The Approximation of the Dissimilarity Projection

Abstract

Diffusion magnetic resonance imaging (dMRI) data allow to reconstruct the 3D pathways of axons within the white matter of the brain as a tractography. The analysis of tractographies has drawn attention from the machine learning and pattern recognition communities providing novel challenges such as finding an appropriate representation space for the data. Many of the current learning algorithms require the input to be from a vectorial space. This requirement contrasts with the intrinsic nature of the tractography because its basic elements, called streamlines or tracks, have different lengths and different number of points and for this reason they cannot be directly represented in a common vectorial space. In this work we propose the adoption of the dissimilarity representation which is an Euclidean embedding technique defined by selecting a set of streamlines called prototypes and then mapping any new streamline to the vector of distances from prototypes. We investigate the degree of approximation of this projection under different prototype selection policies and prototype set sizes in order to characterise its use on tractography data. Additionally we propose the use of a scalable approximation of the most effective prototype selection policy that provides fast and accurate dissimilarity approximations of complete tractographies.

Tractome: a visual data mining tool for brain connectivity analysis

“…In Olivetti et al (2012), the authors proposed the use of the subset farthest first (SFF) (Turnbull and Elkan 2005) algorithm for selecting effective prototypes from tractography data. This procedure is a stochastic scalable approximation of the well known farthest first traversal (FFT) algorithm, which has a computational complexity of O( p|T |).…”

“…Given that the size and complex representation of the brain tractography data is in contrast with the scalability requirement of our interactive tool, we apply a previously proposed vectorial representation for streamlines (Olivetti et al 2012).…”

“…This leads to a vectorial representation of the streamlines, where the domain information is taken into account, provided that an informative streamline-to-streamline distance function is available. A crucial step in the DR approach is the selection of the prototypes (Pekalska and Duin 2005;Olivetti et al 2012). Thus, given the large number of streamlines, a fast prototype selection algorithm is also needed.…”

“…Thus, given the large number of streamlines, a fast prototype selection algorithm is also needed. For this tool, we propose a domain-based heuristic that improves the general prototype selection policy that was proposed in Olivetti et al (2012). This policy is a stochastic approximation of an effective algorithm for prototype selection that scales well with the large collection of streamlines.…”

Multiple-Scale Visualization of Large Data Based on Hierarchical Clustering

“…It is a lossy Euclidean embedding algorithm was previously proposed in [23] for streamlines. The dissimilarity representation is defined as , where d is a distance function between streamlines, and Π = {X 1 , ..., X p } ⊂ X is a set of p streamlines, called prototypes (detail in [23], [24]). …”