Spatial Normalization of the Fiber OrientationDistribution Based on High Angular Resolution DiffusionImaging Data

Xin Hong,1,2* Lori R. Arlinghaus,1,2 and Adam W. Anderson1–3

Comparison of high angular resolution diffusion imaging(HARDI) measurements between subjects or between time-points for the same subject are facilitated by spatial normaliza-tion. In this work an algorithm was developed to transform thefiber orientation distribution (FOD) function, based on HARDIdata, taking into account not only translation, but also rotation,scaling, and shearing effects of the spatial transformation. Thealgorithm was tested using simulated data and intrasubject andintersubject normalization of in vivo human data. All casesdemonstrated reliable transformation of the FOD. This tech-nique makes it possible to compare the intravoxel fiber distri-bution between subjects, between groups, or between time-points for a single subject, which will be helpful in HARDIstudies of white matter disease. Magn Reson Med 61:1520–1527, 2009. © 2009 Wiley-Liss, Inc.

Key words: spatial normalization; fiber orientation distribution;HARDI

Diffusion MRI is a useful tool to study the structure andorganization of human brain white matter. Due to anatom-ical differences between individual brains, spatial normal-ization of the data is usually needed to make comparisonsbetween subjects. However, simple normalization of dif-fusion-weighted (DW) images is not sufficient to retain theorientation information of the underlying structure. Thepreservation of the principal direction (PPD) algorithm(1,2) has been proposed to rotate the diffusion tensor sothat the principal directions of the tensor are preservedrelative to local anatomical structures. Compared with dif-fusion tensor imaging (3), high angular resolution diffu-sion imaging (HARDI) (4–8) is able to provide more accu-rate estimates of the distribution of fiber orientationswithin a voxel. However, effective spatial normalization ofHARDI data has not yet been demonstrated. In this studywe propose an algorithm to transform the fiber orientationdistribution (FOD) function based on HARDI data, takinginto account not only translation, but also rotation, scal-ing, and shearing effects. The algorithm was tested with

transformations of both simulated data and in vivo humandata.

MATERIALS AND METHODS

FOD Transformation

The FOD derived from spherical deconvolution methods(9–11) is a function on the unit sphere. The peaks of theFOD provide information about the underlying fiber com-ponents. The orientation of each peak indicates the orien-tation of a fiber, and the magnitude of each peak is pro-portional to the volume fraction of the corresponding fiber.When sampled in a number of directions evenly distrib-uted over a sphere, the FOD function can be approximatedby the values it takes at those directions. This can berepresented by a set of sampling vectors, the lengths ofwhich represent the fiber volume fractions along the cor-responding directions.

From the deformation field that registers the DW images,a Jacobian matrix J can be derived for each voxel, whichrepresents the local deformation at that point. The pathtaken by a fiber through the voxel can be approximated bya series of line segments, written dri � �dxi,dyi,dzi�

T forthe ith segment. Under the local deformation, these seg-ments transform to drj� � J � dri. Since every fiber passingthrough the voxel is subject to the same Jacobian, the FODmust also be transformed using J. Applying J to the sam-pling vectors of the corresponding FOD will give the neworientations of these vectors, and hence a discrete approx-imation of the transformed FOD.

The integral of the FOD over a unit sphere equals 1,which is a basic property of a distribution function, andshould be retained after transformation. The value of theFOD at polar angle � and azimuthal angle � is writtenP(�,�). The volume fraction of fibers with orientation near(�,�) equals P(�,�) � d�, where d� � sin �d�d� is theelement of solid angle describing the neighborhood (seeFig. 1). The volume fraction of fibers oriented toward thissmall patch d� must remain the same after the patch istransformed:

P��,�d� � P����,��d��

where the unprimed and primed symbols represent thecorresponding quantities before and after transformation,respectively. Equivalently,

P��,�sin�d�d� � P����,��sin��d��d�� [1]

Therefore, the length of the transformed vectors shouldbe adjusted to guarantee that:

