Symmetric Diffeomorphic Registration of Fibre Orientation Distributions

Transcription

1 Symmetric Diffeomorphic Registration of Fibre Orientation Distributions David Raffelt a,b,, J-Donald Tournier c,d, Jurgen Fripp a, Stuart Crozier b, Alan Connelly c,d, Olivier Salvado a a CSIRO Preventative Health National Research Flagship ICTC, The Australian e-health Research Centre, Royal Brisbane and Women s Hospital, Herston, QLD, Australia b Department of Biomedical Engineering, University of Queensland, Brisbane, QLD, Australia c Brain Research Institute, Florey Neuroscience Institutes (Austin), Melbourne, VIC, Australia d Department of Medicine, University of Melbourne, Melbourne, VIC, Australia Abstract Registration of diffusion-weighted images is an important step in comparing white matter fibre bundles across subjects, or in the same subject at different time points. Using diffusion-weighted imaging, Spherical Deconvolution enables multiple fibre populations within a voxel to be resolved by computing the fibre orientation distribution (FOD). In this paper, we present a novel method that employs FODs for the registration of diffusionweighted images. Registration was performed by optimising a symmetric diffeomorphic non-linear transformation model, using image metrics based on the mean squared difference, and cross-correlation of the FOD spherical harmonic coefficients. The proposed method was validated by recovering Correspondingauthor. TheAustralianE-HealthResearchCentre, UQHealthSciences Building, Royal Brisbane and Women s hospital, Herston, Queensland 4029, Australia. Fax: address: david.raffelt@csiro.au (David Raffelt) Preprint submitted to NeuroImage February 1, 2011

2 known displacement fields using FODs represented with maximum harmonic degrees (l max ) of 2, 4 and 6. Results demonstrate a benefit in using FODs at l max =4 compared to l max =2. However, a decrease in registration accuracy was observed when l max =6 was used; this was likely caused by noise in higher harmonic degrees. We compared our proposed method to fractional anisotropy driven registration using an identical code base and parameters. FOD registration was observed to perform significantly better than FA in all experiments. The cross-correlation metric performed significantly better than the mean squared difference. Finally, we demonstrated the utility of this method by computing an unbiased group average FOD template that was used for probabilistic fibre tractography. This work suggests that using crossing fibre information aids in the alignment of white matter and could therefore benefit several methods for investigating population differences in white matter, including voxel based analysis, tensor based morphometry, atlas based segmentation and labelling, and group average fibre tractography. Keywords: Fibre Orientation Distribution, Registration, Spatial Normalization, HARDI, Diffusion MRI, Abbreviations: DWI, diffusion-weighted imaging; HARDI, high angular resolution diffusion-weighted imaging; FOD, fibre orientation distribution; CSD, constrained spherical deconvolution; DT, diffusion tensor; MD, mean diffusivity; FA, fractional anisotropy; T1W, T1-weighted; PSF, point spread function; MSD, mean squared difference; CC, crosscorrelation; F3D, fast free form deformation; MAE, mean angular error; ODF, orientation distribution function; PVF, partial volume fraction; AFD, apparent fibre density; SyN, Symmetric Normalization; ANTS, advanced normalization tools. 2

3 1. Introduction Diffusion-weighted Imaging(DWI) measures the diffusional motion of water molecules and, as a consequence of the interactions between water and cellular structures, provides information about the size, shape and geometry of brain micro-structures. DWI has been widely employed to investigate population differences in shape and diffusion characteristics of white matter bundles (see (Assaf and Pasternak, 2008) for a recent review). To ensure the same anatomy is being compared across individuals correspondence is needed. The most common method for obtaining correspondence involves transforming images from each individual subject into a common co-ordinate or template space (a process called spatial normalisation). Registration is the procedure used to obtain the optimal transformation to spatially normalise data. For scalar valued images, registration requires a transformation model, an image similarity measure and an interpolation scheme. However, for DWI registration the signal is orientation dependent and therefore to preserve alignment between neighbouring anatomy one must also reorient data at each voxel (Alexander et al., 2001). Many previous studies have avoided the need to reorient diffusion data by spatially normalising and investigating differences in rotationally invariant scalar characteristics such as fractional anisotropy (FA) (Basser and Pierpaoli, 1996) or mean diffusivity (MD) computed from the diffusion tensor (DT) model (Basser et al., 1994). To spatially normalise such scalar measures, transformations can for example be computed by co-registering DWI data to T1-weighted (T1W) images that can then be registered to a T1W template (Xu et al., 2003). Potential problems with this approach are the 3

