A Riemannian Framework for the Elastic Analysis of the Spatiotemporal Variability in the Shape and Structure of Tree-like 4D Objects

2025-09-23

AI Summary

Registering and modeling tree-like 4D structures (e.g., plants) is challenging due to their non-rigid deformations, growth-induced shape changes, and topological variations over time. To address this, we propose a unified Riemannian-geometric framework: tree-shaped 3D anatomies are embedded into an elastic shape space, and the Extended Square Root Velocity Function (ESRVF) transformation converts the complex elastic metric into a Euclidean L² metric—enabling efficient geodesic computation and nonlinear optimization. Our method is the first to achieve accurate spatiotemporal registration, mean trajectory estimation, and statistical modeling of dynamic tree trajectories under large deformations and topological differences, while generating biologically plausible synthetic growth sequences. Experiments—including plant growth simulation—demonstrate high accuracy, robustness to noise and topological variation, and strong biological fidelity.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: 3D Computer Vision
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

This paper introduces a novel computational framework for modeling and analyzing the spatiotemporal shape variability of tree-like 4D structures whose shapes deform and evolve over time. Tree-like 3D objects, such as botanical trees and plants, deform and grow at different rates. In this process, they bend and stretch their branches and change their branching structure, making their spatiotemporal registration challenging. We address this problem within a Riemannian framework that represents tree-like 3D objects as points in a tree-shape space endowed with a proper elastic metric that quantifies branch bending, stretching, and topological changes. With this setting, a 4D tree-like object becomes a trajectory in the tree-shape space. Thus, the problem of modeling and analyzing the spatiotemporal variability in tree-like 4D objects reduces to the analysis of trajectories within this tree-shape space. However, performing spatiotemporal registration and subsequently computing geodesics and statistics in the nonlinear tree-shape space is inherently challenging, as these tasks rely on complex nonlinear optimizations. Our core contribution is the mapping of the tree-like 3D objects to the space of the Extended Square Root Velocity Field, where the complex elastic metric is reduced to the L2 metric. By solving spatial registration in the ESRVF space, analyzing tree-like 4D objects can be reformulated as the problem of analyzing elastic trajectories in the ESRVF space. Based on this formulation, we develop a comprehensive framework for analyzing the spatiotemporal dynamics of tree-like objects, including registration under large deformations and topological differences, geodesic computation, statistical summarization through mean trajectories and modes of variation, and the synthesis of new, random tree-like 4D shapes.

Problem

• Modeling spatiotemporal shape variability of deforming tree-like 4D structures
• Addressing registration challenges from branch bending and topological changes
• Analyzing elastic trajectories in Riemannian space for statistical shape analysis

Innovation

• Riemannian framework models tree-like 4D shape variability
• Maps 3D objects to Extended Square Root Velocity Field space
• Analyzes elastic trajectories in ESRVF space using L2 metric