3 Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions Due to the inter-subject anatomical variation, it is necessary to align ODF images of different subjects into a common space so that group-level statistical inference can be performed (see Figure 3.1) In this chapter, we propose a novel registration algorithm to align HARDI data characterized by ODFs across subjects under the Riemannian manifold of ODFs and the LDDMM framework introduced in Chapter Our proposed algorithm seeks an optimal diffeomorphism of large deformation between two ODF fields in a spatial volume domain and at the same time, locally reorients an ODF in a 23 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS manner such that it remains consistent with the surrounding anatomical structure To this end, we first define the reorientation of an ODF when an affine transformation is applied and subsequently, define the diffeomorphic group action to be applied on the ODF based on this reorientation We incorporate the Riemannian metric of ODFs for quantifying the similarity of two HARDI images into a variational problem defined under the LDDMM framework We finally derive the gradient of the cost function in both Riemannian spaces of diffeomorphisms and the ODFs, and present its numerical implementation Both synthetic and real brain HARDI data are used to illustrate the performance of our registration algorithm HARDI data ODF Reconstruction Data Acquisition Subjects ODF images Atlas Generation ODF atlas Registration serve as common space in registration ODF images in common space Statistical Analysis Biomarkers/ Inference Figure 3.1: The role of Chapter in the ODF-based analysis framework 24 3.1 Affine Transformation on Square-Root ODFs 3.1 Affine Transformation on Square-Root ODFs In this section, we discuss the reorientation of the √ ODF, ψ(s), when a non-singular affine transformation A is applied As illustrated in Figure 3.2, we denote the trans√ formed ODF as ψ(s) = Aψ(s), reflecting the fact that an affine transformation induces changes in both the magnitude of ψ and the sampling directions of s We will now show how to derive the analytical form when a non-singular affine transformation acts on an ODF Figure 3.2: Illustration of affine transformation on square-root ODFs (Similar to the shape of ODF, the colors of ODF also indices the relative values of ODF in each direction, where blue stands for low ODF value and red for high value.) First of all, we denote s = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) in Cartesian coordinates and s = (r sin θ cos ϕ, r sin θ sin ϕ, r cos θ) in Cartesian coordinates We first assume that the change of the diffusion sampling directions due to affine transformation A is s = As (3.1) Similar to [56], we assume that the volume fraction of fibers with orientation near direction s equals p(s)dΩ, where dΩ is the small patch Just as in [56], we assume that 25 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS the volume fraction of fibers oriented towards the small patch dΩ remains the same after the patch is transformed That is, p(s)dΩ = p s s dΩ, (3.2) and in polar coordinates, p(θ, ϕ) sin θdθdφ = p(θ, ϕ) sin θdθdϕ (3.3) When s = As in Cartesian coordinates, for the small volume fraction, we have dxdydz = det (A)dxdydz, and in polar coordinates, r2 sin θdθdϕdr = det(A)r2 sin θdθdϕdr Since r = As s r, we obtain As r sin θdθdϕdr = det(A)r2 sin θdθdϕdr, s s ⇒ sin θdθdϕ = det(A) sin θdθdϕ As (3.4) From Eqns (3.3) and (3.4), we get p(θ, ϕ) sin θdθdφ = p(θ, ϕ) sin θdθdϕ = p(θ, ϕ) 26 s det(A) sin θdθdϕ As (3.5) 3.1 Affine Transformation on Square-Root ODFs Removing sin θdθdϕ from both sides yields det(A) s = As s s p p As As Finally, we make a change of variable from s to s → A−1 s, giving the following p s s = det A−1 s A−1 s 3 A−1 s A−1 s p For an ODF, s ∈ S2 , s = 1, and therefore, we have p (s) = ⇒ Aψ(s) = det A−1 A−1 s det A−1 A−1 s A−1 s A−1 s p ψ A−1 s A−1 s (3.6) An alternative way of obtaining the property in Eqn (3.6) is to assume that the change of the diffusion directions due to affine transformation A is s= A−1 s , A−1 s (3.7) where the transformed sampling directions s are normalized