4 lack of structural information due to the homogeneous intensity of T1W white matter, as well as the presence of geometric distortions in echo planar imaging (EPI) based DWI that need to be corrected to allow accurate co-registration with the T1W images. In an attempt to avoid these problems, other studies have chosen to perform registration using rotationally invariant measures of the DT such as their eigenvalues or FA (Jones et al., 2002; Guimond et al., 2002; Gee et al., 2002; Ceritoglu et al., 2009). As different white matter tracts can have different anisotropy these measures are more informative than T1W images. However, registering with rotationally invariant measures discards potentially useful fibre orientation information contained within the DT. Consequently, a number of approaches have utilised this orientation information by using the entire diffusion tensor, or its primary eigenvector, for registration (Alexander and Gee, 2000; Ruiz-Alzola et al., 2002; Park et al., 2003; Leemans et al., 2005; Zhang et al., 2006; Cao et al., 2005; Hecke et al., 2007; Yeo et al., 2009). These methods are more complex due to the required re-orientation of the data during registration. The benefit of diffusion tensor registration was recently demonstrated by Zhang et al. (2007) where it was observed to improve the detection of white matter differences. One limitation of the popular rank-2 diffusion tensor is its inability to adequately model more than a single fibre population. This is a serious problem, as crossing fibres are endemic to DWI. Previous work estimated that one third of white matter voxels contain multiple fibre populations (Behrens et al., 2007), and a more recent study suggested that 90% of white matter voxels are likely to be affected (Jeurissen et al., 2010). 4

5 With the introduction of High Angular Resolution Diffusion-weighted Imaging (HARDI) (Tuch et al., 2002), more advanced models have emerged that better characterise regions with crossing fibre populations. Methods such as Diffusion Spectrum Imaging (Wedeen et al., 2005), Q-ball Imaging (Tuch, 2004), higher order diffusion tensors (Ozarslan and Mareci, 2003), and tensor distribution models (Jian et al., 2007) seek to model the diffusion propagator (or its radial projection) in the presence of multiple fibre orientations. Alternatively, methods such as Spherical Deconvolution aim to estimate the so-called fibre orientation distribution (FOD), a continuous distribution representing the partial volume of the underlying fibres as a function of orientation (Tournier et al., 2004; Anderson, 2005; Alexander, 2005; Tournier et al., 2007; Dell Acqua et al., 2007; Descoteaux et al., 2009). Higher order models provide more information on white matter architecture. An early attempt to utilise this information for deformable registration was proposed by Barmpoutis et al. (2007). This study demonstrated that registration using 4th order DTs gave more accurate spatial normalisation of synthetic data in regions with two fibre populations compared to the traditional rank-2 DT. More recently, methods to perform registration of diffusivity functions(chiang et al., 2008), diffusion orientation distribution functions (ODF) (Geng et al., 2009) and Gaussian Mixture Fields (Cheng et al., 2009) were also proposed. All four of these methods seek to register higher order models of diffusion. However, unlike FODs, peaks of the diffusivity profile do not necessarily correspond to the underlying fibre orientations (Zhan and Yang, 2006; Tournier et al., 2008), which has implications for downstream applications such as fibre tractography. 5

6 Registration and spatial normalisation of FODs is likely to benefit a number of downstream applications. Unlike existing DTI derived atlases (Mori et al., 2008; Hecke et al., 2008; Peng et al., 2009; Zhang et al., 2010b), a FOD-derived atlas could be used to map the location, orientation and partial volume of known fibres bundles. Such atlases would offer more information that may increase the accuracy of automated template-based white matter segmentation. Normalisation of FOD images may also benefit tensor-based morphometry (TBM) (Zhang et al., 2009), as well as existing DWI analysis methods that use fibre tractography performed on group average DTI templates (Yushkevich et al., 2008; Goodlett et al., 2009; Zhang et al., 2010a). FOD spatial normalisation is also critical for investigating differences in the recently proposed Apparent Fibre Density (AFD) measure (Raffelt et al., 2010). In this work we present an extension to the symmetric diffeomorphic normalisation (Avants et al., 2008) method, in which FOD images are registered by employing a point spread function reorientation strategy (Raffelt et al., 2009). The registration method was validated using FODs at various spherical harmonic degrees, and experiments were performed to compare FOD registration against FA based registration using an identical code base and parameters. Finally, we demonstrate a potential application of this method by computing a group average FOD template, which was then to perform group average fibre tractography. 6

