Medical Image Analysis 18 (2014) 953–962

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Medical Image Analysis

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Automated correction of improperly rotated diffusion gradientorientations in diffusion weighted MRI

http://dx.doi.org/10.1016/j.media.2014.05.0121361-8415/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Address: Vision Lab, University of Antwerp (CDE),Universiteitsplein 1, N.1.18, B-2610 Wilrijk, Belgium. Tel.: +32 3 265 24 77; fax:+32 3 265 22 45.

E-mail address: ben.jeurissen@ua.ac.be (B. Jeurissen).

Ben Jeurissen a,⇑, Alexander Leemans b, Jan Sijbers a

a iMinds-Vision Lab, Department of Physics, University of Antwerp, Belgiumb Image Sciences Institute, University Medical Center Utrecht, The Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 December 2013Received in revised form 20 May 2014Accepted 27 May 2014Available online 6 June 2014

Keywords:Diffusion-weighted MRICoordinate frameFiber tractographyDiffusion gradient reorientationStochastic approximation

Ensuring one is using the correct gradient orientations in a diffusion MRI study can be a challenging task.As different scanners, file formats and processing tools use different coordinate frame conventions, inpractice, users can end up with improperly oriented gradient orientations. Using such wrongly orientedgradient orientations for subsequent diffusion parameter estimation will invalidate all rotationally vari-ant parameters and fiber tractography results. While large misalignments can be detected by visualinspection, small rotations of the gradient table (e.g. due to angulation of the acquisition plane), are muchmore difficult to detect. In this work, we propose an automated method to align the coordinate frame ofthe gradient orientations with that of the corresponding diffusion weighted images, using a metric basedon whole brain fiber tractography. By transforming the gradient table and measuring the average fibertrajectory length, we search for the transformation that results in the best global ‘connectivity’. To ensurea fast calculation of the metric we included a range of algorithmic optimizations in our tractography rou-tine. To make the optimization routine robust to spurious local maxima, we use a stochastic optimizationroutine that selects a random set of seed points on each evaluation. Using simulations, we show that ourmethod can recover the correct gradient orientations with high accuracy and precision. In addition, wedemonstrate that our technique can successfully recover rotated gradient tables on a wide range ofclinically realistic data sets. As such, our method provides a practical and robust solution to an often over-looked pitfall in the processing of diffusion MRI.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Diffusion-weighted (DW) MRI is a non-invasive imaging tech-nique that is sensitive to the random microscopic motion of watermolecules (Stejskal and Tanner, 1965). In the brain white matter(WM), which consists of tightly packed bundles of neuronal axons,the regular arrangement of fibers introduces a directional depen-dence of this motion (Moseley et al., 1990). The degree of thisdirectionality provides an indirect measure for the organizationof the underlying tissue, while the preferred diffusion orientationprovides an estimate of the local fiber orientation of the bundles(Basser, 1995). These local fiber orientation estimates can bepieced together to infer the long-range pathways connectingdistant regions of the brain, using a technique called WM fibertractography (Conturo et al., 1999; Mori et al., 1999). DW-MRI is

currently the only imaging technique that allows assessment ofWM organization and connectivity in vivo and it is increasinglyused in clinical studies to investigate brain WM, and its develop-ment and disorders (Johansen-Berg and Behrens, 2006; Mori andZhang, 2006; Assaf and Pasternak, 2008).

Diffusion metrics extracted from DW images can be categorizedinto two categories based on their rotational (in) variance. Rotation-ally invariant diffusion metrics embody intrinsic features of thelocal diffusion pattern and are independent of the orientation ofthe anisotropic structure, the orientation of the specimen withrespect to the laboratory coordinate system, and even the choiceof the laboratory coordinate system (Basser and Özarslan, 2010).Notable examples are the scalar diffusion parameters obtained fromdiffusion tensor imaging (DTI) such as the widely used fractionalanisotropy (FA) and mean diffusivity (MD) (Basser and Pierpaoli,1996). Both parameters are extracted from the eigenvalues of thediffusion tensor, making them rotationally invariant by definition.

Rotationally variant diffusion parameters, on the other hand,embody orientational features of the local diffusion pattern andare dependent of the orientation of the anisotropic structure, the

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orientation of the specimen with respect to the laboratory coordi-nate system, and the choice of the laboratory coordinate system.Notable examples are the principal diffusion vector (PDV)estimated by DTI, diffusion/fiber orientation distribution functionsobtained from high angular resolution diffusion imaging (HARDI)(Tournier et al., 2011) and fiber tractography results.