back into the unit sphere S2 This is analogous to a pullback deformation Notice that for s ∈ S2 , Eq (3.7) defines an invertible function of s and therefore, we can find the ODF Aψ(s) using the change-of-variable technique of PDF Recall the fundamental theorem for PDF: let X be a continuous random variable having probability density function fX (x) Suppose g(x) is one-to-one and differentiable function of x Then the random variable Y defined by Y = g(X) has a probability density function given by fY (y) = fX (g −1 (y))|J(y)| where J is the Jacobian of g −1 (y) Since A is a × matrix, the determinant of the 27 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Jacobian in this case is det A−1 A−1 s In either case, the following theorem is obtained Theorem 3.1.1 Reorientation of ψ based on affine transformation A Let Aψ(s) √ be the result of an affine transformation A acting on a ODF ψ(s) The following analytical equation holds true Aψ(s) = det A−1 A−1 s ψ A−1 s A−1 s , (3.8) where · is the norm of a vector Property Assume A and B to be two matrices of affine transformations and ψ is a square-root ODF The following property holds true B(Aψ)(s) = (A B)ψ(s), (3.9) where (A B) stands for matrix muliplication between A and B Proof Base on the equation (3.8), it yields B(Aψ)(s) = B det A−1 A−1 s ψ = det A−1 det B −1 A−1 s B −1 s = det (A B)−1 (A B)−1 s ψ A−1 s A−1 s ψ A−1 B −1 s A−1 B −1 s (A B)−1 s (A B)−1 s = (A B)ψ(s) The ODF reorientation used in this work ensures that the transformed ODF remains 28 3.2 Diffeomorphic Group Action on Square-Root ODFs consistent with the surrounding anatomical structure and at the same time, not solely dependent on the rotation Rather, by constructing the change-of-variable technique as discussed above, the reorientation takes into account the effects of the affine transformation and ensures the volume fraction of fibers oriented toward a small patch must remain the same after the patch is transformed While [56] computes the ODF reorientation numerically by computing the corresponding Jacobian at each sampling direction via a series of transformations and applying it to transform the orientation, there is in fact an analytical closed form formula for the reorientation as provided by Theorem 3.1.1 Figure 3.3 illustrates how Aψ(s) varies when A is a rotation, shearing, or scaling and ψ(s) is an isotropic ODF, an ODF with a single fiber, or an ODF with crossing fibers From Figure 3.3, one immediately observes that a shearing or scaling introduces anisotropy under the reorientation scheme used here The phenomena is in line with what is observed in [92] By construction, Aψ(s) fulfills the definition of the √ ODF, i.e., Aψ(s) is positive and the integration of (Aψ(s))2 is equal to Hence, the similarity of Aψ(s) to the square-root ODFs can be quantified in the Riemannian structure given in §2.1 for the HARDI registration 3.2 Diffeomorphic Group Action on Square-Root ODFs We have shown in §3.1 how to reorient ψ located at a fixed spatial position x in the image volume Ω ⊂ R3 through an affine transformation In this section, we define an action of diffeomorphisms φ : Ω → Ω on ψ, which takes into consideration the reorientation of ψ as well as the transformation of the spatial volume in Ω ⊂ R3 , as √ illustrated in Figure 3.4 Denote ψ(s, x) as the ODF with the orientation direction s ∈ S2 located at x ∈ Ω We define the action of diffeomorphisms on ψ(s, x) in the 29 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3.3: Examples of local affine transformations on an isotropic ODF in the first row, an ODF with a single orientation fiber in the middle row, and an ODF with crossing fibers in the bottom row From left to right, three types of affine transformations, A, on the ODFs are demonstrated: in panel (a), a rotation with angle θz , where A = [cos θz − sin θz 0; sin θz cos θz 0; 0 1]; in panel (b), a vertical shearing with factor ρy , where A = [1 0; −ρy 0; 0 1]; and in panel (c), a vertical scaling with factor ςy where A = [1 0; ςy 0; 0 1] form of φ · ψ(s, x) = Aφ−1 (x) ψ(s, φ−1 (x)), where the local affine transformation Ax at spatial coordinates x is defined as the Jacobian matrix of