic diffusion signal for different val-ues of diffusion weighting b were obtained along noncol-linear diffusion gradient orientations obtained using theelectrostatic repulsion algorithm and used to reconstructthe ODF. To evaluate the dependence of the reconstructionon SNR, complex-valued independent Gaussian noise wasadded to each sample, and the magnitude of the resultingdata was used to calculate the ODF. SNR was defined asthe ratio of the maximum signal intensity with no diffu-sion weighting to the standard deviation (SD) of the noise,

corresponding to the SNR of the b 0 s mm2 images thatwere obtained experimentally. SNR values for the b 0s mm2 non-DW data range from 10 to 50, which corre-spond to mean SNR values of approximately 2.4 to 12.1 forthe DW data of the simulated two-fiber system computedover all diffusion gradient orientations. This SNR range forthe simulations was deliberately chosen to overlap theSNR range of the experimental HARDI data.

Experimental Data

Whole-brain HARDI was performed on five adult male vol-unteers (2334 years old) using a 3T Signa EXCITE scanner

(GE Healthcare, Waukesha, WI) equipped with an eight-

channel phased-array head coil. The imaging protocol wasapproved by the institutional review board at our medicalcenter, and written informed consent was obtained from allfive participants. A multislice single-shot echo-planar spin-echo pulse sequence was employed to obtain measurementsat a diffusion weighting of b 3000 s mm2, where thediffusion-encoding directions were distributed uniformly

over the surface of a sphere using electrostatic repulsion.Conventional Stejskal-Tanner diffusion encoding was ap-plied with 31.8 ms, 37.1 ms, gmax 40mT m

1, andgeffective 39.5 mT m

1, yielding a q-space radius of534.7 cm1. An additional acquisition without diffusionweighting at b 0 s mm2 was also obtained.

The total scan time for whole-brain acquisition of 131diffusion-encoding directions was 39.6 min with TR/TE 18 s/84 ms, NEX 1, and isotropic 2-mm voxel resolution(FOV 260 260 mm, matrix 128 128, 68 inter-leaved slices with 2-mm slice thickness and no gap). Inthree subjects, whole-brain HARDI was performed at2.2-mm isotropic spatial resolution (FOV 280 280 mm, matrix 128 128, 60 interleaved slices with

2.2 mm slice thickness and no gap) using 55 diffusion-encoding directions with TR/TE 16.4 s/82 ms, for a totalscan time of 15.3 min. Axial slices in both cases wereoriented along the plane passing through the anterior andposterior commissures. Parallel imaging of the DW dataacquired with the eight-channel EXCITE head coil wasaccomplished using the array spatial sensitivity encodingtechnique (GE Healthcare, Waukesha, WI, USA) with anacceleration factor of 2.

Reconstruction and Visualization

The algorithm described in the Theory section and out-lined in Table 1 was used to generate the q-ball ODF from

both simulated and in vivo data. For comparison, thelinear matrix implementation of QBI detailed by Tuch (23)was used to reconstruct the ODF using RBFs. In selectingthe RBF parameters, we used a spherical Gaussian kernelwith width parameter i, calculated to minimize the logcondition number of the RBF interpolation matrix. Thenumber of equator points was k 48. For both methods,values of the ODF were computed at locations correspond-ing to the vertices of a fourfold tessellated icosahedron. Aspherical Gaussian smoothing kernel was also applied tothe RBF reconstructions, where the width of the kernel swas chosen visually to smooth out fluctuations in the ODF,as described by Tuch (23).

To facilitate interpretation of the results, anisotropy was

emphasized by min-max normalization. For display,points corresponding to the desired azimuth and elevationwere plotted as a 3D surface. Points on the surface of theODF were color-coded by direction according to the stan-dard red (leftright), blue (superoinferior), and green (an-teroposterior) convention used to represent the directionof the principal eigenvector in DTI. Since the normaliza-tion step can make relatively isotropic ODFs appear aniso-tropic, the surfaces are overlaid onto a grayscale back-ground corresponding to the generalized fractional anisot-ropy (GFA) within each voxel for visualization of thehuman subject data (23). The more anisotropic ODFs thusappear superimposed on brighter backgrounds, and the

background appears dark in voxels with low GFA.