7 2. Methods 2.1. Data acquisition and processing To investigate the proposed reorientation method, data were acquired from 21 healthy volunteers on a 3T Siemens Trio (60 DW directions, b=3000 s/mm 2, 2.5 mm isotropic voxels, 9 min scan time). We also acquired eight b=0 images interleaved throughout the 60 DW directions in each case. The interleaved b=0 images were used to ensure inclusion only of data with negligible subject movement over duration of the scan. This was performed by computing the mean intensity difference between all interleaved b=0 images to quantitatively estimate the extent of the movement for each subject. Visual inspection of the interleaved scans was also performed to ensure negligible movement. Using these criteria we included data from only 11 of the total 21 volunteer subjects. FODs were computed by Constrained Spherical Deconvolution (CSD) (Tournier et al., 2007) using the MRtrix software package with default parameters(brain Research Institute, Melbourne, Australia; FA maps were computed using a linear least squares approach performed using MRtrix FOD reorientation Spatial normalisation involves a transformation, T, that maps each point, x, in the target image to a corresponding point, x, in the moving image: x = T(x) (1) When transforming scalar valued images, the intensity in the moving image is interpolated at position x, and copied to the corresponding location x, in 7

8 the target image space. However when transforming DW images, data must also be reoriented to ensure orientation information remains anatomically correct with respect to neighbouring voxels (Alexander et al., 2001). This is performed by modifying the interpolated FOD at point x in the moving imageusingatransformationmatrixf. WhenT isarigidoraffinetransform, F is independent of x. When T is a non-linear transformation, a local affine model can be computed at each point x by: F = I+J u (2) where I is the identity matrix and J u is the Jacobian matrix of the displacement vector field u at point x (Alexander et al., 2001). The transformation T maps from the target to the moving image, and consequently the inverse of F must be applied to the FOD interpolated at x as it transformed to the target image space. When reorienting diffusion tensor data, it is undesirable to modify the diffusion properties of the fibre bundle and consequently only a rotation, R, is performed to preserve its shape. Two popular methods for computing R from F are Finite Strain, and Preservation of Principle Direction (Alexander et al., 2001). The reorientation problem is more challenging when spatially normalising higher order models because a single fibre population is no longer assumed. Consequently the entire affine transform must be applied to ensure not only rotation, but shearing and scaling are accounted for. Hong et al. (2009) proposed a method to reorient FODs that preserves both the total FOD integral, and the volume fraction of fibres going through each infinitesimal surface area element on the sphere. Recently, we presented an improved solution that also preserves the to- 8

9 tal FOD integral and partial volume fractions (Raffelt et al., 2009), with a three fold reduction in computation time compared to the Hong et al. (2009) method. FOD reorientation should be performed at every iteration of optimisation and consumes a significant proportion of registration time. For this reason we chose to employ our improved reorientation strategy in this work. We include a brief description of the method here for completeness: 1. Each FOD is approximated as the sum of a number of equally distributed, weighted point spread functions (PSF) represented with the spherical harmonic basis (ie. the spherical harmonic representation of a Dirac PSF). 2. The orientation of each PSF is modified independently according to the local affine transformation, F. 3. The reoriented FOD is then computed by summing up the reoriented PSFs using their original PSF weights. Note that because we assume that the PSF weights are invariant to F, the total FOD integral and partial volume fractions were preserved. Fig. 1 demonstrates this reorientation method using a transformation containing a horizontal shear. The number of PSFs used in the FOD approximation can affect reorientation accuracy. However, too many PSFs result in a large computation time with little or no benefit. Previous experiments by Hong et al. (2009) demonstrated that there was no significant difference in the angular accuracy between 252 and 1002 sampling orientations generated over a unit sphere. For our numerical simulations, 300 sampling vectors were computed over a hemi-sphere (FOD symmetry is assumed) using an electrostatic repulsion 9

10 model (Jones et al., 1999; Papadakis et al., 2000) Registration background A diffeomorphic transformation guarantees a smooth, invertible, one-toone mapping with a positive Jacobian matrix determinant over the entire domain, Ω. This is a desirable property for image registration in general, but particularly so for FOD registration because a positive Jacobian is required for FOD reorientation. In the past decade a number of diffeomorphic registration methods have been proposed (see Holden (2008) for a review). Recently, Avants et al. (2008) proposed a symmetric formulation for diffeomorphic image registration that rated consistently highly in a comparison of 14 non-linear registration algorithms (Klein et al., 2009). Symmetric registration ensures that the geodesic path computed is the same regardless of which image is selected to be the target or moving image. That is, the geodesic path computed from image I to J is the same as the path computed from image J to I. The full details are outlined in Avants et al. (2008), however we include an abbreviated explanation here to aid the description of our FOD image metrics. The diffeomorphic transformation is parameterised by time t [0,1], a spatial coordinate, x, and a velocity field, v(x, t). The correspondence maps φ(x,t), are computed at t=1 by integrating the velocity field in time φ(x,1) = φ(x,0)+ 1 0 v(x,t)dt, where x = φ(x,t). In Avants et al. (2008), symmetry was enforced by splitting the problem into two parts φ 1 and φ 2, which displaces both input images along opposite equal length paths to meet at a mid point. This converts the standard mapping of φ(x,1)i = J to the symmetric formulation of φ 1 (x,t)i = φ 2 (x,1 t)j, where x is the corre- 10