For rotationally variant parameters to be meaningful, it isimportant that the coordinate system of the diffusion sensitizinggradient orientations and the spatial coordinate system of the vox-els in the diffusion weighted data set coincide. In theory, the metadata generated by MRI scanner software should provide all thenecessary information to properly align the gradient orientationswith the subject coordinate frame. In practice, however, thisprocedure is error-prone, as the appropriate steps are dependenton the coordinate system conventions of the scanner software,the file format and even the post-processing tool. For example,while both MRtrix (Tournier et al., 2012) and FSL (Jenkinsonet al., 2012) support DWI data in the NifTI format and while bothrepresent the gradient vectors as simple text files, MRtrix assumesthat the gradient vectors are provided with respect to real-world/scanner coordinates, whereas FSL assumes that the gradient vec-tors are provided with respect to the image axes.

While large misalignments (e.g. flipping or permutations of thegradient orientation coordinate axes) can be detected by visualinspection of PDV maps or tractography results, small rotationsof the gradient table (e.g. due to angulation of the acquisitionplane), are much more difficult to detect. Nevertheless, even smallrotations have a large impact on tractography results due to anaccumulation of errors.

In this work, we propose a data-driven method that does notrely on header information to automatically detect and correctfor the rigid misalignment of the gradient table and the DW data,using a metric based on whole brain tractography.

2. Methods

The use of incorrect gradient orientations will result in the pre-mature termination of fiber trajectories, which can be measuredusing whole brain fiber tractography. By transforming the gradienttable and measuring the average fiber trajectory length, one cansearch for the transformation that results in the best global ‘con-nectivity’. Finding the correct DW gradient settings then becomesakin to a registration problem (Fig. 1). The gradient orientations(moving entity) are subjected to a 3D rotation (transformation),such that they are aligned with the DW images (reference entity).

2.1. Transform

The 3D rotation R of the gradient table can be parameterized assuccessive rotations around the main orthogonal axes:

Fig. 1. Schematic overview of the gra

RðhÞ ¼ RxðhxÞRyðhyÞRzðhzÞ ð1Þ

resulting in a parameter vector h ¼ ðhx; hy; hzÞ with 3 transformationparameters. Note that, since the gradient table is rotated as a whole,this is a rigid body transformation. Also note that translations donot have to be taken into account, as they do not affect the gradientorientations.

2.2. Metric

The metric, which provides a measure of how well the gradientorientations match the DW images, is defined as the average tra-jectory length f ðh;nÞ, measured with whole-brain streamline DTItractography, where h parametrizes the 3D rotation of the gradienttable and n is the number of seed points, evenly distributed overthe whole brain. As fiber tractography is more reliable in structureswith high FA, the length contribution of each step is weighted bythe local FA value. To ensure robust fiber tractography even on dataof low quality, the DWI images are smoothed with a 3D Gaussianfilter prior to tractography.

The metric exploits the fact that the brain is a complicated 3Dnetwork of fiber bundles. While rotating the gradient orientationsaway from the true orientations may result in a few slightly longerfalse trajectories in some locations, it will at the same time reducethe length of many true trajectories.

Since finding the optimal gradient table requires many repeatedevaluations of the metric, it is imperative that the computation off ðh;nÞ is fast. However, as whole-brain tractography is relativelytime-consuming, several optimizations were implemented tospeed up the standard streamline DTI tractography algorithm. Tra-ditionally, DTI tractography is performed as follows (Basser et al.,2000). First, the DW signal at the current position of the trajectoryis obtained using trilinear interpolation. Next, the diffusion tensoris estimated from the interpolated signal. Then an eigenvaluedecomposition is performed on the diffusion tensor and the eigen-vector corresponding to the largest eigenvalue is assumed to be thecurrent fiber orientation. Finally, the trajectory is advanced by afixed step size along this orientation and the whole process isrepeated. The most time consuming operations in this algorithmare the interpolation, the diffusion tensor fit and the eigenvaluedecomposition.