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In addition to computing the q-ball ODFs for the humansubject data, the HARDI measurements were used to cal-culate the FA and orientation of the principal eigenvectoraccording to the standard DT model. Color FA maps usingthe standard red-blue-green directional encoding conven-tion were generated in DTIstudio v2 (Johns Hopkins Uni-versity, Baltimore, MD, USA) in order to compare the

orientation of the principal eigenvector with the q-ballODFs. All q-ball computations were performed using cus-tom software written in MATLAB version 7.0 (MathWorks,Natick, MA, USA).

RESULTS

Numerical Simulations

Figure 2a shows ODFs computed using the proposed methodfrom simulated data with SNR 10. Illustrated are represen-tative ODFs corresponding to harmonic orders L {4,6,8,10},

generated using m 131 diffusion measurements. Data weresynthesized using a two-fiber model with a 90 angle ofintersection and b 3000 s mm2. The effects of noise areminimal when L 4 but become increasingly more conspic-uous as the model order increases, as manifested by artifac-tual peaks that could erroneously be interpreted as distinctfiber populations. Low-frequency harmonics provide thelargest contribution to the structure of the ODF. As a conse-

quence, the directional information encoded by the ODFremains the same with larger harmonic orders. Only theshape of the ODF becomes more distorted by the noise-dominated high-frequency harmonics. By forcing a low har-monic order for the reconstruction, as done empirically forour in vivo data, high-frequency fluctuations due to measure-ment noise are avoided. For larger values of b, however, theselection of a smaller harmonic order comes at the expense ofangular resolution, such that fiber crossings at shallow angleswill not be reliably discerned when the harmonic order is toolow.

Figure 2b plots the root-mean square error (RMSE) be-tween the min-max normalized ODF and the noise-free ODFas a function of harmonic series order, averaged over 1000

trials of noise. Simulated data were again generated withSNR 10. The RMSE was calculated for diffusion weightingvalues ranging from b 1000 s mm2 to b 4000 s mm2

from the absolute difference between the noisy and noise-free reconstructions. By analogy with a linear Fourier series,noise in the unscaled ODF varies as the square root of thenumber of angular frequencies in the harmonic series. How-ever, as illustrated by the plot, RMSE following min-maxnormalization grows more rapidly as a function of har-monic order. The approximately linear relationship be-tween harmonic series order and RMSE is maintained overa large range ofb-values, with differences in magnitude ateach order L caused by the intrinsically lower SNR of data

that are more heavily diffusion weighted.

FIG. 3. Effect of sampling density on q-ball ODF reconstruction

from noisy simulated data using RBFs and spherical harmonics.

ODFs were generated using the L 4 spherical harmonic recon-

struction (first row) and the RBF reconstruction (second row) using

(a) 55, (b) 95, and (c) 131 data points. For each sampling density, the

reconstructions were computed from data with SNR 30 (first

column), SNR 20 (second column), and SNR 10 (third column).

Simulated data were generated for fibers crossing at 90 using b

3000 s mm2. The radial basis interpolation and smoothing param-

eters were (a) i 4.66 and s 18; (b) i 6.17 and s 12;

and (c) i 6.37 and s 10.

FIG. 4. Plots of the average RMSE between the computed noisy

ODF and noise-free ODF as a function of SNR. Min-max normalized

reconstructions were again generated using the two-fiber system

from Fig. 3 using m 55 samples. The RMSE is shown for harmonic

orders L 4 (red), L 6 (green), and L 8 (blue), and for theconventional QBI algorithm (black). The spherical Gaussian interpo-

lation and smoothing kernel parameters were i 4.66 and s

18 for RBF reconstruction.

110 Hess et al.


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Figure 3 compares example q- ball reconstructions fortwo fibers crossing at 90 over a range of SNRs that aretypical for in vivo HARDI. Reconstructions are depicted

for both the proposed approach and the conventional RBFimplementation. With sufficiently high SNR and largernumbers of measurements, estimation of both harmonicand RBF parameters is numerically stable and the resultsare qualitatively similar. However, the accuracy of thereconstructions becomes more dependent on SNR whenthe number of diffusion measurements is reduced. Notethat with a small number of measurements, as illustratedfor m 55, reconstruction of the ODF using the harmonicseries is less sensitive to SNR. Reconstructions using theproposed method are similar in appearance over a widerrange of SNR, with the primary effect of noise being per-turbation of the true widths and locations of the ODF

peaks.To more quantitatively assess the global effects of noiseon the reconstruction, we calculated the RMSE betweenODFs computed from noisy measurements and noise-freedata using the two-fiber system shown in Fig. 3. In thissimulation, 55 diffusion-encoding directions were used.Plots of the RMSE, averaged over 1000 independent trialsof noise, are shown in Fig. 4 as a function of SNR. Forsmaller harmonic orders ofL 4 and L 6, reconstructionusing the spherical harmonic basis is less sensitive tonoise than the reconstruction from RBFs in terms of RMSE.However, when the model order is increased to L 8, theproposed method becomes more sensitive to noise thanthe RBF implementation. As noted above, however, the

high angular frequency information contributes little tothe reconstruction at the tested b-values.