φ evaluated at x, i.e., Ax = Dx φ According to Eq (3.8), the action of diffeomorphisms on ψ(s, x) can be computed as φ · ψ(s, x) = det Dφ−1 φ D φ−1 φ −1 −1 s ψ (Dφ−1 φ (Dφ−1 φ −1 −1 s s , φ−1 (x) (3.10) Property (The law of composition for diffeomorphic group action on ODF) As√ sume φ and ϕ to be two diffeomorphisms and ψ(s, x) as the ODF with the orientation 30 3.2 Diffeomorphic Group Action on Square-Root ODFs direction s ∈ S2 located at x ∈ Ω The following property holds true ϕ · (φ · ψ)(s, x) = (ϕ ◦ φ) · ψ(s, x) , (3.11) where ◦ stands for composition between φ and ϕ Proof Based on the equation (3.10), it yields ϕ · (φ · ψ)(s, x) = ϕ · (Ax ψ(s, x)) ◦ φ−1 (x) = Bx (Ax ψ(s, x)) ◦ φ−1 ◦ ϕ−1 (x) = Bφ(x) (Ax ψ(s, x)) ◦ φ−1 ◦ ϕ−1 (x) , where we denote Ax = Dx φ, Bx = Dx ϕ, and Bφ(x) = Dφ(x) ϕ Using the property from the equation (3.9), we have ϕ · (φ · ψ)(s, x) = Bφ(x) Ax ψ(s, x) ◦ φ−1 ◦ ϕ−1 (x) = [Dx (ϕ ◦ φ) ψ(s, x)] ◦ (ϕ ◦ φ)−1 (x) = (ϕ ◦ φ) · ψ(s, x) For the sake of simplicity, we denote φ · ψ(s, x) as φ · ψ(s, x) = Aψ ◦ φ−1 (x) , (3.12) where it will be used in the rest of the chapter √ Since φ · ψ(s, x) is in the space of ODF, the Riemannian distance given in §2.1 31 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3.4: Illustration of diffeomorphic group action on square-root ODFs can be directly used to quantify the similarity of φ · ψ(s, x) to other √ ODFs, which we employ in the HARDI registration described in the following section 3.3 Large Deformation Diffeomorphic Metric Mapping for ODFs The previous sections equip us with an appropriate representation of the ODF and its diffeomorphic action Now, we state a variational problem for mapping ODFs from one volume to another We define this problem in the “large deformation” setting of Grenander’s group action approach for modeling shapes, that is, ODF volumes are modeled by assuming that they can be generated from one to another via flows of diffeomorphisms ˙ φt , which are solutions of ordinary differential equations φt = vt (φt ), t ∈ [0, 1], starting from the identity map φ0 = Id They are therefore characterized by time-dependent velocity vector fields vt , t ∈ [0, 1] We define a metric distance between a target volume ψtarg and a template volume ψtemp as the minimal length of curves φt · ψtemp , t ∈ [0, 1], in a shape space such that, at time t = 1, φ1 · ψtemp = ψtarg Lengths of such curves are computed as the integrated norm vt V of the vector field generating the transformation, where vt ∈ V , where V is a reproducing kernel Hilbert space with kernel kV and norm 32 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS 3.4 Synthetic Data We first illustrate that the HARDI model is useful to align crossing fibers, especially when crossing fibers have equal orientation distributions To so, we construct two synthetic datasets, template and target, where there are two identical fibers perpendicularly crossing each other (Figure 3.6 (a, b)) The orientations of the two crossing fibers differ from the template image (Figure 3.6 (a)) to the target image (Figure 3.6 (b)) We will compare the performance of LDDMM-ODF to the LDDMM algorithm based on DTI (LDDMM-DTI) [96] We refer the reader to [96] for detailed mathematical derivation for LDDMM-DTI In the HARDI model, such orientation differences are encoded by the ODFs, while in the DTI model, the diffusion tensors of both the template and target data look like disks, where the first two eigenvalues being equal and the third eigenvalue being almost zero Although the overall image shapes are the same in both the template and target HARDI data, the LDDMM-ODF algorithm is able to characterize the orientation difference of the ODFs between them by generating the deformation shown in Figure 3.6 (d) with the help of term (B) of Eq (3.18) Note that a mask is used here and only the ODFs in the circle is transformed LDDMM-DTI fails to find any deformation (Figure 3.6 (e)) even though the reorientation of the tensor is taken into account in the tensor mapping 44 3.5 HARDI Data of Children Brains Figure 3.5: The first and second rows respectively illustrate the original HARDI and their enlarged images Compared to the