# A Riemannian Approach for Spatiotemporal Analysis and Generation of 4D Tree-shaped Structures

arXiv: https://arxiv.org/abs/2408.12443 · HTML: https://mlsanity.com/paper/2408.12443/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 24 references that are in the map.
- First posted on arXiv 22 Aug 2024.

## Builds on (most cited first, 6 of 6)

- [Fast and Robust Iterative Closest Point](https://mlsanity.com/paper/2007.07627/) (2020)
- [H4D: Human 4D Modeling by Learning Neural Compositional Representation](https://mlsanity.com/paper/2203.01247/) (2022)
- [A Survey on Non-rigid 3D Shape Analysis](https://mlsanity.com/paper/1812.10111/) (2018)
- [4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data](https://mlsanity.com/paper/2101.09403/) (2021)
- [LoRD: Local 4D Implicit Representation for High-Fidelity Dynamic Human Modeling](https://mlsanity.com/paper/2208.08622/) (2022)
- [Elastic Shape Analysis of Tree-like 3D Objects using Extended SRVF Representation](https://mlsanity.com/paper/2110.08693/) (2021)

## Built on by (most cited first, 4 of 4)

- [TreeSRNF: Square-Root Normal Fields for Generative Modelling of the Geometric and Structural Variability in Tree-like 3D Objects](https://mlsanity.com/paper/2607.13456/) (2026)
- [GrowFields: Compositional 4D Neural Fields for Topology-Changing Plant Growth](https://mlsanity.com/paper/2607.03330/) (2026)
- [Structure-aware Riemannian Growth Fields for 4D Plant Modeling](https://mlsanity.com/paper/2608.13007/) (2026)
- [Growing a Stand, Not a Tree: Joint Canopy Generation Reproduces Crown Shyness](https://mlsanity.com/paper/2608.28692/) (2026)

## Abstract (arXiv)

We propose the first comprehensive approach for modeling and analyzing the spatiotemporal shape variability in tree-like 4D objects, i.e., 3D objects whose shapes bend, stretch, and change in their branching structure over time as they deform, grow, and interact with their environment. Our key contribution is the representation of tree-like 3D shapes using Square Root Velocity Function Trees (SRVFT). By solving the spatial registration in the SRVFT space, which is equipped with an L2 metric, 4D tree-shaped structures become time-parameterized trajectories in this space. This reduces the problem of modeling and analyzing 4D tree-like shapes to that of modeling and analyzing elastic trajectories in the SRVFT space, where elasticity refers to time warping. In this paper, we propose a novel mathematical representation of the shape space of such trajectories, a Riemannian metric on that space, and computational tools for fast and accurate spatiotemporal registration and geodesics computation between 4D tree-shaped structures. Leveraging these building blocks, we develop a full framework for modelling the spatiotemporal variability using statistical models and generating novel 4D tree-like structures from a set of exemplars. We demonstrate and validate the proposed framework using real 4D plant data.