11 sponding point in image J FOD registration In this work, we implemented a FOD registration algorithm using the Insight Segmentation and Registration Toolkit (ITK) that extends the aforementioned symmetric image normalization (SyN) method ( edu/ants/). Unless otherwise noted, all experiments performed within this work used the following registration parameters. Regularisation was performed by Gaussian smoothing on the update field (variance of 5 mm) and displacement field (variance of 1 mm). Akin to the registration comparison performed by Klein et al.(2009), SyN registration was performed using only a single time step in the forward and reverse directions, which reduces the computation time and memory usage significantly. Optimisation was performed using gradient descent until convergence using an initial gradient step size of 0.12 for the mean squared difference metric, and 0.20 for the cross-correlation metric. The gradient step size is a multiplicative factor applied to the gradients computed on the image metric and controls the rate of convergence. The two image metrics being compared contain gradients of different magnitudes, and will converge at different rates if the same gradient step size is used; this may differentially affect the ability to locate the global minimum. To ensure a fair comparison we performed the registration described in experiment 1 over a range of different step sizes (using 0.02 increments), and selected the values that gave optimal results. A two-level multiple resolution approach was employed by down-sampling by a factor of two at the first level, then employing the full resolution at the second level. Between levels the displacement field was up-sampled using 11

12 tri-linear interpolation. Non-linear registrations were initialised with affine transformations computed using FA data and a block matching approach (Ourselin et al., 2001). Image similarity was computed within a target image brain mask. Reorientation of FODs was performed using the aforementioned PSF method at each iteration based on the current estimate of the displacement field. FODs were interpolated by tri-linear interpolation of the SH coefficients during registration. Final transformations were applied using cubic B-spline interpolation of the SH coefficients. Note that although the CSD method used to compute FODs from DWI data is non-linear, the mapping from FOD to DW signal is linear. Interpolating over the SH coefficients is therefore a good approximation equivalent to interpolating over the raw DW signal Mean squared difference image metric FODs are represented with real valued spherical harmonic (SH) orthonormal basis functions and therefore form a vector space with a defined L 2 norm. The distance between two FODs, f 1 and f 2, with corresponding even degree SH coefficients, c and d respectively, can be computed by: D(f 1,f 2 ) = f 1 f 2 = lmax l (c m l d m l ) 2 (3) l=0,2... m= l Where c m l, and d m l represents the coefficient for the real valued SH function of degree l and order m. To register FOD images we employed the squared L 2 norm as a metric, summed over all voxels: l max l Ω l=0,2... m= l [ R ( φ 1 (x,0.5)i ) m 12 l R ( φ 2 (x,0.5)j ) ] 2 m dx (4) l

13 where Rl m describes the SH coefficient of degree l and order m of the reoriented FOD at positions, x and x in the warped images I and J respectively. We refer to this metric here in as mean squared difference (MSD). To update the diffeomorphism, the gradient of the cost function must be computed. This was performed by averaging the gradients for all SH coefficients, computed using the Thirions s Demons optical flow update (Thirion, 1998). Notethis gradient isonlyapproximate asit doesnottake into account the FOD reorientation Cross-correlation image metric Unlike the mean squared difference image metric, the normalised crosscorrelation similarity measure can adapt to local variations in intensity (for example due to a bias field) (Bajcsy and Kovacic, 1989). Avants et al. (2008) define a cross-correlation (CC) metric for symmetric diffeomorphic image registration along with the Euler Lagrange equations for this problem. In their work, a scalar image cross-correlation metric was shown to outperform the mean squared difference metric when assessed by segmentation accuracy. Here we adapt the cross-correlation measure to the problem of minimising the difference between corresponding FOD coefficients. We compute the CC over a local image region for each SH coefficient. The warped SH coefficient image at order l and degree m are defined as Il1 m = Im l (φ 1 (x,0.5)) and Jl2 m = Jl m (φ 2 (x,0.5)) respectively. We also define variables to represent each SH coefficient image with its local mean, µ, subtracted. Īl m (x) = Il1 m(x) µ 1(x) and J l m (x ) = Jl2 m(x ) µ 2 (x ). All experiments in this work use a radius of 3 13