The computational complexity of the DTI tractographyalgorithm can be reduced significantly using the Log Euclidianframework (Arsigny et al., 2006), as follows. Prior to tractography,the log diffusion tensor field is calculated for the whole brain. Then,the log diffusion tensor at the current position of the trajectory canbe obtained using trilinear interpolation directly on the log diffu-sion tensor field. Next, an eigenvalue decomposition is performedon the diffusion tensor and the eigenvector corresponding to thelargest eigenvalue is assumed to be the current fiber orientation.Finally, the trajectory is advanced by a fixed step size along this

dient table reorientation method.

Fig. 2. Cost function values f for a phantom data set with correctly orientedgradient table and SNR = 10 (defined on the b ¼ 0 s=mm2 images) as a function of hx

for selected values of n (a). A zoomed-in plot near the optimum is provided in (b).

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orientation and the whole process is repeated. Following thisapproach, only a single diffusion tensor fit is required and the num-ber of interpolations at each tractography step is limited to 6 ascompared to the number of diffusion weighted images for the tra-ditional method, greatly reducing computation time and memoryconsumption.

Instead of rotating the gradient table and recalculating the logdiffusion tensor at each metric evaluation, our algorithm directlyrotates the log tensor, requiring only a single diffusion tensor fitfor repeated evaluations of the metric. To avoid unnecessary calcu-lations outside the brain, a rough skull stripping was performed onthe b ¼ 0 s=mm2 images, separating brain from background.

The only remaining bottleneck is the eigenvalue decompositionwhich needs to be performed at each tractography step. To reducethe computational load of this operation, the analytical eigenvaluedecomposition proposed in Hasan et al. (2001) was used, whichexploits the fact that the diffusion tensor is a 3 � 3 symmetricand semi-positive-definite matrix, and which is approximately 40times faster than standard iterative decomposition techniques.

Finally, to reduce computation time even further, whole braintractography is not performed from all N voxels of the whole brain,but from a set of n < N seed points distributed over the wholebrain, allowing a trade-off between metric calculation speed andmetric precision.

For reasons outlined below, we also define a stochastic costfunction (Robbins and Monro, 1951) ~f ðh;nÞ which is the approxi-mate average tract length resulting from whole brain tractographywith rotation angles h and n the number of seed points randomlydistributed over the whole brain according to a uniform distribu-tion. As a consequence, each time ~f ðh;nÞ is called, it probes a differ-ent set of seed points, resulting in a different metric outcome.

2.3. Optimization

We provide two optimization approaches, depending on whattype of gradient corruptions are present in the data.

The first approach assumes that the gradient table is onlycorrupted by means of column order permutations and/or columnsign changes, a situation that often occurs in practice, due to differ-ent coordinate system conventions. The number of possible col-umn order permutations is 3! ¼ 6, while the number of possiblesign changes is 4 (sign change in the first column, the second col-umn, the third column and no sign change), resulting in 6� 4 ¼ 24possible solutions in total. This optimization problem can be solvedin a brute force fashion by exhaustively evaluating the metricf ðh;nÞ. From this list of evaluations fi (with i ¼ 1; . . . ;24) we selectthe permutation and/or sign change that results in the longestaverage fiber length. To avoid selecting false solutions in the caseof insufficient signal-to-noise ratio (SNR), instead of just evaluatingf ðh;nÞ, we repeatedly evaluate ~f ðh;nÞ and the solution is onlydeemed optimal if it is consistently selected as having the longestaverage fiber orientation across the stochastic function evalua-tions. If the same solution is not found consistently across stochas-tic function evaluations, the method reports that the solution isundecided due to insufficient SNR. In that case, the method is rerunwith twice the amount of seed points n, increasing the metric pre-cision. This way, the method dynamically adjusts the number ofseed points required for precise metric evaluation to the qualityof the data and avoids false solutions.

The second approach assumes that the gradient table is cor-rupted by means of a general 3D transformation. Note that, dueto the antipodal symmetry of the diffusion signal, this approachalso resolves the column order permutations and column signchanges resolved with the first approach. Since a reliableevaluation of the metric requires a large number of seed pointsn, repeated evaluation has a very high computational cost. In

addition, even with high n, the metric still exhibits spurious max-ima, which would cause deterministic optimization routines toreport suboptimal solutions (Fig. 2).

