  How can one analyze detailed 3D biological objects, such as neurons and
botanical trees, that exhibit complex geometrical and topological variation? In
this paper, we develop a novel mathematical framework for representing,
comparing, and computing geodesic deformations between the shapes of such
tree-like 3D objects. A hierarchical organization of subtrees characterizes
these objects -- each subtree has the main branch with some side branches
attached -- and one needs to match these structures across objects for
meaningful comparisons. We propose a novel representation that extends the
Square-Root Velocity Function (SRVF), initially developed for Euclidean curves,
to tree-shaped 3D objects. We then define a new metric that quantifies the
bending, stretching, and branch sliding needed to deform one tree-shaped object
into the other. Compared to the current metrics, such as the Quotient Euclidean
Distance (QED) and the Tree Edit Distance (TED), the proposed representation
and metric capture the full elasticity of the branches (i.e., bending and
stretching) as well as the topological variations (i.e., branch death/birth and
sliding). It completely avoids the shrinkage that results from the edge
collapse and node split operations of the QED and TED metrics. We demonstrate
the utility of this framework in comparing, matching, and computing geodesics
between biological objects such as neurons and botanical trees. The framework
is also applied to various shape analysis tasks: (i) symmetry analysis and
symmetrization of tree-shaped 3D objects, (ii) computing summary statistics
(means and modes of variations) of populations of tree-shaped 3D objects, (iii)
fitting parametric probability distributions to such populations, and (iv)
finally synthesizing novel tree-shaped 3D objects through random sampling from
estimated probability distributions.