1Department of Biomedical Engineering, School of Engineering, VanderbiltUniversity, Nashville, Tennessee.2Institute of Imaging Science, Vanderbilt University, Nashville, Tennessee.3Department of Radiology and Radiological Sciences, Vanderbilt University,Nashville, Tennessee.Grant sponsor: National Institutes of Health (NIH)/National Institue of Biomed-ical Imaging and Bioengineering (NIBIB); Grant number: R01-EB02777.*Correspondence to: Xin Hong, Vanderbilt University Medical Center, Instituteof Imaging Science, AA-1105 MCN, Nashville, TN 37232-2310. E-mail:xin.hong@vanderbilt.eduReceived 26 April 2008; revised 12 September 2008; accepted 14 November2008.DOI 10.1002/mrm.21916Published online 7 April 2009 in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 61:1520–1527 (2009)

© 2009 Wiley-Liss, Inc. 1520

P����,�� � P��,�sin�d�d�

sin��d��d��[2]

According to the substitution rule for multiple variables,

d��d�� � �det(J�)�d�d� [3]

where � � � denotes absolute value, det( � ) denotes the

determinant, and J� � ���

�

��

���

�

��

��, which is the Jaco-

bian of the angular transformation from (�,�) to (��,��).Substituting this relation into Eq. [2], we have:

P����,�� � P��,�sin�

sin��

1�det(J�)� [4]

To calculate J�, let r � P(�,�) and r� � P�(��,��) be theFOD magnitudes along the original direction (�,�) and thecorresponding transformed direction (��,��), respectively.Since J is most naturally expressed as a 3 � 3 matrix inCartesian space, we must convert from spherical to Carte-sian coordinates,

� x � rsin�cos�y � rsin�sin�z � rcos�

[5]

before applying the spatial normalization Jacobian J, thenconvert the results back to spherical coordinates,

�r� � �x�2 � y�2 � z�2

�� � cos�1�z�

r���� � tan�1�y�

x��[6]

The Jacobians of these two transformations are:

J1 � �xr

x�

x�

yr

y�

y�

zr

z�

z�

�� sin�cos� rcos�cos� � rsin�sin�

sin�sin� rcos�sin� rsin�cos�cos� � rsin� 0

[7]

and

J3 � �r�x�

r�y�

r�z�

��

x�

��

y�

��

z���

x�

��

y�

��

z�

�� �

x�

r�y�

r�z�

r�x�z�

�x�2 � y�2 � r�2

y�z�

�x�2 � y�2 � r�2

� �x�2 � y�2

r�2

� y�

x�2 � y�2

x�

x�2 � y�2 0� [8]

respectively. The total transformation consists of 3 parts:

�r,�,��O¡J1

�x,y,z�O¡J2 � J

�x�,y�,z��O¡J3

�r�,��,��� [9]

so that:

�r,�,��O¡Jtotal

�r�,��,���, where Jtotal � J3 � J2 � J1

� �r�r

r��

r��

��

r��

�

��

���

r��

�

��

�

� [10]

Therefore, for a direction (�,�), the angular Jacobian, J�,of the total transformation to (��,��) is given by the lowerright 2 � 2 part of Jtotal.

In summary, the transformation of an FOD is performedin four steps:

1. A Jacobian matrix J for each voxel is derived from thedeformation field of the spatial normalization trans-formation applied to the DW images.

2. Each FOD is represented by a number of samplingvectors evenly distributed over a sphere. For eachsampling vector, J is applied to transform the orien-tation from (�,�) to (��,��).

3. For each sampling vector, �det(J�̂)� is calculated for the

FIG. 1. A small patch on the unit sphere before (a) and after(b) transformation, with solid angles d� � sin �d�d� and d�� �sin ��d��d��, respectively.

Spatial Normalization of FOD 1521

pair of directions (�,�) and (��,��) and the length ofthe vector is adjusted according to Eq. [4].

4. Finally, the transformed FOD is approximated by theset of transformed, length-adjusted sampling vectors.

Note that noise and truncation artifacts may cause theFOD to have negative values along some orientations.These negative vectors are transformed in the same way aspositive vectors in order to maintain the unit integral ofP(�,�).

