The geometry and statistics of geometric trees
Aasa Feragen (May 24, 2012)
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Anatomical tree-structures such as airway trees from lungs, blood vessels or dendrite trees in neurons, carry information about the organ that they are part of. Anatomical trees can be modeled as geometric trees, which are combinatorial trees whose edges are endowed with edge attributes describing their geometry. We consider edge attributes which take continuous scalar or vector values, leading to a continuum of trees rather than a discrete set of trees.
We shall discuss different ways of building spaces of such geometric trees, all with the goal of obtaining a geodesic space of trees where statistical parameters can be computed with the help of geodesics. For geometric trees of any size, we can define a geodesic space of trees, but geodesic computations are NP complete and the space has nowhere bounded curvature, which means that many statistical tools are not readily available. By adding restrictions on size, admissible topologies, branch order and/or branch labeling, we can regularize the space in order to obtain spaces which have nicer properties in terms of computational complexity and statistical applications. We shall discuss the positive effect of these assumptions on the solvability of statistical problems along with their negative effect on the ability to model real anatomical trees. Finally, we shall present some recent results from experiments on airway trees from lung CT scans.