In order to keep the computation time of the metric to a mini-mum and allow accurate estimation of the rotation parameters, wechose to use the stochastic gradient ascent (Robbins and Monro,1951) which has previously been used successfully for the purposeof image registration (Klein et al., 2007). The stochastic gradientascent method follows the same scheme as the deterministic gra-dient ascent, with the distinction that the derivative of the costfunction is replaced by a fast stochastic approximation. By usinga new, randomly selected subset of seed points every iteration ofthe optimization process, a bias in the approximation error isavoided. More specifically, in order to find the optimal set of rota-tion angles h, a stochastic gradient ascent is defined as follows:

hkþ1 ¼ hk þ ckr~f ðhk;nÞ with k ¼ 0; . . . ;K ð2Þ

with ck the scalar gain factor or ‘‘step size’’ and ~f ðhk;nÞ the approx-imate average tract length resulting from whole brain tractographywith rotation angles hk and n randomly distributed seed points. Theapproximated gradient does not necessarily vanish close to thesolution h, in contrast to the exact derivative that satisfiesrf ðhÞ ¼ 0. Thus, convergence must be forced by ensuring ck ! 0as k! K . In this work, K ¼ 150 and ck � cðkÞ ¼ 100e�0:068k. Usingthe stochastic gradient ascent makes the optimization routinerobust to spurious maxima, and also allows the number of seedpoints n to be small, significantly speeding up the metric evaluation(Klein et al., 2007). Unless stated otherwise, n ¼ 200 seed pointsrandomly distributed over the whole brain, were used in all subse-quent experiments. While a small number of seed points results in~f ðhk;nÞ having low precision, the repeated evaluations of ~f ðhk;nÞwillconverge to the true optimum. Since the gradient ascent can onlyfind the local minimum, we use a multi-start approach startingfrom 30 points distributed evenly in the space of 3D rotations.

Fig. 3. Illustration of the image quality at different simulated SNR levels, for a single DW image and for the resulting DEC FA map. Note that SNR is defined on theb ¼ 0 s=mm2

Fig. 4. Average metric values (over repeated stochastic metric evaluations) for the different combinations of gradient table column order permutations and sign changes, fordifferent SNRs (single realization). The correct answer is the top left cell, which corresponds to no permutation and no sign changes.

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2.4. Simulations

To evaluate the proposed method quantitatively, a noiselessfull-brain DWI data set consisting of 6 b ¼ 0 s=mm2 and 60b ¼ 1200 s=mm2 images and with a voxel size of2:4� 2:4� 2:4 mm3 was simulated (Leemans et al., 2005). Fromthis data set, noisy Rician distributed data sets were obtained for

different noise levels, with SNR in the b ¼ 0 s=mm2 image rangingfrom 1 to 30 and 100 noisy realizations for each noise level(Gudbjartsson and Patz, 1995).

2.4.1. Gradient column order permutations and sign changesFor each noisy realization, we checked whether the method

could distinguish the correct column order permutation and sign

Fig. 5. Percentage of correct (green), no (blue) and wrong (red) solutions as afunction of SNR, over 200 noise realizations. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. The number of seed points required by the method to restore the gradienttable as a function of SNR, over 200 noise realizations.

Fig. 7. The recovered gradient orientations (colored dots) vs the true gradient orientationgradient orientations are given the same color to aid visual clustering. (For interpretatioversion of this article.)

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changes from the wrong ones. We also investigated the amount ofseed points required in the metric to make this distinction.

2.4.2. General 3D rotationsFor each noisy realization, the gradient table was corrupted by a

random 3D rotation. After correction with the proposed method,quality of the corrected gradient table was assessed with respectto accuracy and precision as follows. For each gradient orientation,an average gradient orientation was obtained from the 100 cor-rected orientations by extracting the first eigenvector of the meandyadic tensor (Basser and Pajevic, 2000) and the minimum anglessubtended between each estimated gradient orientation and theaverage gradient orientation were recorded. A measure of accuracywas obtained by calculating the angular bias of the average gradi-ent orientation compared to the ground truth gradient orientation(Basser and Pajevic, 2000). A measure of precision was obtained bycalculating the 95% confidence interval (CI) of the one-tailed distri-bution of minimum angles between the average gradient orienta-tion and the individual gradient orientations (Jones, 2003).

