# Structure-Aware Trail Bundling for Large DTI Datasets
## 3 Method
### DTI Bundling
#### 3.2.2 Anisotropy-Constrained Bundling  
Figure 4d shows that 3D bundling considerably decreases occlusion, allowing one to see structures created by fibers deep inside the trail set. However, a problem immediately appears: All manifold-like structures, such as known to exist in DTI fiber tracts, are now broken into a set of parallel bundles, separated by gaps. This is a serious problem as the resulting rendering does not faithfully convey the brain anatomy, as also reported in other works that use DTI fiber contraction [16]. Figure 5 shows this. Image (a) shows an unbundled trail set. Given that this is a dense dataset (100 K trails), the manifold structure of the _corpus callosum_ is well captured, albeit occluded by the outer fiber tracks. Image (b) shows the isotropic bundling computed by the constrained advection explained earlier (Section 3.2.1). Occlusion is massively reduced, but the _corpus callosum_ manifold is broken into a set of bundles separated by gaps.  
We address this problem by modulating the advection, i.e., replacing the advection step \(\mathbf{d}^{\perp}\) (Equation (4)) by \(\kappa\mathbf{d}^{\perp}\), where  
\[\kappa(\mathbf{x}_{j})=\begin{cases}1,&\text{if }\alpha(\mathbf{x}_{j})\geq \alpha_{0}\\ 0,&\text{otherwise},\end{cases} \tag{5}\]  
where \(\alpha(\mathbf{x}_{j})\) is any suitable DTI anisotropy metric computed at sample point \(\mathbf{x}_{j}\) and \(\alpha_{0}\) is a threshold thereof. Figure 5c shows the effect of using fractional anisotropy (FA) [51] for \(\alpha\), defined as  
\[FA=\sqrt{\frac{1}{2}}\frac{\sqrt{(\lambda_{1}-\lambda_{2})^{2}+(\lambda_{2}- \lambda_{3})^{2}+(\lambda_{3}-\lambda_{1})^{2}}}{\sqrt{\lambda_{1}^{2}+ \lambda_{2}^{2}+\lambda_{3}^{2}}}, \tag{6}\]  
where \(\lambda_{i}\) are the major, medium, and minor eigenvalues of the diffusion tensor evaluated at the current sample point \(\mathbf{x}_{j}\). Figure 5c shows the effect of constraining bundling by \(FA\). Fibers with a value \(FA>\alpha_{0}\), where \(\alpha_{0}=0.7\), are in regions of strong anisotropy indicating tube-like structures, and are as such strongly bundled. These are color-coded in the image by their \(FA\) values. \(FA\) values below \(\alpha_{0}\) indicate planar anisotropy or isotropic regions, and are as such not bundled. These fibers are marked gray in the image. Image (d) shows the anisotropic bundling, color-cored directionally, for an easier comparison with the isotropic bundling (image (b)). As visible, anisotropic bundling reduces occlusion _and_ also keeps manifold-like structures unbroken. Similar results can be obtained by using the \(c_{l}\) and \(c_{p}\) anisotropy metrics of [PERSON] et al. [46], as shown further by the examples in Section 5.  
Figure 4: (**a**) Constrained advection scheme for trail endpoints. (**b**) Unbundled fiber set. (**c**) Unconstrained bundling. (**d**) Bundling with constrained advection.  
Figure 5: Anisotropic bundling. (**a**) Unbundled trail set. (**b**) Isotropic bundling. (**c**) Anisotropic bundling color-coded by \(FA\) metric. (**d**) As (**c**), with directional color coding. See Section 3.2.2.