More important than the global effect of noise on thereconstruction is how noise alters the orientation of indi-vidual peaks in the ODF. To assess this directional esti-mation error, we again used the two-fiber system from Fig.3 and numerically ascertained the coordinates of the larg-est peaks of the ODF. Table 2 shows how the averageangular distance between the true and estimated peaklocations varied as a function of SNR over 1000 trials ofnoise for both the harmonic and RBF q-ball reconstructionmethods. Results are also given for harmonic reconstruc-tion using Tikhonov regularization. Note that for both

choices of basis, the mean and SD of the error increase as

the SNR falls. For low values of SNR and small numbers ofdiffusion-encoding directions, the regime in which QBImay be applied in the clinical setting, the proposed

method using low harmonic order yields higher accuracyof fiber orientations than RBF reconstruction. As with theglobal RMSE, this relative advantage is lost when the orderof the harmonic series is increased. However, by regular-izing the harmonic decomposition matrix, errors due tonumerical instability are attenuated and the accuracy inestimating fiber orientations can be significantly improvedwith larger harmonic orders.

As a final simulation, Fig. 5 illustrates the results of

Table 2

Error in Fiber Orientation Estimates as Calculated from the Maxima of the ODFa

SNR10 SNR20 SNR30 SNR40

Harmonic no regularization L4 5.9 2.7(10.3) 2.5 1.7(4.5) 1.1 1.3(4.2) 0.4 0.9(2.3)

L6 8.2 3.3(13.8) 3.7 1.9(6.8) 2.2 1.5(4.5) 1.4 1.4(4.2)

L8 15.1 5.8(22.9) 12.9 5.6(20.9) 11.0 5.4(20.2) 9.7 5.4(19.3)

Harmonic with regularization L4 5.7 2.5(10.0) 2.5 1.6(4.4) 1.1 1.3(4.2) 0.4 0.9(2.3)

L6 6.5 3.2(11.8) 2.8 1.8(6.0) 1.5 1.5(4.3) 0.7 1.1(2.6)L8 8.3 3.8(15.3) 3.9 2.0(6.8) 2.3 1.6(5.1) 1.4 1.4(4.2)

RBF i4.66;

s18

8.1 4.4(17.0) 5.4 2.0(7.9) 5.0 1.2(6.2) 4.9 0.9(5.9)

aThe SNR is increased from 10 to 40 from left to right, and the rows show the error in peak location estimation for the proposed harmonic

method with L4, L6, and L8 (both with and without regularization) and for the conventional RBF implementation of QBI. Reconstruction

was performed for fibers crossing at 90 using 55 diffusion measurements. Errors in estimated peak locations are reported as mean SD

(95% confidence interval) in degrees, computed over 1000 independent trials of noise at each SNR. Selection of the RBF parameters was

performed as described in Ref. 23.

FIG. 5. Bar graph showing power contributions of coefficients at

orders L 2, 4, 6 and 8 to harmonic series. Harmonic series

coefficients up to order L 10 were computed from m 282 data

points at b 3000 s mm2 for a single fiber (blue), a two-fiber

system (green), and a three-fiber system (red). For each simulated

architecture, 1000 realizations were generated, with individual fiber

orientations selected randomly from a uniform distribution over the

sphere. The graph depicts the fractional energy of each angular

frequency for all three simulated architectures, averaged over the

random fiber orientations. Note that the isotropic (l 0) harmonic is

excluded from the energy calculation because it does not contribute

to ODF anisotropy.