14 voxels to compute µ. The cross-correlation measure was then computed by: CC(I,J) = l max l l=0,2... m= l ( Im l, Jm l 2 I m l J m l The gradient of cost function required to update both diffeomorphisms, φ 1 and φ 2 were calculated by averaging the gradients computed on each SH coefficient using the Euler-Lagrange equations as defined by equations (6) and (7) in Avants et al. (2008) Experiment 1: Registration validation This experiment was designed to validate the proposed method and to investigate the hypothesis that additional information provided by higher degree harmonics improves image registration accuracy. This was performed by recovering a known displacement field obtained using real subject data (see below). Both image metrics were validated using FODs with a maximum SH degree (l max ) of 2, 4 and 6. FODs at l max =2 are represented using 6 SH coefficients. As seen in the simulated FODs in Fig. 2, this permits only a single maximum and therefore does not allow accurate representation of more than a single fibre population. Multiple fibre populations can be described by FODs at l max =4 containing 15 SH coefficients; however, FODs reconstructed at higher degrees (such as l max =6) allow multiple fibre populations to be resolved at more acute angles, at the expense of increased susceptibility to noise. Due to computer memory constraints when using the cross-correlation metric, the highest harmonic degree tested was l max =6. We also included FA driven registration using an identical code base and parameters. For FA registration we employed scalar valued equivalent metrics ) (5) 14

15 to our FOD metrics, namely mean square difference and cross-correlation as described in Avants et al. (2008). Validation of both FOD and FA registration was undertaken as follows: 1. First, 10 realistic ground truth displacement fields were created by registering the FAimage of a single volunteer (subject A) to each of the 10 other volunteers. For this step, affine registration (Ourselin et al., 2001), followed by Fast Free Form Deformation (F3D) was used, which employs a b-spline transformation model with a Normalised Mutual Information metric (Modat et al., 2010). 2. The ground truth displacement fields from step 1 were applied to FA and FOD images (with PSF reorientation) of subject A to create a series of 10 target images. 3. FOD and FA images of subject A were registered to each of the 10 target images in an attempt to recover the applied displacement fields. Registrations were run until convergence. 4. The displacement field error was computed as the mean Euclidean distance between the ground truth displacement field vectors and recovered displacement field vectors, averaged within a brain mask. 5. A two-tailed paired t-test was employed to determine if any differences between data types or metrics were significant. We also performed this experiment using a different acquisition of subject A as the moving image in step 3. The different acquisition was acquired during the same session but with slightly tilted and shifted images axes. This resulted in the moving and target images having different image slicing and noise realisations, therefore allowing registration to be evaluated in a more 15

16 realistic scenario than if the same acquisition was used to produce the moving and target images. To initialise the registration, the rigid transformation that relates both subject A acquisitions was composed with the affine transform used in generating the target images Experiment 2: Inter-subject registration comparison The validation method described in experiment 1 was performed by recovering known displacement fields between images derived from the same subject. We performed an additional experiment to compare the performance of the proposed metrics and data types using inter-subject registration. The comparison was performed as follows: 1. Each of the previously mentioned 11 volunteer subjects was registered to all other subjects, using FA data and FODs at l max =2, 4 and 6, with both MSD and CC image metrics (880 registrations in total). 2. Computed warps were applied to spatially normalise FOD images at l max =8. This is the highest l max attainable using our acquired 60 direction DWI data without super-resolved spherical deconvolution(tournier et al., 2007). 3. Registration performance was assessed using the mean angular error (MAE) between corresponding peaks in the template and normalised FODs. The MAE for each voxel was computed as follows: (a) The orientations of the three largest peaks in the template FOD were found using a gradient ascent approach (see appendix A). (b) The number of fibre populations was estimated by applying a simple threshold to the peak amplitudes determined as 0.75 times 16

17 the amplitude of an expected peak in a FOD with three fibres and equal partial volume (0.45 with our data). Note that this approach results in a more conservative estimate of voxels with multiple fibre populations than the more complex bootstrapping method used in Jeurissen et al. (2010). (c) The MAE was computed as the average angle between peak orientations in the template and spatially normalised FOD. The MAE was assumed to be the minimum MAE of all possible fibre correspondence combinations. (Note: When a mismatch between the number of fibres exists, an angle of 90 degrees is assigned to the addtional fibre therefore penalising the mis-registration). 4. The MAE was computed for all 880 registrations. The different registration methods were compared by averaging the MAE across common templates. Significant differences in the average MAE between methods were determined using a paired Students t-test. To demonstrate the requirement of FOD reorientation, we also applied the same non-linear warps without performing PSF reorientation and computed Experiment 3: Group average fibre tractography To illustrate a potential application of the proposed registration method, we generated an unbiased group template of the 11 aforementioned volunteer subjects using an iterative averaging approach. To obtain the initial FOD template, FA maps were affine registered to the same space and resulting transformations were applied to FOD images, which were then averaged. Each of the 11 subjects was non-linearly registered to the initial template 17

