Fiber tracts of high angular resolution diffusion MRI are easily segmented with spectral clustering

L. Jonasson, P. Hagmann, J-P. Thiran, V. J. Wedeen Signal Processing Institute, EPFL, Lausanne, Vaud, Switzerland, Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital and the Harvard Medical School, Boston, MA, United States Introduction Fiber tractography on High Angular Resolution Diffusion Imaging (HARDI) data, such as Diffusion Spectra Imaging (DSI) or q-ball imaging, results in a large set of fiber tracts with a very complex geometry. The higher complexity compared to Diffusion Tensor MRI (DT-MRI) is due the numerous intersections between fibers that can be resolved, that is to say separated, using HARDI. Even though fiber bundles can be well separated in orientation space they intertwine and mix in the 3D position space. Spectral clustering techniques are methods that aim at obtaining new data representations to separate clusters with significant overlaps by creating a new feature space in which the clusters are clearly distinct from each other. This new space is constructed from the eigenvectors of a local affinity matrix representing the data and any classical clustering algorithm can then be applied on the eigenvectors [1, 2]. Considering the character of fiber tract data, spectral clustering seems like a highly appropriate segmentation technique. To test this hypothesis we have constructed a simple algorithm for counting the number of intersections between fibers and run a spectral segmentation algorithm on the co-occurrence matrix obtained from the counting. The proposed method is an unsupervised clustering technique, applicable for large sets of fiber tracks. It is easy to implement and has a low computational cost. Method The first step of our algorithm is to create a 3D Euclidean space of an appropriate resolution. An intuitive choice would be to simply return to the voxel space from which the fibers have originally been generated. However, we have chosen a coarser resolution by reducing the image size by half along each axis. As Brun et al. remarks in [3], mapping the fibers into Euclidean space might seem a bit crude but works really well. Every fiber will then be assigned a number of identification and a list of all fibers passing through will be saved in every voxel. In the second step we create an N by N large co-occurrence matrix, where N is the number of the fibers to cluster. The co-occurrence matrix contains the number of times two fibers share the same voxel. Since the number of fibers that we wish to segment is very large this co-occurrence matrix will also be very large. Due to the nature of the HARDI data the matrix is also sparse since most of the fibers never cross which makes it possible to handle it in Matlab despite its size. The matrix will be made even sparser by removing the influence of fibers that only have a few voxels in common by setting their values in the co-occurrence matrix to zero. It is this simple co-occurrence matrix that will serve as our affinity matrix and represent our data set. The sparse function in Matlab makes it possible to calculate the six largest eigenvalues and their corresponding eigenvectors of the affinity matrix. In the last step we run a k-means clustering on these eigenvectors and the resulting clusters will correspond to fiber bundles. Material The method has been tested on a set of fibers obtained from DSI data using a tractography method described in Hagmann et al. [4]. The diffusion images were obtained on a healthy volunteer with a 3T Allegra scanner (Siemens, Erlangen, Germany). We used a twice-refocused spin echo EPI sequence with TR/TE/∆=3000/154/66 ms, bmax = 1700mm2/s and a spatial resolution of 3mm3. Data were acquired using 515 different diffusion encoding directions sampling on a sphere of radius r=5 grid units.