image on panel (a), the image on panel (b) has the same ODFs but a different ellipsoidal image shape, while the image on panel (c) shows different ODFs but the same circular image shape Panels (d) and (e) show the deformations (grid) and the corresponding momenta (arrows), calculated using ∇φ1 E in Eq (3.19), for mapping the image on panel (a) to panels (b) and (c), respectively Panels (f) and (g) show the deformations and the corresponding momenta, calculated using the gradient in our previous work [2], for mapping the image on panel (a) to panels (b) and (c), respectively 45 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3.6: Comparison between the LDDMM-ODF and LDDMM-DTI algorithms Panels (a, b) respectively show the template and target HARDI and their enlarged images, where the ODF or diffusion tensor at each location contains two crossing fibers with equal orientation distribution Panel (c) illustrates the template HARDI image transformed via the deformation given in panel (d), the result of the LDDMM-ODF algorithm Panel (e) illustrates no deformation found via the LDDMM-DTI algorithm and thus the template HARDI image remains 46 3.5 HARDI Data of Children Brains 3.5 HARDI Data of Children Brains 3.5.1 Comparison of LDDMM-FA, LDDMM-DTI and LDDMMODF In this section, we apply our proposed algorithm to real HARDI data We evaluate the mapping accuracy of our LDDMM-ODF algorithm by comparing it with the LDDMMimage mapping based on FA (LDDMM-FA) and the LDDMM-DTI mapping based on diffusion tensors using the brain datasets of 26 young children (6 years old) All three algorithms are developed under the LDDMM framework as given in §3.3 with the exception that the matching functional, E, is the least square difference between two image intensities for the image mapping, LDDMM-FA, and the Frobenius norm between two tensors for the DTI mapping, LDDMM-DTI More precisely, LDDMM-FA is based on the method developed by [97] and LDDMM-DTI is based on the method developed by [96] In our implementation however, we optimize the deformation with respect to the momentum rather than the velocity (see [93]) It is important to note that all three mapping algorithms used in the following evaluation have the same numerical scheme, such that any potential errors due to numerical related issues are avoided and we can make a fair comparison Our image data are acquired using a 3T Siemens Magnetom Trio Tim scanner with a 32-channel head coil at the National University of Singapore Diffusion weighted imaging protocol is a single-shot echo-planar sequence with 55 slices of 2.3mm thickness, with no inter-slice gaps, imaging matrix 96 × 96, field of view 220 × 220mm2 , repetition time=6800ms, echo time=89ms, flip angle 90◦ 61 diffusion weighted images with b=900s/mm2 , baseline (b0) images without diffusion weighting are acquired Notice 47 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS that the b-value used in our acquisition is relatively low when compared to HARDI acquisition where b > 1000s/mm2 typically This is because the water diffusivity is in general faster in young children’s brain than in adults’ brain [98] The large b-value could result in significant loss of diffusion signals In addition, our dataset is for the purpose of the comparison between the HARDI and DTI models Thus, the b-value is determined by balancing the needs of both HARDI and DTI acquisition In the data processing, DWIs of each subject are first corrected for motion and eddy current distortions using affine transformation to the b0 image (where there is no diffusion weighting) We randomly select one subject as the template in this study and first align the remaining subjects to this template using the affine transformation computed based on the b0 images of the subject and the template Existing literature [92, 99] have proposed different ways of reorienting the diffusion gradients In this chapter, the diffusion gradients are reoriented using the method proposed in [92] Briefly speaking, if gi is the ith diffusion gradient, then the reoriented diffusion gradient after the affine transformation A is simply A − gi A − gi Then, the DTI is computed using