18 with FODs at l max =4 using the CC metric. The resulting normalised images were averaged and used to update the template for the following iteration. This process was repeated until convergence. One potential application of the group average FOD template is to obtain group-wise tractography results. Such an approach was first explored for DTI by Jones et al. (2002), and has been recently employed for group based analysis of individual white matter structures (Yushkevich et al., 2008; Goodlett et al., 2009; Zhang et al., 2010a). One advantage of performing tractography on the group average data is the reduction in noise, a possible source of error. Group-wise tractography using higher order models has not been feasible to date. Therefore to explore group-wise tractography using FODs, we applied the displacement fields computed by registration of FODs at l max =4, to FOD images at l max =8. The spatially normalised FOD images were then averaged, and used to perform whole brain probabilistic tractography using MRtrix ( with default parameters. We seeded tracts within a probabilistic white matter mask computed using MRtrix. 3. Results 3.1. Experiment 1: Registration validation Prior to performing registration, the mean error between intial displacement fields (identity) and all ground truth displacement fields was computed to be 2.03mm. After registration the displacement field error was reduced as illustrated in Fig. 3. FOD registration performed significantly better than FA when the MSD metric was used to register moving and target images 18

19 created from the same acquisition (Fig. 3a). A small but significant difference was observed when registration was performed with FODs at l max =4 compared to l max =2. No significant difference was observed between l max =4 and 6. A similar trend was observed with the CC metric (Fig. 3b), however it performed worse than the MSD metric for all data types (p<0.05). Validation using different acquisitions to compute the moving and target images resulted in different image slicing and noise. This provided a more realistic validation of registration performance because both FA and FOD reconstruction are affected by noise and variations in fibre partial volume fractions. As expected, using different acquisitions resulted in an overall increase in error compared to registration performed using the same acquisition (Fig. 3c-d). FODs at l max =4 were observed to perform significantly better than FODs at l max =2, and FA data. However, results obtained with FODs at l max =6 showed a significant increase in error compared to l max =4. This trend was observed in both the MSD (Fig. 3c) and the CC metric registrations (Fig. 3d). In this case there was no significant difference between metrics Experiment 2: Inter-subject registration comparison Fig. 4 shows the results of the inter-subject registration comparison performed using the MSD metric (Fig. 4a) and CC metric (Fig. 4b). Registration using FODs resulted in a reduced MAE compared to FA driven registration for both metric types. In a similar trend to that observed in experiment 1, optimal results were obtained for both metrics using FODs at l max =4. An increase in MAE was observed when FODs were represented at l max =6. A small but significant reduction in MAE was observed when using 19

20 the CC metric compared to the MSD metric for all data types (p<0.0001). As shown in Fig. 4, when FOD reorientation is not used when transforming FOD images, we observe an increase in the MAE (by approximately 2 degrees) Experiment 3: Group average fibre tractography As demonstrated by the axial slice shown in Fig. 5a, group-wise registration of FOD images from 11 subjects resulted in a sharp group average template. The group average template remained stable after 10 iterative updates. This slice was selected as it demonstrates the existence of many voxels with multiple fibre populations within deep white matter tissue. Fig. 5b illustrates group average whole brain probabilistic fibre tractography results performed on data from Fig. 5a. Displayed are tracts within a 4mm slab centred on the axial slice shown in Fig. 5a. A sagittal view of all tracts generated is shown in Fig. 5c. To qualitatively illustrate the registration results, and to demonstrate that image topology is well preserved, registration results from three of the subjects used to create the group average are illustrated in Fig. 6. FOD images (at l max = 8) are overlaid on FA maps. As shown, the spherical harmonic L 2 Norm difference computed between the template image and the affine registered image (column 4) is reduced after non-linear registration (column 5). 4. Discussion In this work we have presented a novel FOD registration method based on the previously reported registration algorithm by Avants et al. (2008). 20

21 Two different image metrics were used to compare registration with FODs at different harmonic orders and FA data. FOD registration performed significantly better than FA driven registration in both experiment 1 and 2. This result was expected given the inferior white matter contrast provided by FA data. As hypothesised, FOD registration driven with the extra crossing fibre information provided by l max =4 enhanced the recovery of the displacement field compared to FODs at l max =2, which are only capable of modelling a single peak fibre orientation (using 6 SH coefficients). There was no further gain in using l max =6 over l max =4 when the target and moving images were generated from the same acquisition (Fig 3a-b). This might be explained by the fact that most major fibre bundles cross at reasonably large angles (see Fig. 2), and therefore FODs at l max =6 offer relatively little additional information for registration. As might be expected, an overall increase in registration error was observed when validation was performed with a different acquisition to generate the moving and target images. However, under these conditions where images contained different noise and slicing, FODs at l max =4 were still able to achieve a sub-voxel displacement field error of 0.4mm (Fig. 3c-d). In this experiment we also observed a significant increase in registration error at l max =6 compared to l max =4. Given that no significant difference was observed between l max =4 and l max =6 when the same moving and target images were used, we hypothesise that the increase in error seen when different moving and target images are used is likely caused by noise in the higher frequency SH coefficients. 21

