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 R 3 , and (2) a random number of secondary branches, each one of them a parameterized curve in R 3 , 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 composite curves made up of: a main branch and a continuum of side branches along the main branch with each branch being a curve in R 3 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 geometric 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.
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