least square fitting [100] and the FA is calculated from the DTI, and the ODF, ψaffine transformed , is estimated using the approach proposed in [48] We then respectively employ the LDDMM-FA, LDDMM-DTI, and LDDMM-ODF algorithms to register all subjects to the template To ensure a fair comparison, we fix the general setting of LDDMM with kernel σV = (Eq (3.15)) so that the global smoothness of the deformation field is the same for all three methods As the metrics of LDDMM-FA, LDDMM-DTI and LDDMM-ODF have different units and ranges, this is taken into consideration by normalizing their magnitude to the same level For LDDMM-FA and LDDMM-DTI, based on the diffeomorphic mappings computed in each case, we apply the diffeomorphic group action defined in Eq (3.10) to ψaffine transformed to obtain the 48 3.5 HARDI Data of Children Brains registered ODFs To evaluate the mapping accuracy for the whole brain, we compute both the symmetrized Kullback-Leibler divergence (sKL) and the mean squared error of spherical harmonics coefficients (MSE of SH) of the ODFs between the deformed subject and the template The sKL has been used as a metric for comparing ODFs in [51] and is defined as sKL(p1 , p2 ) = s∈S2 p1 (s) log p1 (s) ds + p2 (s) s∈S2 p2 (s) log p2 (s) ds, p1 (s) (3.23) for two ODFs p1 (·) and p2 (·) The MSE of SH has also been used previously In [57], instead of using the Riemannian metric as we have done in this chapter, they used the Euclidean metric of ODFs Furthermore, [57] represented an ODF p with spherical harmonics which is a set of orthonormal basis functions, i.e., p(s) = lmax l=0 cl Yl (s) where Yl (s) is the SH basis and cl are the SH coefficients Therefore, the Euclidean distance can be computed much more efficiently by taking the MSE of SH as follows: lmax dist(p1 , p2 ) = (cl − cl )2 , (3.24) l=0 where the 4-th order SH is used here, lmax = 15 Lower sKL and MSE of SH indicates that the ODF of the subjects are better aligned Figure 3.7 illustrates the sKL map and Figure 3.8 the MSE of SH map, averaged across all 25 subjects when affine, LDDMM-FA, LDDMM-DTI, or LDDMM-ODF are respectively applied This figure suggests that LDDMM-ODF is the best mapping among all studied in this chapter as it has the least amount of variation, even though we not use the sKL or MSE of SH distances in LDDMM-ODF Figure 3.9 also shows 49 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS the cumulative distributions of sKL and MSE of SH across the image space from each mapping Kolmogorov-Smirnov tests on the cumulative distributions also suggest that the LDDMM-ODF significantly reduces both distances against the other three methods (p < 0.001) Figure 3.7: Panels (a-d) respectively show the maps of mean symmetrized Kullback– Leibler (sKL) divergence of the ODFs between the template and the subjects deformed via affine, LDDMM-FA, LDDMM-DTI, and LDDMM-ODF For each voxel, the red (sKL=0.5) indicts the difference between the template and the deformed subjects is large, while the blue (sKL=0) indicts the two corresponding ODFs are equal We now evaluate the mapping accuracy of individual white matter tracts using 1) sKL of the ODF between the template’s and deformed subject’s tract and 2) Dice overlap ratio to quantify the percentage of the overlap volumes between the template and deformed subject’s tracts We extract three major white matter tracts, including the corpus callosum (CC) and bilateral corticospinal tracts (CST-left, CST-right), using 50 3.5 HARDI Data of Children Brains Figure 3.8: Panels (a-d) show the maps of mean squared error of spherical harmonics coefficients (MSE of SH) of the ODFs between the template and the subjects deformed via affine, LDDMM-FA, LDDMM-DTI, and LDDMM-ODF respectively For each voxel, the red (MSE=0.1) indicts the difference between the template and the deformed subjects is large, while the blue (MSE=0) indicts the two corresponding ODFs are equal probabilistic tractography with the help of Camino [100] The probabilistic tractography is performed on the q-ball reconstruction using spherical harmonic representation up to order with the number of directions for each ODF limited to and the maximum