On the Statistical Analysis of Complex Treeshaped 3D Objects
Abstract
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 treelike 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 SquareRoot Velocity Function (SRVF), initially developed for Euclidean curves, to treeshaped 3D objects. We then define a new metric that quantifies the bending, stretching, and branch sliding needed to deform one treeshaped 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 treeshaped 3D objects, (ii) computing summary statistics (means and modes of variations) of populations of treeshaped 3D objects, (iii) fitting parametric probability distributions to such populations, and (iv) finally synthesizing novel treeshaped 3D objects through random sampling from estimated probability distributions.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.08693
 Bibcode:
 2021arXiv211008693W
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Computational Geometry;
 Computer Science  Graphics;
 Statistics  Machine Learning