We present a novel visualisation-based strategy for the assessment of a recently proposed clustering technique [1] for raw DWI volumes. By leveraging spherical harmonics to represent the DWI data, one can use rotationally-invariant features for the clustering step, thereby obviating the need for choosing a diffusion model. In addition, the clustering step collates voxels whose statistical properties are similar, and which can therefore be inferred to be comprised of similar microstructural compartments. The validity of the division of all brain tissue voxels into such classes was assessed using the recently developed streamlets visualisation technique, which aims to represent brain fibres by collections of many short streamlines as opposed to only ones that succeed in propagating between seed and target regions. It was originally proposed that the length of a streamlet be parameterised by some property of the seed region. However, here we use fixed length streamlets in all seed clusters. The benefit of applying streamlets, as opposed to a simple hard truncation or removal of streamlines that propagated outside of their seed cluster, is the maintenance of a visual perceptual flow across the cluster boundaries. As such, it is easy to see if streamlines from a cluster begin to cross their boundaries. We used the pre-processed DWI data from the Human Connectome Project [1,2] Q3 release. Further processing of the DWI data was handled in Matlab, Python and FSL [3], whilst streamline tractography was done using MRtrix [4].For noise-reduction of the DWI data prior to processing, we applied a formalised scale-space decomposition (sigma = 0.5). Thereafter we computed the SH coefficients according to [5] and [6].The rotationally-invariant features were then obtained from these spherical harmonic (SH) representations. As SH are eigen functions of the Laplace-Beltrami operator on the unit sphere $$$\mathbb{S}^2$$$, they provide an orthogonal eigen-subspace decomposition of $$$L^2(\mathbb{S}^2)$$$. Each $$$L^2$$$-spherical function $$$f$$$ can be expanded as $$f(x) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell c_\ell^m(f) Y_\ell^m(x)$$ Rotations of the sphere act via unitary transforms on each eigen-subspace. So although the SH decomposition is not rotationally invariant, the squared-norms of SH coefficient vectors $$$\|c_\ell(f)\|^2 = \sum_{m=-\ell}^{\ell}|c_\ell^m(f)|^2$$$ are, for a given eigen-subspace. Recent work by Caruyer and Verma [7] has laid the groundwork for deriving such rotationally-invariant features as homogenous polynomials of the SH coefficients. Herein we use the simplest such feature, derived from the squared-norms:$$L(f) = \left(\log \|c_\ell(f)\|^2\right)_{\ell=0\dots L},\quad \ell\text{ even}$$ The set of voxelwise features was subsequently used with a Gaussian mixture model to generate 8 clusters which could then act as seeding masks for the tractography. The recent adaptation of conventional streamline visualisation, streamlets [8], to instead display shorter, overlapping streamlines segments, was then applied to each cluster in turn. The goal here was to assess how well perceptual tracing could occur across the boundaries of the clusters, under the assumption that streamlines seeded with a cluster would stay within it, as this is where the microstructural properties would be most similar. 100,000 streamlets were generated randomly within each cluster using MRtrix, with a maximum length of 7mm (corresponding to approx. 6 voxels). The streamlets were then visualised in Paraview (Kitware Inc, NY).

Figure 1 shows an example Gaussian mixture model clustering of the rotationally-invariant features derived from the SH decomposition of an HCP subject, using 8 clusters and a prior scale-space smoothing at sigma 0.5.

Figures 2-4 show the streamlets displayed in various 3D views in Paraview. The streamlets seeded in the same cluster are all rendered in the same colour (8 in total). Note that the white matter clusters generally are more homogenous, with less intermixing of streamlets between neighbouring clusters.

The work presented here is a visual aid to the assessment of the clustering performance of rotationally-invariant metrics used to classify voxels. Although there are bound to be many quantitative metrics that could be constructed, we believe that it is important to be able to visualise how well algorithms perform. The technique described herein serves as an example as to how novel ways of considering visualisation during the development phase of a new processing method could highlight potential errors early. When the data is as complex to handle, as challenging and numerous as streamline trajectories, such tools are important.The use of a subset of recently-derived rotationally-invariant features based upon the spherical harmonic representation of DWI datasets shows promise for its ability to classify voxels according to their microstructural properties. Herein we see how this is also reflected in the maintenance of streamlets within their original seeding cluster. We conclude that streamlets could prove to be a useful visualisation tool.