allowed turning angle limited to 70◦ To generate the fiber masks from probabilistic tractography, we have empirically selected the threshold of 0.001 for each tract across all subjects We adopt the anatomical definition of the CC, CST-left and CST-right given in [101] and define three regions of interest (ROI) such that each tract is comprised of all fibers passing through these three ROIs Figure 3.10 shows the sKL maps for the 51 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3.9: sKL and MSE of SH cumulative distributions across the whole brain image and averaged over all 25 subjects are shown in blue for affine, cyan for LDDMM-FA, yellow for LDDMM-DTI, and red for LDDMM-ODF, respectively three tracts, suggesting that, again LDDMM-ODF provides the best alignment for the ODFs of these three tracts when compared to affine, LDDMM-FA, and LDDMMDTI Figure 3.11 shows the average sKL values for the CC, CST-left, and CST-right Moreover, Figure 3.12 shows the averaged Dice overlap ratios across all 25 subjects for the CC and bilateral CST One-sample t-tests shows that LDDMM-ODF significantly improves the alignment of local fiber directions for three fiber tracts against the other methods in terms of sKL (p < 0.001) In addition, the one-sample t-tests between any two mapping algorithms suggest that all the non-linear methods show significant improvement against affine in terms of Dice overlap ratio (p < 0.001) for the three tracts, and LDDMM-ODF shows significant improvement against LDDMM-FA and LDDMM-DTI (p < 0.001) In the comparison between LDDMM-FA and LDDMMDTI, the only significant difference is found in the CST-left (p < 0.05), while no significant differences are found in the CC and CST-right 3.5.2 Comparison with existing ODF registration algorithm In this section, we compare the performance of our algorithm with an existing ODF registration algorithm proposed by Raffelt et al in [57] As the implementation of 52 3.5 HARDI Data of Children Brains Raffelt’s approach is not available, we implement it under our LDDMM framework This gives a fair comparison on different metrics and reorientation schemes proposed in this chapter and in [57] As mentioned in §3.5.1, [57] used the Euclidean distance computed based on the spherical harmonics coefficients given in Eqn (3.24) Therefore, instead of minimizing the cost function in Eqn (3.15), the following objective function is minimized, lmax J(mt ) = inf ˙ mt :φt =kV mt (φt ), φ0 =Id mt , kV mt dt + λ x∈Ω l=0 (cl (x) − cl (x))2 dx, temp targ (3.25) where cl temp is calculated from the transformed ODF ptemp obtained by applying the affine reorientation scheme in [57] to the ODF ptemp Furthermore, in line with what is proposed in [57], the gradient term is computed only using the gradient of the sum of squared SH coefficients and is not dependent on the reorientation of ODF We will denote this method as LDDMM-Raffelt Similar to what has been done in §3.5.1, all the other parameters are kept the same for both methods to ensure a fair comparison To evaluate the mapping accuracy here, we will use the mean squared error of spherical harmonics coefficients as given in Eqn (3.24), which is essentially the Euclidean distance between ODFs One-sample t-test based on MSE of SH for each subjects indicates that there is no significant difference between LDDMM-Raffelt and LDDMM-ODF (p > 0.05) across the whole brain and all subjects It is worth noting that although LDDMM-Raffelt directly minimizes MSE of SH, LDDMM-ODF achieves a similar average accuracy Nevertheless, for ROI comparisons, we further examine how the MSE of SH differs spatially in the image volume, as shown in Figure 3.13 Figure 3.13 suggests that 53 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS the MSE of SH for LDDMM-Raffelt and for LDDMM-ODF is different in the region where the CC and CST intersect each other Therefore, to extract this region, we first perform probabilistic tractography on the atlas space and create binary mask for CST-left, CST-right and CC with the threshold 0.001 as showed in Figure 3.14 Based on these binary masks, we extract the intersecting region between CC and CST where it is common