2.5. Experiments with in vivo human brain data

The method was also tested on a set of real human brain DWIdata sets with different characteristics:

� Data set A is a DTI data set acquired on a 3T Siemens Trioscanner, using a 12-channel receiver head coil. Diffusionweighting was applied along 30 gradient directions withb ¼ 1000 s=mm2. Additionally, 2 non-DW images wereacquired. Other imaging parameters were: TR/TE: 8100/116 ms; voxel size: 2:5� 2:5� 2:5 mm3; matrix: 96� 96;slices: 54; NEX: 1.� Data set B is a HARDI data set acquired on a 3T GE HDx Signa

scanner, using an 8-channel receiver head coil and cardiac gat-ing (Jeurissen et al., 2013). Diffusion weighting was applied

s (black crosses) for 100 noisy realizations of the data set. Corresponding correctedn of the references to color in this figure legend, the reader is referred to the web

Fig. 8. Accuracy (a) and precision (b) of the recovered gradient tables as a functionof SNR. The dashed red lines indicate the points at which sub-degree accuracy (a)and sub-degree precision (b) are achieved. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

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along 60 gradient directions with b ¼ 1200 s=mm2. Addition-ally, 6 non-DW images were acquired. Other imaging parame-ters were: TR: 20 R-R intervals; TE: 86 ms; voxel size:2:4� 2:4� 2:4 mm3; matrix: 96� 96; slices: 60; NEX: 1.� Data set C is a HARDI data set with high spatial resolution

acquired on a 3T Philips Achieva scanner, using an 8-channelreceiver head coil (Tax et al., 2014). Diffusion weighting wasapplied along 60 gradient directions with b ¼ 2500 s=mm2.Additionally, 1 non-DW image was acquired. Other imagingparameters were: TR/TE: 10265/107 ms; voxel size:2� 2� 2 mm3; matrix: 112� 112; slices: 70; NEX: 1.� Data set D is a multi-shell DW data set acquired with high

spatial resolution on a 3T GE Discovery MR750 scanner, usingan 8-channel receiver head coil. Diffusion weightings of

Fig. 9. Example of the gradient correction on in vivo data set A with a gradient table colu(a), after gradient column permutation and sign correction (b), after the full correction

b ¼ 0;700;1200 and 2800 s=mm2 were applied in 5, 25, 45and 75 directions, respectively. Other imaging parameterswere: TR/TE: 9500/100 ms; voxel size: 2� 2� 2 mm3; matrix:120 � 120; slices: 68; NEX: 1.

We deliberately corrupted the gradient tables of these data setsusing a random column permutation, a random column signchange, a small random 3D rotation and a large random 3D rota-tion. Subsequently, the corrupted data sets were processed usingour correction method, to test whether it could correctly restorethe gradient orientations.

3. Results

3.1. Simulations

Image quality for the different simulated SNR levels can beappreciated from Fig. 3. Note that SNR is defined on theb ¼ 0 s=mm2 image and that for realistic clinical studies it is inthe range of [10–30]. However, to ensure proper functioning ofour method even on low quality data, we also simulated SNRvalues well below 10.

3.1.1. Gradient column order permutations and sign changesFig. 4 shows the results of the exhaustive evaluation of the

different combinations of gradient table column order permuta-tions (rows) and sign changes (columns), for a single realizationof different SNR levels. The correct answer is the top left cell, whichcorresponds to no permutation and no sign changes. Note that onecan clearly distinguish the correct solution up till SNR = 5. How-ever, at SNR = 2, the method is still able to repeatedly recognizethe first solution as the optimum, over repeated stochastic evalua-tions of the metric f. At SNR = 1, no solution stands out from therest and the method reports no answer.

Fig. 5 shows the percentage of correct, no, and wrong solutionsas a function of SNR. Note that the method achieves a 100% successrate all the way up to SNR ¼ 2. Starting from SNR ¼ 1, the methodreports that it can no longer reliably estimate the correct gradientorientations in all cases. Note that the method makes no falseclaims about the correct gradient orientations, even at low SNR.

Fig. 6 shows the number of seed points required by the methodto restore the gradient table as a function of SNR. Note that 50 seedpoints were sufficient all the way down to SNR = 4. Droppingbelow SNR = 4, the method gradually requires more seed points.

mn permutation. FA, DEC and a whole brain tractogram are shown before correction(c) and for the original data set (d).

Fig. 10. Example of the gradient correction on in vivo data set B with a gradient table column sign change. FA, DEC and a whole brain tractogram are shown before correction(a), after gradient column permutation and sign correction (b), after the full correction (c) and for the original data set (d).

Fig. 11. Example of the gradient correction on in vivo data set C with a gradient table subject to a large 3D rotation. FA, DEC and a whole brain tractogram are shown beforecorrection (a), after gradient column permutation and sign correction (b), after the full correction (c) and for the original data set (d).