Numerical Simulations

The proposed algorithm was tested using numerical sim-ulations. Four intravoxel fiber structures with crossingangle varying from 0° to 90° were chosen. The correspond-ing FODs were simulated using the FORECAST sphericaldeconvolution method (11) through 6th order. Two trans-formations were applied to the FODs: a horizontal stretch

1.5 0 00 1 00 0 1

and a vertical shear �1 0 0

� tan�

10� 1 0

0 0 1�

(Fig. 2a). The integrals of the transformed FODs weretested against the expected value 1. Performance was eval-uated by the angular error between the peaks of the trans-formed FODs and the true transformed fiber orientations.Four different numbers of sampling points, 92, 252, 1002,and 4049, all generated by icosahedral tessellation of theunit sphere (12), were tested in each case.

Image Acquisition and Registration

In addition to numerical simulations, this algorithm wasalso tested by transformation of in vivo human data. Twointrasubject experiments and one intersubject experimentwere performed. All scanning procedures were approvedby the Vanderbilt University Institutional Review Boardand performed on a 3T Philips Achieva (Best, the Nether-

lands) scanner. In the first intrasubject experiment, ahealthy subject rotated his head in the scanner betweenscans, first around the left–right, then around the superi-or–inferior axis. HARDI images were acquired in all threehead positions using the following imaging parameters:b � 1000s/mm2, 92 diffusion-sensitizing directions (the 46directions given by 3rd order icosahedral tessellation of asphere (12) and their opposites), 96 � 96 � 53 matrix size,and 2.5 mm isotropic voxel size. For each head orienta-tion, high-resolution T2-weighted (T2-W) images (512 �512 � 53 matrix size, 0.45 � 0.45 � 2.5 mm voxel size)were also obtained at the same slice positions for registra-tion purposes. The total imaging time was about 1 hr. Asecond intrasubject experiment was performed on anotherhealthy subject using the same imaging parameters, butdata were collected in only two head orientations whichdiffered by rotation around the anterior–posterior axis. Forthe intersubject experiment, HARDI images were acquiredfrom a group of 23 schizophrenia outpatients and20 healthy controls using the following imaging parame-ters: b � 1000s/mm2, 92 diffusion-sensitizing directions,96 � 96 � 55 matrix size, and 2.5 mm isotropic voxel size.High-resolution T2-W images (512 � 512 � 55 matrix size,0.45 � 0.45 � 2.5 mm voxel size) and T1-weighted (T1-W)images (256 � 256 � 176 matrix size, 1 � 1 � 1 mm voxelsize) were also obtained for registration purposes. Eddycurrent distortion correction (13) was performed on eachset of HARDI images prior to further analysis.

The b � 0 images in each HARDI dataset have higherimage contrast than the individual DW images; therefore,they were used in the image normalization process and theresulting transformation was then applied to the DW im-ages for each diffusion direction. Image normalization foreach experiment was performed in a series of steps, in-volving registration of the b � 0 images to their corre-sponding anatomical image volumes and then registrationof the anatomical volumes to a common image space. Bothlinear (14) and nonlinear (15) registration algorithms wereused. The process for each experiment is described below.

For the intrasubject experiment with three head posi-tions, one head position was designated as the target towhich the remaining positions (denoted as source images)were registered through the following steps. First, the b �0 images for each head position were registered to thecorresponding high-resolution T2-W volume by a nonlin-ear transformation, initialized using parameters from rigidregistration between the two image sets. Then a similartwo-step transformation was carried out between the T2-Wimages of each source and the target. Finally, the twotransformations between the source b � 0 images and thesource T2-W images, the two transformations between thesource T2-W images and the target T2-W volume, and theinverse of the two transformations between the target b �0 images and the target T2-W volume were combined toform the total transformation between the source b � 0images to the target b � 0 images. The b � 0 images fromthe two head positions in the second intrasubject experi-ment were coregistered in the same manner, where oneposition was selected as the target, and the other positionwas the source image.

