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:如何分析展现复杂几何和拓扑变化的三维生物对象,例如神经元和植物树木?本文提出了一种新的数学框架,用于表示、比较和计算这类树状三维物体的形状变化和距离。这些对象基于一个分级结构组织,每个子树都由主干和附加的侧枝组成,对于有意义的比较需要在对象之间匹配这些结构。我们提出了一种新的表示方法,将初始用于欧几里得曲线的平方根速度函数(SRVF)扩展到树状三维物体上。我们还定义了一种新的度量方法,该方法量化将一个树状三维物体变形成另一个所需的弯曲、拉伸和支干滑动等形变。与当前度量方法,如商数欧几里得距离(QED)和树编辑距离(TED)相比,所提出的表示和度量方法捕捉了枝条完全弹性(即弯曲和拉伸)以及拓扑变化(即树枝的生死、滑动),完全避免了QED和TED的折叠和节点分裂操作导致的收缩。我们展示了该框架在比较、匹配和计算神经元和植物树木等生物对象方面的实用性。该框架还应用于各种形状分析任务: (i)树状三维物体的对称分析和对称化,(ii)计算树状三维物体群体的概括统计信息(均值和变化模式)、(iii)将参数概率分布拟合到这样的群体,以及(iv)通过从估计的概率分布中进行随机抽样来合成新的树状三维物体。