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At SNR = 2, the method requires at most 200 seed points, which isstill reasonable.

3.1.2. General 3D rotationsFig. 7 shows the gradient orientations recovered with our

method after random 3D rotations for 100 noisy realizations ofthe data set, together with the true gradient orientations. ForSNR values all the way down to 5, the corrected gradient orienta-tions are tightly clustered around the true gradient orientations,with no discernible bias. For SNR = 2, precision of the correctedgradient orientations is markedly decreased, but the corrected gra-dient orientations are still clustered around the true orientations,suggesting only a small bias. Note that an SNR as low as 2 in theb ¼ 0 s=mm2 images, is extremely low and does no longer repre-sent realistic clinical data.

Fig. 8 quantifies both accuracy and precision as a function ofSNR. Sub-degree accuracy is obtained starting from SNR = 2. Sub-degree precision is obtained, starting from SNR = 4. This is wellwithin the boundaries of clinically realistic SNR.

3.2. Experiments with in vivo human brain data

3.2.1. Data set AFig. 9 shows an FA map, a DEC (directionally encoded color) FA

map (Pajevic and Pierpaoli, 1999) and a whole brain tractogram fordata set A, after corruption with a column order permutation (a),

after correcting for column order permutations and column signchanges (b), after correction of a full 3D rotation (c), and withoutcorruption (d). Note that column order permutation can easily bedetected by visual inspection of the DEC FA map and causes a sig-nificant disruption of the tracts in the whole brain tractogram (a).Both our correction strategies correctly restore the DEC FA mapand whole brain tractogram (b–d).

3.2.2. Data set BFig. 10 shows an FA map, a DEC FA map and a whole brain trac-

togram for data set B, after corruption with a column sign change(a), after correcting for column order permutations and columnsign changes (b), after correction of a full 3D rotation (c), and with-out corruption (d). Note that column sign changes can no longer bedetected by visual inspection of the DEC FA map (a). However, thecolumn sign change still causes a significant disruption of thetracts in the whole brain tractogram (a). Both our correction strat-egies correctly restore the DEC FA map and whole brain tractogram(b-d).

3.2.3. Data set CFig. 11 shows an FA map, a DEC FA map and a whole brain trac-

togram for data set C, after corruption with a large, random 3Drotation (a), after correcting for column order permutations andcolumn sign changes (b), after correction of a full 3D rotation (c),

Fig. 12. Example of the gradient correction on in vivo data set D with a gradient table subject to a subtle 3D rotation. FA, DEC and a whole brain tractogram are shown beforecorrection (a), after gradient column permutation and sign correction (b), after the full rigid correction (c) and for the original data set (d).

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and without corruption (d). Note that the large rotation candetected by visual inspection of both the DEC FA map and thewhole brain tractogram (a). However, manual realignment is nowno longer feasible. Correcting for column order permutations andcolumn sign changes only marginally improves the DEC FA mapand the whole brain tractogram, and cannot completely restore it(b). The full 3D correction on the other hand results in a correctlyrestored DEC FA map and whole brain tractogram (c–d).

3.2.4. Data set DFig. 12 shows an FA map, a DEC FA map and a whole brain trac-

togram for data set D, after corruption with a subtle, random 3Drotation (a), after correcting for column order permutations andcolumn sign changes (b), after correction of a full 3D rotation (c),and without corruption (d). Note that the small rotation is virtuallyimpossible to detect by visual inspection on both the DEC FA mapand the whole brain tractogram and will go unnoticed (a). Onlyvery subtle differences exist between the corrupted and the truetractogram (a, d). Correcting for column order permutations andcolumn sign changes comes to no avail (b). The full 3D correctionon the other hand results in a correctly restored DEC FA map andwhole brain tractogram (c–d).

4. Discussion and conclusion

For rotationally variant diffusion parameters to be meaningful,the diffusion gradient orientations should use the same coordinateframe as the voxels in the DW data set. The meta data generated byMRI scanners should provide all the necessary information to prop-erly align the gradient table with the subject coordinate frame. Inpractice, however, this procedure can be error-prone and smallangulation errors can easily go unnoticed. As such, wrongly ori-ented gradient tables are an often overlooked and hard to detectpitfall in the processing of diffusion MRI (Jones and Cercignani,2010; Leemans and Jones, 2009).