The data collected for the intersubject experiment wereoriginally coregistered for use in a voxel-based analysis of

FIG. 2. Simulated intravoxel fiber structures (a) and the correspond-ing FODs (b). a: From top to bottom the crossing angle betweenfibers increases from 0° to 30°, 60°, and 90°. The left column givesthe original fiber structure. The middle and right columns illustratethe fiber structures after a horizontal stretch and a vertical shear,respectively (denoted by the black arrows). b: Corresponding FODswith the true fiber orientations plotted in the solid lines.

1522 Hong et al.

fractional anisotropy (FA) measures derived from a diffu-sion tensor analysis of the HARDI data. The normalizationprocess consisted of two main steps: creation of a study-specific FA template and coregistration of the individualFA maps to the FA template. This was done in an attemptto minimize potential bias in the normalization results dueto selecting a single subject as the target image. The study-specific template was created by a set of three registrationsteps. First, nonlinear registration, initialized by rigid reg-istration, between the b � 0 images of each subject and thecorresponding high-resolution, slice-matched T2-W im-ages was performed to reduce image distortion due tosusceptibility artifacts. Second, rigid registration betweenthe T2-W images and T1-W images of each subject wasperformed. Third, nonlinear registration, initialized byrigid registration, between the T1-W images of each subjectand the T1-W image volume of a target subject (chosenfrom the control group) was performed. The resultingtransformations were combined to create a single transfor-mation for each subject, which was applied to the subject’sFA map. The normalized FA maps from all subjects wereaveraged to create the study-specific template. Finally,each subject’s original FA map was then registered to theFA template through both rigid registration with scalingand nonlinear registration. The resulting transformationswere combined to create the transformation applied to theDW images in this study. Compared to simpler schemes,this multistep approach was found to provide more robustintersubject registration.

In each experiment the total transformation was appliedto the DW images for each diffusion direction. Based onthe transformed HARDI images, FODs were calculated us-ing the FORECAST spherical deconvolution methodthrough 6th order. To reduce the effects of noise, the FODswere regularized by minimizing the negative values (16)with a fixed weighting factor (� � 0.3, which was chosento optimize reproducibility based on Monte Carlo simula-

tions). The Jacobian matrix for each voxel was calculatedbased on the deformation field of the total transformationand was used to transform the FODs as described above.For the intrasubject experiments the FODs in the targetimage, and in the transformed source image before andafter adjusting the FOD orientation and shape, were com-pared. The similarity between two FODs was evaluated bythe angular correlation coefficient (ACC) (11) and the rootmean square (RMS) error of the point-wise FOD values.For the intersubject experiment, the transformed FODsfrom different subjects were compared. The mean andstandard deviation of the transformed FODs over the con-trol group (excluding the target subject) were derived, andthe ACC and RMS error for each subject relative to thegroup mean were also calculated.

RESULTS

Simulation results demonstrate the ability of the proposedmethod to handle nonrigid transformations in spatial nor-malization which cannot be fully accounted for by a sim-ple rotation. The transformed FOD is able to provide reli-able estimates of the transformed fiber orientation whenthe number of sampling points is sufficient. In our test, 92sampling points gives poor performance in terms of highangular error and large deviation from the unit integral.When the number increases to 252 and above, the angularerror drops to acceptable levels (for example, a mean errorof 5° in the horizontal stretch of the 60°-crossing case), andthe unit integral is preserved. There is no significant dif-ference in the angular accuracy achieved when the numberof sampling points increases beyond 252. Parts of thesimulation results are shown in Fig. 2.

An example of the intrasubject transformation is shownin Fig. 3. It is obvious that the FODs derived from thetransformed HARDI images maintain their orientations inthe original source image, and do not agree with the fiber

FIG. 3. Example of intrasubject nor-malization of data acquired after in-plane head rotation. The FA is shown inone slice of the target (a), source (b),and transformed source (c) datasets.FODs in the ROI (highlighted in the yel-low box) overlaid on the FA map in thetarget image (d), in the transformed im-age before (e), and after (f) FOD trans-formation. ACC between source andtarget FODs before (g) and after (h)transformation. RMS error with respectto the target FODs for the source FODbefore (i) and after (j) transformation.Note that a white matter mask wasapplied (FA � 0.25). The size of eachFOD is scaled by the corresponding FAvalue.