9/14

simulations that tested the representation efficiency of theharmonic series. In these simulations a large number ofmeasurements (m 282) were used to compute the har-monic series coefficients of one-, two-, and three-fiber ar-chitectures using harmonic orders up to L 10. The con-tribution of each angular frequency to the power spectrumwas computed according to

El

1

ET mll

42

Pl02

clm

2

, [13]

where ET denotes the total power of the tenth-order har-monic representation for all orders l 2. The l 0 har-monic is excluded from the calculation because it does notcontribute to anisotropy in the reconstruction and varieswith normalization of the ODF. For all three architectures,the power spectrum was averaged over 1000 uniformlydistributed random orientations for each of the componentfibers. The graph summarizes the fraction of the totalpower that each angular frequency contributes to the har-monic series for the different simulated tissue architec-tures. Because the spectral energy is concentrated at low

angular frequencies, accurate reconstruction of the ODF ispossible using a small order L. With b 3000 s mm2,harmonics up to order L 4 represent greater than 99.3%of the spectral energy of the series for all three syntheticfiber architectures. Larger values of b (results not shown)give rise to an ideal ODF that more closely resembles adelta function, resulting in a harmonic representation inwhich there is a more uniform distribution of energy overthe angular frequency. For b 5000 s mm2 and b 7000 s mm2, for example, the fraction of total energycontributed by harmonic orders up to L 4 falls to 96.9%and 92.9%, respectively. Note that the power spectrum isconcentrated similarly for each fiber system tested, as ex-

pected from the linearity of the harmonic representation.

Experimental Data

For the 131-direction in vivo HARDI data acquired at2-mm isotropic voxel resolution, the average SNR of theb 3000 s mm2 DW images was calculated to be 3.1 inthe thalamus, 2.6 in the corpus callosum, and 3.4 in thecentrum semiovale. For the corresponding b 0 s mm2

T2-weighted images, the mean SNR was estimated to be15.8 in the thalami, 16.4 in the corpus callosum, and 16.3in the centrum semiovale. For the 55-direction HARDI

data set acquired at 2.2 mm isotropic voxel resolution, themean SNR for the b 3000 s mm2 DW images wasestimated to be 4.2 in the thalami, 4.2 in the corpus callo-sum, and 4.7 in the centrum semiovale. For the b 0 s mm2 T2-weighted image it was estimated to be 21.2in the thalami, 19.4 in the corpus callosum, and 21.9 in thecentrum semiovale. For the 55-direction data set, Fig. 6shows the b 0 s mm2 T2-weighted image and severalb 3000 s mm2 DW images with different diffusion-encoding directions from one volunteer for a single axialslice location at the level of the foramen of Monro. Becauseparallel imaging was used in our experiments, we calcu-lated the effective SNR measurements from two separateacquisitions using identical experimental parameters (37).

Axonal pathways can be broadly categorized as one ofthree general types: 1) association fibers, which connectdifferent regions of the cerebral cortex within the samecerebral hemisphere; 2) commissural fibers, which passthrough the midline of the brain to connect cortical regionsin the left and right hemispheres; and 3) projection fibers,which connect the cerebral cortex with subcortical struc-tures and include such functionally important pathwaysas the pyramidal tract and somatosensory radiations, bothof which course in the centrum semiovale. Directionally-encoded color FA images, such as the one shown in Fig.7a, differentiate between association, projection, and com-missural fibers by virtue of their distinct topologies. Since

commissural fibers connect the left and right hemispheres

FIG. 6. Representative HARDI images from

one normal adult volunteer. T2-weighted

(top left) and b 3000 s mm2 DW imagescorresponding to seven diffusion gradient

orientations are shown at the level of the

foramen of Monro. A total of 55 diffusion

gradient orientations were acquired at

2.2-mm isotropic voxel resolution in this vol-

unteer. See Materials and Methods for de-

tails of the experimental parameters.

112 Hess et al.


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of the brain, they are oriented in a leftright direction andtherefore appear red on color FA images. Since projectionfibers connect the cerebral cortex with subcortical struc-tures, most projection tracts are oriented superoinferiorlyand appear blue. Long association fiber tracts tend to con-nect regions in the front of a hemisphere with regions in

back of the same hemisphere, and therefore these fibersappear green due to their anteroposterior orientation.

