A Prototype Representation to Approximate White Matter Bundles with Weighted Currents

Abstract

Abstract. Quantitative and qualitative analysis of white matter fibers resulting from tractography algorithms is made difficult by their huge number. To this end, we propose an approximation scheme which gives as result a more concise but at the same time exhaustive representation of a fiber bundle. It is based on a novel computational model for fibers, called weighted currents, characterised by a metric that considers both the pathway and the anatomical locations of the endpoints of the fibers. Similarity has therefore a twofold connotation: geometrical and related to the connectivity. The core idea is to use this metric for approximating a fiber bundle with a set of weighted prototypes, chosen among the fibers, which represent ensembles of similar fibers. The weights are related to the number of fibers represented by the prototypes. The algorithm is divided into two steps. First, the main modes of the fiber bundle are detected using a modularity based clustering algorithm. Second, a prototype fiber selection process is carried on in each cluster separately. This permits to explain the main patterns of the fiber bundle in a fast and accurate way.

Joint Morphometry of Fiber Tracts and Gray Matter Structures Using Double Diffeomorphisms

“…In order to deal with the considerable amount of fibers resulting from tractography algorithms, we rely on the approximation scheme introduced in [4]. Fiber bundles are approximated with weighted prototypes represented as "tubes".…”

“…The templates of cortex and putamen have been initialised as the average of the vertices. For the fiber bundle template, we have first gathered the fibers of all subjects in a single bundle which has then been approximated as a set of weighted prototypes [4]. Both Fig.3 and Fig.4 show the first mode of PCA based on the estimated covariance matrix of the deformation parameters of the first diffeomorphism W applied to the fiber bundle template keeping fixed the templates of cortex and putamen.…”

“…Weighted Prototypes A fiber bundle B is approximated with a set of weighted prototypes {τ i M i } chosen among the fibers [4]. The prototype M i is modelled as a weighted current and its weight τ i is linked to the number of fibers approximated.…”

“…The norm of the difference between two meshes is defined as the sum of squared differences between pair of vertices. White matter Fiber bundles are modelled as weighted currents [4]. Let X and Y be two fibers which can be modelled as polygonal lines of Q and Z segments respectively.…”

“…They are chosen among the fibers and their radius is related to the number of fibers approximated. This new representation is based on the metric of weighted currents [4], an extension of the framework of currents. As usual currents, it does not require point-correspondence between fibers or fiber-correspondence between bundles.…”