Spatial Normalization of FOD 1523

tracts in the target image. After transformation, the FODsindicate the correct orientation of the corpus callosum andthe cingulum bundle, and the similarity to the correspond-ing target FODs becomes higher. The mean ACC across allthe white matter voxels in the slice is raised from 0.53before transformation to 0.70 after transformation. Themean RMS error is lowered from 0.26 to 0.20. Figure 4shows an example of different fiber tracts (the right inter-nal capsule) and rotation axis from the other intrasubjecttransformation, where the mean ACC increases from 0.56before transformation to 0.78 after transformation and themean RMS error decreases from 0.25 to 0.17.

The transformation results between subjects are demon-strated in Figs. 5 and 6. Figure 5 compares the FODs alongpart of the left cingulum from two subjects before and aftertransformation. It is clear that the FODs derived from thetransformed DW images without adjustment still take theorientation of the fiber before registration, not the trans-formed fiber. After transformation, the FODs are moreconsistent with the transformed cingulum bundle.

Figure 6 shows the mean and standard deviation of thetransformed FODs over the control group of 19 subjects.The region of interest (ROI) was chosen to include voxelscontaining single fiber and multiple fiber orientations. Thevariation of the transformed FODs across subjects is small,but relatively higher at the boundary between white matterand cerebrospinal fluid, or the boundary between twodistinct fiber bundles.

DISCUSSION

Spatial normalization of brain MR images, especially thosebetween subjects, usually involves scaling and shearingdue to anatomic differences between individual brains.These factors, in addition to the relatively simple case ofrotation, may change not only the orientation, but also the

FIG. 4. Example of intrasubject nor-malization of data acquired afterthrough-plane head rotation. The FAis shown in one coronal slice of thetarget (a), source (b), and trans-formed (c) image. Note that imageswere acquired axially. FODs in theROI (highlighted in the yellow box)overlaid on the FA map in the targetimage (d) and in the source imagebefore (e) and after (f) FOD transfor-mation. ACC between source andtarget FODs before (g) and after (h)transformation. RMS error with re-spect to the target FODs for thesource FODs before (i) and after (j)transformation. Note that a whitematter mask was applied (FA � 0.25).The size of each FOD is scaled by thecorresponding FA value.

FIG. 5. Example of intersubject normalization. a–d: Results for sub-ject A. FA map in the native (a) and common (b) space, withcorresponding ROI highlighted in the yellow box. FODs in the ROIbefore (c) and after (d) transformation, overlaid on the FA map. e–h:Results from subject B in the same ROI: native (e) and common (f)space FA maps and ROI FODs before (g) and after (h) transforma-tion. The similarity between subject A and B FODs in the ROI isshown in ACC (i) and RMS deviation (j) maps. The size of each FODis scaled by the corresponding FA value.

1524 Hong et al.

overall shape of the fiber orientation distribution within animaging voxel. Therefore, it is necessary to take into ac-count scaling and shearing effects in order to obtain accu-rate transformed FODs. This is demonstrated by the nu-merical simulations and the high similarity achieved inour between-subject normalization results.

Recently, Chiang et al. (17) proposed a fluid registrationmethod for HARDI data, in which the principal directionof the diffusivity function is estimated using principalcomponent analysis (PCA), and its reorientation is per-formed using a rotation derived from the PPD algorithm.This method is limited in that the principal direction ofthe diffusivity function is not able to fully describe multi-ple fiber orientations within a voxel. Also, a rotation is notsufficient to represent the nonrigid transformations in-volved in the normalization process. By contrast, the algo-rithm proposed in our study not only takes advantage ofHARDI data by working on the FOD functions, but alsosuccessfully addresses the problem of nonrigid deforma-tion.

Although the normalization algorithm was demon-strated using the FOD derived from the FORECAST model,the method is applicable to FOD functions derived fromother spherical deconvolution approaches as well. Simi-larly, the orientation distribution function (ODF) derivedfrom diffusion spectrum imaging (18) or q-ball imaging(19) gives the probability that spin displacement lies in aparticular direction. This is also an angular distributionfunction on the unit sphere that can be transformed ac-cording to the proposed method.

