TreeSRNF: Square-Root Normal Fields for Generative Modelling of the Geometric and Structural Variability in Tree-like 3D Objects

2026-07-15

AI Summary

This work proposes the first unified framework for jointly modeling the geometric morphology and branching topology of tree-like 3D objects. By extending the Square-Root Normal Field (SRNF) representation to structures with branches, the authors construct a novel Riemannian space of tree-shaped surfaces that enables point-wise and branch-wise correspondence computation, geodesic path generation, and shape alignment. The framework effectively quantifies variations arising from surface bending, stretching, and topological changes, thereby facilitating statistical modeling of shape populations and high-quality sample synthesis. Experiments on both real and synthetic plant datasets demonstrate that the proposed method significantly outperforms existing approaches in computing shape means, analyzing shape variability, and generating realistic geometric instances.

Abstract

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.

Problem

• tree-shaped 3D objects
• geometric variability
• structural variability
• generative modelling
• 3D geometry

Innovation

• Square-Root Normal Fields
• Tree-shaped 3D Objects
• Riemannian Shape Space
• Generative Modeling
• Geometric and Structural Variability