Spherical harmonic QBI using L 4 and conventionalQBI using RBFs yielded comparable ODF reconstructions

for the 131-direction HARDI data sets from all volunteers,although the RBF reconstruction produced generally nois-ier results. Figure 7b and c compare harmonic q- ball re-construction and RBF q-ball reconstruction of ODFs fromthe supratentorial WM of one representative volunteer,

based on 131-direction HARDI. Figure 7d and e were ob-tained from a 2 2 array of voxels in the subcortical WMof the right cerebral hemisphere, where there are crossingsamong commissural fibers of the corpus callosum, associ-ation fibers of the superior longitudinal fasciculus, andprojection fibers of the centrum semiovale. Both conven-tional QBI and spherical harmonic QBI at L 4 yield ODFsthat successfully depict the multimodal diffusion. How-

ever, as predicted by the simulations of Fig. 3, RBF q-ball

reconstruction yields poorer results for the 55-directionHARDI data shown in Fig. 8b, whereas harmonic q-ballreconstruction at L 4 still accurately depicts the complexsupratentorial WM architecture even at this lower sam-pling density. Additional in vivo results using 55-direc-tion HARDI are shown in Fig. 9 for the infratentorial WMof the brainstem. The two-way fiber crossing within theupper pons between superoinferiorly oriented projectionfibers of the pyramidal tract and transversely orientedpontocerebellar fibers is clearly demonstrated with spher-

ical harmonic QBI at L 4, but not as well by RBF q-ballreconstruction.

DISCUSSION

For visualizing the complex WM anatomy of the humanbrain, high spatial resolution and high SNR of the in vivoHARDI data are both desirable. The use of 3T improved theSNR in this study, and parallel acquisition helped to mit-igate susceptibility artifacts around the air-filled paranasalsinuses and mastoid sinuses, and reduce the eddy-currentdistortion that is often associated with heavy diffusionweighting. The SNR achieved at 2-mm isotropic voxel

resolution compares favorably with that obtained in an-

FIG. 7. In vivo comparison of QBI using spherical

RBFs with spherical harmonic QBI for 131-direction

HARDI in a 34-year-old normal volunteer. a: Coronal

directionally-encoded color FA image derived from

DTI of a 131-direction DW data set, obtained with 3Tparallel imaging at b 3000 s mm2. The cingulum

bundle (CB), centrum semiovale (CS), corpus callo-

sum (CC), and superior longitudinal fasciculus (SLF)

are labeled. b: 19 12 voxel array of ODFs derived

from RBF q-ball reconstruction of the 131-direction

diffusion data set, within the region of the right su-

pratentorialWM enclosed in the yellow box inset of a.

Each voxel is cubic and 2 mm on a side. The ODF in

each voxel is superimposed on a grayscale back-

ground modulated by the GFA in that voxel (black:

GFA 0; white: GFA 1). The color shading of the

ODFs indicates orientation using the same three-

color scheme as in DTI. The interpolation value i

3.03 optimized the condition number of the RBF

reconstruction matrix, and smoothing kernel width s 15 was used for best visual results. c: Corre-

sponding 19 12 voxel array of ODFs derived from

spherical harmonic QBI with model order L 4. d:

Enlarged view of a 2 2 array of RBF q-ball ODFs

from thevoxels indicatedby the yellow boxinsetof b,

without the grayscale background, depicting cross-

ings of association fibers of the SLF, projection fibers

of the CS, and commissural fibers of the CC. e:

Corresponding 2 2 array of spherical harmonic

q-ball ODFs from the yellow box inset of c, showing

the same fiber crossings as in d.



11/14

FIG. 8. In vivo comparison of QBI using spherical

RBFs with spherical harmonic QBI for 55-direction

HARDI in the supratentorial WM of a 29-year-old

healthy volunteer. a: Coronal directionally-en-

coded color FA image derived from DTI of a 55-

direction DW data set, obtained with 3T parallel

imaging at b 3000 s mm2. The cingulum bun-

dle (CB), centrum semiovale (CS), corpus callosum

(CC), and superior longitudinal fasciculus (SLF) are

labeled. b: 20 12 voxel array of ODFs derived

from RBF q-ball reconstruction of the 55-direction

diffusion data set, within the region enclosed in the

yellow box inset of a. Each voxel is cubic and

2.2 mm on a side. All conventions are as in Fig. 7.

The interpolation value i 2.36 optimized the

condition number of the RBF reconstruction ma-

trix, and smoothing kernel widths 15 was used

for best visual results. c: Corresponding 20 12voxel array of ODFs derived from spherical har-

monic QBI with model order L 4 shows improved

reconstruction of the in vivo fiber orientations. d:

Magnified view of the 2 2 array of RBF q-ball

ODFs from the voxels indicated by the yellow box

inset of b. e: The corresponding 2 2 array of

spherical harmonic q-ball ODFs, from the yellow

box inset of c, better visualizes the crossings of

association fibers of the SLF, projection fibers of

the CS, and commissural fibers of the CC.