For simple spatial normalization where only rigid rota-tion is involved, the reorientation of the FOD function can

be achieved in two ways: calculating the FOD based onrotated diffusion gradient directions, or simply rotatingthe FOD calculated using the original gradient directions.These two are equivalent because the relationship betweenthe FOD and measured DW signal is linear and shift-invariant, where shifts on a sphere are equivalent to rota-tions. For nonrigid spatial transformation, however, thefirst approach does not work even if the original deconvo-lution kernel is used (results not shown). This is becausenonrigid transformations are not shift-invariant on the unitsphere and therefore do not preserve the convolution re-lationship between the FOD and measured signal. Simi-larly, transforming the diffusion gradient directions doesnot give the correctly transformed ODF of q-ball imaging,since nonrigid transformation does not preserve Funk–Radon transform relationships.

Since the FOD function is expressed in terms of spher-ical harmonic (SH) expansion coefficients, the highest ex-pansion order determines its angular resolution. One fac-tor that will possibly affect the accuracy of the transformedFOD is the SH expansion order. Due to the scaling andshearing effects possibly included in the image registrationprocess, especially for intersubject cases, the transformedFOD may take a different shape (for example, a narrowerpeak), which may require higher-order SH expansion inorder to be fully described. Insufficient SH order may leadto error in both the shape and orientation of the resultingFOD. In this case, expressing the FOD through a higherorder may be helpful. Take the example in the simulationstudy where a voxel with two fibers crossing at 60° under-goes a stretch (see Fig. 2b, row 3 column 2) and results ina smaller crossing angle. The transformed FOD fitted

FIG. 6. Example of group datatransformed to the commonspace. a: The mean (opaque) andmean � SD (transparent) FODsover a group of 19 subjects nor-malized to a common space. Theinset shows the ROI (yellow box)in the averaged FA map and en-larged FODs from five represen-tative voxels highlighted in col-ored boxes. The mean (b) and SD(c) ACC maps in the ROI werecalculated across all subjects (rel-ative to the group mean FOD).The mean (d) and SD (e) RMSerror maps are also shown. Notethat in all cases a white mattermask was applied (averagedFA � 0.25).

Spatial Normalization of FOD 1525

through 6th order gives a mean angular error of 4.6°, whichis reduced to 1.5° when fitted through 8th order. Note thata higher SH order can provide higher angular resolution atthe cost of introducing high-frequency noise (which mayalso affect the accuracy of FODs), somewhat longer com-putational time, and larger data storage space. The optimalSH order depends on both the original fiber configurationand the nature of the spatial normalization. Here in our invivo data, we chose to use the same SH order for both thetarget and transformed FODs in order to facilitate the com-parison between them.

The number of sampling points is another factor thatmay affect the accuracy of the transformation algorithm.According to the simulations, 252 sampling points evenlydistributed over a unit sphere are able to achieve satisfac-tory results. A higher number improves the performanceslightly, but at the cost of longer computation time.

Although the transformation is successful in most vox-els (high similarity between the transformed source andthe target FODs), there are some regions where the trans-formation is less accurate. One example is the small lateralregion in the left hemisphere (right side of the image) inFig. 3 (see part j, for example), where the ACC between thetransformed source and target FODs is relatively low andthe RMS error is relatively high. The low similarity stemsfrom shape and orientation differences between the FODs:the target FODs clearly show two fiber components withdifferent orientations, while most of the original and trans-formed source FODs contain only one fiber along the meanorientation of the two fibers in the target image (details notshown here). Generally speaking, the discrepancy couldresult from three possible causes. It may be due to any oneof them, or more likely, a combination of some or all ofthem. First, FODs from either the target or the sourceimage or both might not be reliable due to limitations ofthe FORECAST model. This model assumes a single radialdiffusivity for all fibers within a voxel (i.e., a single kernelfor spherical deconvolution within a voxel) (11). Violationof this assumption may cause errors in the estimated fibervolume fractions. Based on simulations, the error could beabout 25% at the current signal-to-noise ratio (SNR) levelof 30�50 under the imaging protocols used in this study(results not shown). Also, errors in the estimated FOD maybe due to noise or image artifacts, which were not com-pletely removed by the eddy current distortion correctionand the regularization process. Since interpolation is in-volved in the image registration process, the transformedimage is in fact spatially smoothed, resulting in a higherSNR level (and partial volume averaging) than the targetimage, and hence likely requires a smaller regularizationweighting factor. Even though the regularization weightingfactor was chosen carefully based on Monte Carlo simula-tions and the chosen value is considered proper for theSNR range here, the fixed value might still underregularizethe target FODs and overregularize the transformed FODs.Alternative methods were tested to objectively determinethe regularization weighting factor for each voxel, such asthe generalized cross-validation (GCV) (20) and the L-curve method (21). Both methods gave discrepant resultsbetween the target and the transformed source, similar tothose using the fixed factor, and none of them gave betteroverall regularization than the fixed value. A better FOD