to have the crossing fiber tracts Figure 3.15 shows the mean square error of spherical harmonics coefficients of these regions across all subjects The mean values of LDDMM-Raffelt is 0.0273 for CST-left and 0.0313 for CST-right, whereas LDDMM-ODF gives the mean of 0.0223 for CST-left and 0.0246 for CST-right Using the one-sample t-test based on MSE of SH for each subjects indicts that LDDMM-ODF gives a significantly lower mean than LDDMM-Raffelt (p < 0.001) 3.5.3 Computational complexity of LDDMM-FA, LDDMM-ODF and LDDMM-Raffelt The computational complexity for the calculation of the trajectory and the regulation term of LDDMM-ODF and LDDMM-Raffelt is of the same order as LDDMM-FA However, as the computational cost for the data attachment term is much higher in LDDMM-ODF, it takes − hours for LDDMM-ODF to process each subject on average whereas LDDMM-FA takes − hours typically As for LDDMM-Raffelt, it takes 2.5 − 3.5 hours to process each subject on average The additional time LDDMMODF takes is due to the additional computation on the gradient of reorientation 54 3.6 Summary 3.6 Summary We presented a novel diffeomorphic metric mapping algorithm for aligning HARDI data in the setting of large deformations Our mapping algorithm seeks an optimal diffeomorphic flow connecting one HARDI to another in a diffeomorphic metric space and locally reorients ODFs due to the diffeomorphic transformation at each location of the 3D HARDI volume in an anatomically consistent manner We incorporated the Riemannian metric for the similarity of ODFs into a variational problem defined under the LDDMM framework The diffeomorphic metric space combined with the Riemannian metric space of ODF provides a natural framework for computing the gradient of our mapping functional We demonstrated the performance of our algorithm on synthetic data and real brain HARDI data This registration approach will facilitate atlas generation and group analysis of HARDI for a variety of clinical studies 55 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3.10: Panels (a-h) show the maps of mean symmetrized Kullback–Leibler (sKL) divergence of the ODFs between the template and the subjects deformed via affine, LDDMMFA, LDDMM-DTI, and LDDMM-ODF for the three major white matter tracts of the corpus callosum (CC) and bilateral corticospinal tracts (CST-left, CST-right) For each voxel, the red (sKL=0.5) indicts the difference between the template and the deformed subjects is large, while the blue (sKL=0) indicts the two corresponding ODFs are equal 56 3.6 Summary 0.4 affine LDDMM−FA LDDMM−DTI LDDMM−ODF 0.35 0.3 sKL 0.25 0.2 0.15 0.1 0.05 CC CST−left CST−right Figure 3.11: sKL averaged over all 25 subjects are shown for the corpus callosum (CC) and bilateral corticospinal tracts (CST-left, CST-right) when affine (blue), LDDMM-FA (cyan), LDDMM-DTI (yellow), or LDDMM-ODF (red) are applied 0.7 affine LDDMM−FA LDDMM−DTI LDDMM−ODF 0.6 Dice overlap ratio 0.5 0.4 0.3 0.2 0.1 CC CST−left CST−right Figure 3.12: Dice overlap ratios averaged over all 25 subjects deformed by affine (blue), LDDMM-FA (cyan), LDDMM-DTI (yellow), or LDDMM-ODF (red) 57 DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3.13: Spatial distribution of mean squared error of spherical harmonics coefficients across all subjects For each voxel, the red (MSE=0.1) indicts the difference between the template and the deformed subjects is large, while the blue (MSE=0) indicts the two corresponding ODFs are equal Figure 3.14: Illustration of three fiber tract masks, CST-left, CST-right and CC Figure 3.15: Mean squared error of spherical harmonics coefficients (MSE of SH) in the region where the CC and CST intersects each other 58 ... MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3. 13: Spatial distribution of mean squared error of spherical harmonics... DIFFEOMORPHIC METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS the cumulative distributions of sKL and MSE of SH across the... METRIC MAPPING OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING BASED ON RIEMANNIAN STRUCTURE OF ORIENTATION DISTRIBUTION FUNCTIONS Figure 3. 3: Examples of local affine transformations on an isotropic