22 When we investigated the performance of FOD and FA data for intersubject registration, the results supported the trend observed in the intrasubject validation experiment (Fig. 4). Registration with FODs at l max =4 resulted in the minimum MAE. Even though FODs at l max =6 are more similar to the l max =8 FOD images used to compute the MAE, the results were worse than with FODs at l max =4. Again, this result is likely explained by noise negatively impacting registration performance, with little or no gain provided by l max =6 information. In this experiment the cross-correlation metric performed better than mean squared difference. It is possible that the benefit of cross-correlation was only observed for inter-subject and not intra-subject registration due to bias field differences between subjects. If so, this suggests bias field correction should be included as a pre-processing step for spherical deconvolution; future work will investigate this. It should be noted that the observed differences in MAE between all methods appear smaller than the differences seen during the intra-subject registration comparison. It is possible that the MAE measure is not as informative as the displacement field error because a descrepancy in a given voxels location, may not necessarily translate to a large error in fibre orientation if it is still located within the same fibre bundle. We also note that the observed differences in MAE between methods are <1 degree, for downstream applications such as tractography small errors in fibre orientation may compound to large errors in brain connectivity. Therefore in our opinion the additional computationalexpense inusing FODsatl max =4(anadditional 1-2hoursdepending on the metric and parameters) is warranted for such applications even on the evidence of differences observed in MAE. 22

23 In this study we included FA data within our l max comparisons due to its current widespread use for normalising DWI data, and because it could be easily performed using an identical code base. However, most state of the art diffusion registration algorithms are based on the whole tensor and future studies could include a comparison of this with FOD registration. Nevertheless, to investigate whether higher order models are superior to DTI for driving DWI registration, ideally an identical code base with similar parameters should be used to ensure the metric can be investigated independently of the registration algorithm. Another consideration for such a comparison is the b-value used to acquire the DW images. Higher order methods such as Spherical Deconvolution and Q-ball benefit from the increased angular contrast provided by large b-values. However, such large b-values are suboptimal for DTI reconstruction due to low signal-to-noise ratios. For these reasons a comparison of FOD data and DTI is not straightforward, and is beyond the scope of the current study. Irrespective of any differences that might be observed between FOD and DTI registration, we argue on a theoretical basis that FOD registration should provide improved correspondence, since the FOD is a more accurate characterisation of the underlying white matter architecture than the diffusion tensor (especially in crossing fibre regions). For example, a popular application is the use of FA to detect white matter differences between population groups. Performing registration using the diffusion tensor, or the derived FA measure, may bias results when comparing populations on the basis of tensor derived measures. Similar to work by Zhang et al. (2007), a future study could be conducted to investigate if registration using FOD 23

24 data enables better group separation than registration with DTI data. As seen in Fig. 5, using the proposed method we were able to compute an unbiased group average template. This type of procedure is the first step required in building a white matter atlas. Current state of the art DWI atlases are DTI based and are consequently limited to accurately representing only major white matter bundles in voxels with single fibre populations(mori et al., 2008; Hecke et al., 2008; Peng et al., 2009; Zhang et al., 2010b). For applications such as automated template-based white matter parcellation, an atlas derived using FODs would provide more accurate information about the location, orientation and partial volume of white matter bundles, and is therefore likely to improve segmentation accuracy. The results illustrated in Fig. 5a-b show the extent of crossing fibres within the group average template. Previous studies for investigating population differences have used fibre tractography derived from group average DTI templates to identify and compare known white matter structures (Yushkevich et al., 2008; Goodlett et al., 2009; Zhang et al., 2010a). This type of analysis may benefit from FOD group template fibre tractography, however the DT measures used for comparison in these studies are difficult to interpret in regions with crossing fibres (Wheeler-Kingshott and Cercignani, 2009; Jones, 2010). In recent work by Jbabdi et al. (2010), a method for whole brain voxelbased analysis was presented aimed at investigating differences in a multifibre context. A crossing fibre model (Behrens et al., 2007) was used to compare partial volume fractions (PVF) of fibre bundles between groups; this is based on the premise that PVF are more readily interpretable than 24