regularization method will be helpful in the validation ofthe transformation technique. However, the problem ofregularization is beyond the scope of this study.

Second, the discrepancy may come from local imageregistration errors. Registration based on FA maps (ratherthan b � 0 images) was tested, but this gave similar results.Note that the region in question is at the edge of the brain,containing complex gyral and sulcul structures, whichmay vary drastically between subjects. This presents amajor challenge for accurate registration, even with thesophisticated nonlinear registration method used in thisstudy. The Adaptive Bases Algorithm (15) uses regularlyspaced radially symmetric basis functions to model thedeformation field and works on a multiresolution scale,allowing fine adjustments within local regions. The initial-ization parameters for the algorithm (a total of 14 levelswith number of basis functions increasing from 3 to 40along each dimension) were selected based on prior expe-rience with the algorithm to provide the best possiblematch throughout the brain without introducing registra-tion artifacts such as tearing and folding. Close inspectionof the corresponding FODs in the target and source imagesand the neighborhood indicates that this is likely onecause of the FOD discrepancy in this region since trans-formed FODs with high similarity to the correspondingtarget FODs can be found in the near neighborhood (about1�2 voxels away, details not shown). More advanced im-age registration methods may improve the results of ourmethod in the future.

Third, the discrepancy may be due to sampling differ-ences between the two acquisitions. For example, considerthe possible sandwich-like topology of fiber mixing, whichis not positioned exactly in the same way in the target andsource image voxels. Suppose one fiber bundle lies inferiorto another with different orientation and the mixing area isthin relative to the slice thickness. One axial slice of thetarget image is centered exactly between the two bundlesand thus the FODs fully capture the fiber mixing, while inthe source image two neighboring slices happen to lie justsuperior and inferior to the mixing plane, and thus theFODs in each slice reveal just one of the fiber bundles (asillustrated in Fig. 7). In this case, even if the FODs givereliable orientations of the fiber tracts and the image reg-istration is accurate, the transformation algorithm cannotprovide satisfying results since the source and target FODscontain different fiber populations. Inspection of theneighboring slices indicates that this is likely anothercause of the FOD discrepancy in this region (data notshown).

CONCLUSION

We developed an algorithm to perform spatial normaliza-tion of the FOD function derived from HARDI data, whichmakes it possible to compare the intravoxel fiber distribu-tion between subjects, between groups, or between twoimage sets acquired at different timepoints for the samesubject. This is an important step forward from spatialnormalization of scalar images and diffusion tensors, sincethe FOD function derived from HARDI provides moredetailed information about intravoxel fiber structure thanscalar measures such as FA, and the diffusion tensor. This

1526 Hong et al.

technique will likely be helpful in clinical studies thatmake use of HARDI data to assess white matter disease.

ACKNOWLEDGMENT

The authors thank Dr. Xia Li for helpful discussions onimage registration.

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FIG. 7. Illustration of the possible FOD sampling differences. Leftcolumn: FODs in the target image where the voxel in the middle slicecaptures the fiber crossing. Right column: FODs in the source imagewhere the neighboring slices miss the fiber crossing.

Spatial Normalization of FOD 1527