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How can one analyze detailed 3D biological objects, such as neurons andbotanical trees, that exhibit complex geometrical and topological variation? Inthis paper, we develop a novel mathematical framework for representing,comparing, and computing geodesic deformations between the shapes of suchtree-like 3D objects. A hierarchical organization of subtrees characterizesthese objects -- each subtree has the main branch with some side branchesattached -- and one needs to match these structures across objects formeaningful comparisons. We propose a novel representation that extends theSquare-Root Velocity Function (SRVF), initially developed for Euclidean curves,to tree-shaped 3D objects. We then define a new metric that quantifies thebending, stretching, and branch sliding needed to deform one tree-shaped objectinto the other. Compared to the current metrics, such as the Quotient EuclideanDistance (QED) and the Tree Edit Distance (TED), the proposed representationand metric capture the full elasticity of the branches (i.e., bending andstretching) as well as the topological variations (i.e., branch death/birth andsliding). It completely avoids the shrinkage that results from the edgecollapse and node split operations of the QED and TED metrics. We demonstratethe utility of this framework in comparing, matching, and computing geodesicsbetween biological objects such as neurons and botanical trees. The frameworkis also applied to various shape analysis tasks: (i) symmetry analysis andsymmetrization 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 fromestimated probability distributions.

Elastic Shape Analysis of Tree-like 3D Objects using Extended SRVF Representation