发表机构
Murdoch University; University of Western Australia; Johns Hopkins University（莫道克大学; 西澳大利亚大学; 约翰霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对树状3D对象几何与结构变异性问题，将平方根法向场表示推广到此类对象，构建黎曼树状空间及度量，开发相关算法，可计算统计摘要并合成新对象，性能显著优于现有技术。
AI 中文摘要
我们引入了一个新颖的数学框架，用于分析和生成复杂的树状3D对象，如植物树木和植物，它们在3D几何形状和分支结构上都会变形。与之前只考虑树状对象骨骼结构或用分支厚度近似其3D几何形状的工作不同，该框架能精确地对树枝的3D几何形状及其相互连接方式进行建模。本文首先将最初为零亏格曲面统计分析提出的平方根法向场（SRNF）表示推广到树状3D对象。接着将树状三维物体视为一个新的黎曼树状空间上的点，该空间配备了一种新的黎曼度量，用以测量表面弯曲和拉伸的程度以及将一个3D树形与另一个对齐所需的结构变化。这样，变形就成为这个新树状空间中的轨迹。我们分析了这个新树状空间及其相应度量的理论性质，并开发了用于计算复杂3D树之间逐点及分支对应和测地线的算法。最后展示了如何使用这些构建模块来计算树状3D对象集合的统计摘要，即变异均值和模式，以及通过从拟合到树状3D对象群体的概率分布中采样来合成新的树状3D对象。我们在真实和合成的植物及树木上展示了该框架的性能和实用性，并表明它显著优于现有技术。
英文摘要
We introduce a novel mathematical framework for analyzing and generating complex tree-shaped 3D objects, such as botanical trees and plants, which deform both in their 3D geometry and branching structure. Unlike previous works, which either consider only the skeletal structure of tree-like objects or approximate their 3D geometry using branch thickness, the proposed framework accurately models both the 3D geometry of the tree branches and the way they are interconnected. In this paper, we first generalize the Square Root Normal Fields (SRNF) representation, originally proposed for the statistical analysis of genus-0 surfaces, to tree-shaped 3D objects. We then treat tree-shaped 3D objects as points on a novel Riemannian tree-shape space equipped with a novel Riemannian metric that measures the amount of surface bending and stretching, and structural changes one needs to apply to one 3D tree-shape to align it with another. This way, deformations become trajectories in this novel tree-shape space. We analyze the theoretical properties of this novel tree-shape space and the corresponding metric and develop algorithms for computing point-wise and branch-wise correspondences and geodesic paths between complex 3D trees. We finally show how to use these building blocks for (1) computing statistical summaries, \ie means and modes of variation, of collections of tree-shaped 3D objects, and (2) synthesizing novel tree-shaped 3D objects by sampling from probability distributions fitted to a population of tree-shaped 3D objects. We demonstrate the performance and utility of the proposed framework on real and synthetic plants and botanical trees and show that it significantly outperforms the state-of-the-art.
CommentsECCV 2026
