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.

翻译：如何分析展示复杂几何与拓扑变异的三维生物对象（如神经元与植物树木）？本文提出一种新颖的数学框架，用于表征、比较并计算此类树状三维物体形状间的测地线变形。这些物体由子树的层级组织特征描述——每棵子树包含附着若干侧枝的主枝——需跨对象匹配这些结构以实现有意义的比较。我们提出一种将最初为欧几里得曲线开发的平方根速度函数（SRVF）扩展至树状三维物体的新型表征方法。随后定义一种新度量，用以量化将一个树状物体变形为另一个时所需的弯曲、拉伸与枝干滑动。相较于当前度量（如商欧几里得距离（QED）与树编辑距离（TED）），所提出的表征与度量能完整捕捉枝干的弹性（即弯曲与拉伸）及拓扑变异（即枝干消亡/新生与滑动），完全避免了QED与TED度量中因边塌缩与节点分裂操作导致的收缩问题。我们展示了该框架在神经元与植物树木等生物对象比较、匹配及测地线计算中的实用性。该框架还应用于多种形状分析任务：（i）树状三维物体的对称性分析与对称化，（ii）树状三维物体群体的汇总统计量（均值与变异模态）计算，（iii）对此类群体拟合参数化概率分布，（iv）最终通过从估计概率分布中随机采样合成新型树状三维物体。