In this work, we have introduced a method that allows rigidregistration of the DW gradient table to the corresponding DWimages, using a metric based on whole brain DTI tractography.Our tool enables researchers to check, and, if necessary, correctthe gradient tables used in their study in a fully automated way.

The proposed method uses whole brain fiber tractography as ameasure of goodness-of-fit, which means it takes into account the

global, long range, alignment patterns of the local diffusion tensors,something which cannot be achieved with methods that simplycheck how well the tensors are aligned with their neighboring ten-sors. Indeed, rotating a set of aligned tensors, will keep themaligned, only with respect to a different local orientation.

To enable a fast metric calculation, we have combined severalalgorithmic optimizations from the literature, that help speedingup the process of whole brain tractography. Despite these optimi-zations, repeated metric calculations can still require a significantamount time when many seed points are used. Since a reliableevaluation of the metric requires a large number of seed points,repeated evaluation will still have a significant computational cost.In addition, even with a large number of seed points, the metricstill exhibits spurious maxima, due to the discrete nature of trac-tography, which would cause deterministic optimization routinesto report suboptimal solutions. In order to keep the computationtime of the metric to a minimum and allow accurate estimationof the rotation parameters, we chose to use the stochastic gradientascent. The stochastic gradient ascent method follows the samescheme as the deterministic gradient ascent, with the distinctionthat the derivative of the cost function is replaced by a fast stochas-tic approximation. By using a new, randomly selected subset ofseed points every iteration of the optimization process, a bias inthe approximation error is avoided.

Indeed, our simulations show that, by adopting a stochasticoptimization approach, we are able to obtain corrected gradientorientations with sub degree accuracy and precision for SNR valuesdown to 4, which is much lower than the SNR reported in the bulkof clinical studies. We have also shown that the proposed methodworks on a range of real clinical data sets without requiring theuser to change any parameters of the metric function or the opti-mization routine. While changing the default number of randomseed points of 200 to a lower value will reduce the metric compu-tation time proportionally, it will potentially result in the metricbecoming so unreliable that the stochastic gradient descent fails.Also, while reducing the default number of gradient descent itera-tions of 150 will reduce the computation time, it is not advised todo so as this could prevent the gradient descent from converging tothe optimum.

An additional benefit of the stochastic gradient descent is that itcan potentially avoid getting converging to local minima due to thepresence of local anomalies in the fiber bundles (e.g. injured ormissing tracts). As the stochastic gradient descent selects a small,

Fig. A.1. Gradient correction on in vivo data sets with a gradient table subject to: (a) a column permutation (data set A); (b) a column sign change (data set B); (c) a large,random 3D rotation (data set C); and (d) a subtle 3D rotation (data set D). True gradient orientations are displayed as black circles; corrupted gradients as red squares;gradient orientations corrected by column permutation and sign correction as blue plus signs; and gradient orientations corrected by the full rigid correction as green crosses.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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different set of random seed points at each iteration. This way, adifferent random part of the brain is sampled at each iteration,avoiding getting stuck in the local optima that could result fromlocal anomalies, should the complete brain be sampled.

Using our current Matlab routines, the column permutation andsign correction took on average 23 s, while the full correction of 3Drotations required on average 27 min (3.5 GHz quad core desktopprocessor). Note that the current algorithm was written in Matlab,which is not the ideal platform to perform an iterative algorithmsuch as fiber tracking, and further speed ups could be obtainedby moving to C or C++.

Note that our current method uses DTI tractography, which issub optimal in the presence of crossing fibers (Tuch et al., 2002).While using a tractography method that resolves crossing fiberswill clearly result in different metric values, we expect theoptima to be found at the same locations. However, DTI tractog-raphy can generally be performed much faster than any of theHARDI alternatives and is compatible with a large set of acquisi-tion schemes (simple DTI schemes (Jones et al., 1999), HARDIschemes (Tuch et al., 2002), multi-shell schemes such as thoseused for diffusion kurtosis imaging (Jensen et al., 2005), andeven full blown Cartesian-sampling (Wedeen et al., 2005)). Sodespite the fact that our approach uses DTI tractography, therecovered gradient orientations can be applied in general DWIpost-processing.

Acknowledgements

This work was supported by the Interuniversity Attraction PolesProgram (P7/11) initiated by the Belgian Science Policy Office andby the Fund for Scientific Research-Flanders (FWO).

Appendix A

See Fig. A.1.

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