Geometric Analysis of Axonal Tree Structures

This paper develops a framework for shape analysis of tree-like structures with the following common features: (1) a main branch viewed as a parameterized curve in R3, and (2) a random number of secondary branches, each one of them a parameterized curve in R3, emanating from the main branch at arbitrary points. In this framework, comparisons of objects is based on shapes-scales-orientations of the curves involved, and locations and number of the side branches. The objects are represented as compos- ite curves made up of: a main branch and a continuum of side branches along the main branch with each branch being a curve in R3 itself (including the null curve, or zero curve). Extending the previous work on elastic shape analysis of Euclidean curves, the space of these composite curves is endowed with a natural Riemannian metric, using the SRVF representation, and one computes geodesic paths in the quotient space of this representation modulo the re-parameterization function. As a result, appropriate geomet- ric structures are optimally matched across trees, and geodesic paths show deformations of main branches into each other while either deforming/sliding/creating/destroying the side branches. We present some preliminary results using axonal trees taken from the Neuromorpho database.