FIG. 9. In vivo comparison of conventional QBI

and spherical harmonic QBI in the infratentorial

WM of a 23-year-old normal volunteer, using 55-

direction HARDI. a: Coronal directionally-encoded

color FA image derived from DTI of a 55-direction

DW data set, obtained with 3T parallel imaging at

b 3000 s mm2. The pyramidal tract (PT), and

transverse pontocerebellar fibers (TPF) are labeled.

b: 10 12 voxel array of ODFs derived from con-

ventional QBI, corresponding to the yellow boxed

inset of a, and displayed with the same conven-

tions as in Fig. 7. The interpolation value i 2.36optimized the condition number of the RBF recon-

struction matrix, and smoothing kernel width s

15 was used for best visual results. c: The corre-

sponding 10 12 voxel array of ODFs derived

from spherical harmonic QBI with model order L

4. d: Magnified view of the 2 2 array of conven-

tional q-ball ODFs from the voxels indicated by the

yellow boxed inset of b, without the grayscale

background, depicting two-way crossings of the

PT with the TPF. e: Corresponding spherical har-

monic q-ball ODFs, from the yellow boxed inset of

c, better visualizes the same two-way fiber cross-

ings as in d.

114 Hess et al.


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other recent 3T HARDI investigation with 2.2-mm spatialresolution (23), and the spatial resolution exceeds that ofmany reported DTI studies.

Prior research on analysis of ADC profiles has shownthat spherical harmonics provide a useful representationfor detecting multimodal diffusion in regions of complexWM architecture (15,16). In the absence of noise, isotropic

diffusion of the ADC profile is described by the zero-orderspherical harmonic, and single fiber diffusion is character-ized by the second-order harmonics. Similarly, the zero-order component of the harmonic series represents theisotropic component of the ODF, and higher-order har-monics of the ODF characterize anisotropic diffusion.Whereas peaks of the ADC profile do not necessarily cor-respond to individual fiber orientations, however, severalpapers have shown through both simulations and experi-ments that the q-ball ODF peaks coincide with well-de-fined principal directions of diffusion (27,28).

Since the greatest fraction of the total spectral energy ofthe ODF is concentrated in the lower-frequency harmonicsfor typical b-values, the harmonic series representation isparsimonious. The ability to recover the ODF from a rela-tively small number of measurements is particularly im-portant in the clinical setting, where scan times exceeding20 min are unlikely to be useful for evaluating patients. Inour experiments, accurate q- ball ODF reconstructionswere performed using L 4 with as few as 55 diffusion-encoding directions, and required only 15 min of scantime for a whole-brain acquisition at 2.2-mm isotropicvoxel resolution.

The results of Monte Carlo simulations, such as thoseillustrated in Fig. 4 and Table 2, indicate that when thenumber of measurements is small, the spherical harmonicrepresentation for the ODF with L 4 has better SNR

performance and less angular error of the reconstructedfiber orientations than does RBF q-ball reconstruction. Al-though this low harmonic order necessarily sacrifices an-gular resolution, our simulations show there is minimaldegradation in angular resolution for b-values up to4000 s mm2. Figures 8 and 9 show that there are inter-esting features in the complex WM of the human brain thatcan be revealed within a clinically feasible whole-brainacquisition time using the proposed approach. For in vivoHARDI data with hundreds of diffusion measurements andlarge b-values, as may be employed in scientific studies ofanimal models or normal volunteers, the RBF representa-tion may be a better choice than the harmonic representa-

tion. At these large q-space radii, higher-order harmonicsare necessary to capture the greater angular resolution, andthe noise sensitivity of the spherical harmonic representa-tion escalates rapidly with the model order.

While optimal compression of HARDI data remains anactive area of research, it is worthwhile to note that theharmonic representation enables a significant reduction instorage requirements. Specifically, as an alternative to stor-ing each point of the raw HARDI data or the calculatedvalues of the ODF, only the harmonic series coefficientsneed be stored for each voxel. For a given harmonic modelof order L, the (L1)(L2)/2 complex-valued coefficientscan be stored as the same number of real-valued coeffi-

cients by making use of conjugate symmetry.