25 FA in regions with crossing fibres. Using FODs for registration may improve the alignment of these PVFs due to the higher order information available. In addition to PVF analysis, the work presented in this paper will be important for investigating differences in the in the FODs themselves, as in the recently proposed Apparent Fibre Density (AFD) measure (Raffelt et al., 2010). In this work, information provided by FODs enables voxel wise comparisons to be made over all orientations. Any differences in the FOD amplitude along a given orientation can be attributed to differences in the relative amount of underlying axons aligned with this orientation: the apparent fibre density (AFD). Investigating white matter by analysing PVF or AFD will permit differences to be attributed to a single fibre population within a voxel containing multiple fibres. Different fibre populations in voxels with crossing fibres connect functionally different areas of the cortex and therefore analysis in a multiple fibre context will provide more information about the disease under investigation. Although the proposed method focuses on registration of FODs, it may be similarly applied to register other models represented using the SH basis, such as the diffusion orientation distribution function (dodf) computed using Q- Ball imaging. We also note that other previous work to register dodfs by Geng et al. (2009) could be applied to register FODs. However the major difference with this method lies in the reorientation approach used: in Geng et al. (2009), only the rotation component of the Jacobian is applied to rotate the dodfs. We believe that if a single fibre population can not be assumed, the entire local affine transform must be applied to ensure that not only rotation, but shearing and scaling are properly accounted for. 25

26 A limitation of the proposed FOD registration method is the approximation of the gradient computed to update the diffeomorphism, which does not take into account the FOD reorientation. Other previous DT registration methods have considered reorientation within the gradient calculation (Cao et al., 2005; Zhang et al., 2006; Yeo et al., 2009). This has been shown to improve registration results (Yeo et al., 2009). However, given the considerably more complicated nature of FOD reorientation, incorporating it into analytic gradient computation poses a significant challenge. Future work will investigate this within the current registration framework. 5. Conclusion In this work we have presented a novel method for symmetric diffeomorphic registration of FODs. Our experiments suggest that using higherorder crossing fibre information enables more accurate image registration. We showed that the cross-correlation metric was beneficial for inter-subject registration. The utility of the proposed method was demonstrated by computing a group average unbiased FOD template, which was used to perform group-based whole brain fibre tractography. Accurate spatial alignment using higher order models will be increasingly important in methods such as tensor based morphometry, white matter atlas generation, atlas based segmentation and labelling, group average fibre tractography, and voxel based analysis in a multiple fibre context. 26

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38 Figure 1: FOD point spread function (PSF) reorientation. 1) Each FOD is approximated as the sum of a number of equally distributed weighted PSFs. 2) The orientation of each PSF is modified independently according to the local affine transformation, F. 3) The reoriented FOD is then computed by summing up the reoriented PSFs using the original PSF weights. 38

39 Figure 2: Simulated FODs used to demonstrate the ability of larger harmonic degrees (l max ) to resolve fibre populations at more acute angles. We hypothesised that additional information provided by higher order harmonics would improve image registration accuracy. 39

40 Figure 3: Intra-subject FOD registration at l max =2, 4 and 6 compared with FA registration. Each line displays the mean displacement field error for registration to each of 10 different target images. The mean displacement field error for all registrations is shown in red. Significant differences computed using a paired t-test are indicated by the p-values shown. (A) Same moving and target acquisition registered with the mean squared difference metric. (B) Same moving and target acquisition, cross-correlation metric. (C) Different moving and target acquisition, mean squared difference metric. (D) Different moving and target acquisition, cross-correlation metric. 40

41 Figure 4: Inter-subject registration comparison using FA data and FODs at l max =2, 4 and 6. Each line displays the mean angular error (MAE) between FODs in one image (the target) and the 10 other spatially normalised images. The average MAE across all templates is shown in red. Significant differences computed using a paired t-test are indicated by the p-values shown. (A) Registrations performed using the mean squared difference metric(b) Registrations performed using the cross-correlation metric. (C) MAE with no FOD reorientation when applying the MSD metric computed warps. (D) MAE with no FOD reorientation when applying the CC metric computed warps. 41

42 Figure 5: Unbiased group average FOD template and group-wise whole brain probabilistic fibre tractography. (A) An axial slice of the unbiased FOD template generated by spatially normalising and averaging FODs at l max =8. FODs are colour coded according to orientation (red: left-right, green: anterior-posterior, blue: inferior-superior) overlaid on the average FA map. (B) Probabilistic fibre tractography results performed on group average FOD data in A. (C) Whole brain fibre tractography results (20000 tracts). 42

43 Figure 6: Qualitative illustration of registration results from 3 subjects towards the group average template image (column 3). Shown here are FOD images (l max = 8) overlaid on FA maps. Images were transformed using affine transformations (column 1) and nonlinear warps (column 2) computed using FODs at l max = 4. Shown right are maps of the spherical harmonic L 2 Norm difference computed between the template image and both the affine (column 4) and non-linear (column 5) warped FOD images. 43