A weakness of spherical harmonics is the tendency toproduce false peaks when the harmonic series is truncatedprematurely or when the SNR is very low, which may beproblematic for experiments that employ heavy diffusionweighting (large q0). This deficiency of the harmonic seriesis a consequence of the fact that the harmonic basis ele-ments are monotonic, and may thus be slow to converge

when there are sharp angular features in the ODF. In thisscenario, it may be advantageous to employ a non-mono-tonic basis, such as spherical Gabor functions or wavelets.Such basis sets may also be analytic with respect to great-circle integration, and, if so, an analytic solution for theODF would remain possible.

The issue of sampling also merits further investigation.In theory, there exist a minimum number of measurementsthat must be obtained in order to capture the intrinsicangular information encoded by samples at a fixed radiusin q-space. When the minimum sampling requirements arenot met, aliasing may arise in the reconstruction (38). Withthe b-values in our simulations and in vivo experiments,the angular variation of the underlying diffusion was suf-

ficiently smooth to avoid aliasing. An additional potentialavenue for further research on the effect of samplingwould be to determine the optimum orientations for thediffusion-encoding gradients. Sampling in the presentwork was based on electrostatic repulsion. However, theq-ball reconstruction places no restriction on how gradientorientations are distributed on the sphere. It may be ad-vantageous to acquire samples on a subsurface of thesphere, such as a hemisphere, or nonuniformly on thesphere. The implications of such sampling schemes willlikely depend upon the choice of basis functions.

APPENDIX

In this appendix we show how great circle integration overspherical harmonics can be performed analytically. Theproof follows that originally given by Backus (39) for ap-plication to geophysical data.

The integral along the great circle orthogonal to thevector u in the reference frame (x,y,z) can be rewritten asan equivalent integral in which the vector u coincides withthe z axis in a rotated coordinate system (x,y,z). Thischange of coordinates places the equator of the great circlein the (x,y) plane, such that

qu

Ylmqdq0

2

Ylm2,d, [A.1]

where is the azimuthal coordinate in the (x,y) plane,and Yl

m denotes the rotated spherical harmonic function.This change of coordinates allows the great circle integralto be more readily calculated as a linear integral definedover azimuthal angles . Note that the integration is alsonow independent of , the elevation parameter in the(x,y,z) coordinate system.

Because the spherical harmonics for any order l com-prise an orthonormal basis for the (2l1)-dimensional

space spanned by lth degree spherical polynomials (29),



13/14

any rotated spherical harmonic Ylm can be uniquely ex-

pressed as a linear combination of spherical harmonics ofthe same degree. That is, there exist complex coefficientsalmn(u) such that

Ylm ,

nl

l

almn(u)Yl

n, [A.2]

and

nl

l

almn(u)al

mn(u) mn. [A.3]

While the proof does not require their calculation, it isworthwhile to note that the coefficients al

mn(u) correspondto rotational harmonics, defined in terms of the rotationthat places u onto the z-axis (32). Making use of this

expression, Eq. [A.1] can be written as

qu

Ylmqdq

nl

l

almnu

0

2

Yln2,d. [A.4]

By inspecting the expression for spherical harmonics asdefined in Eq. [4] in the text, one can appreciate that theintegral on the right is nonzero only for n 0, due to thesymmetry of the phase term. As a result,

qu

Ylm(q)dq 2al

m0(u)Yl02,0, [A.5]

and only the coefficient alm0(u) has to be calculated in

order to complete the proof. From Eqs. [A.2] and [A.3], it isalso possible to express any harmonic in the rotated frameas a linear combination of harmonics with the same orderin the reference frame:

Yln,

ml

l

almn(u)*Yl

m,. [A.6]

A well known property of spherical harmonics is that theysatisfy an addition theorem, which states that for any twodirections (,) and (,) separated on the sphere by anangle (,),

Yl0,0

1

Yl00,0

ml

l

Ylm ,Yl

m , . [A.7]

Comparing terms in Eqs. [A.6] and [A.7], the coefficient

alm0

(u) is derived as

alm0

Ylm(,)

Yl0(0,0)

. [A.8]

Substituting this expression into Eq. [A.5] yields

quYl

m(q)dq 2

Yl0/2,0

Yl0

0,0 Yl

m,, [A.9]

where (,) now denotes the the angle between q and u.Finally, noting that these two vectors are orthogonal, andcomputing the values of the harmonics at the necessarypoints on the sphere, we have

qu

Ylm(q)dq 2Pl0Yl

m(u). [A.10]

As discussed in the text, this equation allows the harmoniccoefficients of the q-ball ODF to be derived directly fromthose of the diffusion signal.

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