# Elastic Shape Analysis of Tree-like 3D Objects using Extended SRVF Representation

arXiv: https://arxiv.org/abs/2110.08693 · HTML: https://mlsanity.com/paper/2110.08693/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 50 references that are in the map.
- First posted on arXiv 17 Oct 2021.

## Builds on (most cited first, 4 of 4)

- [Image-based 3D Object Reconstruction: State-of-the-Art and Trends in the Deep Learning Era](https://mlsanity.com/paper/1906.06543/) (2019)
- [Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces](https://mlsanity.com/paper/1610.04531/) (2016)
- [4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data](https://mlsanity.com/paper/2101.09403/) (2021)
- [Modelling brain-wide neuronal morphology via rooted Cayley trees](https://mlsanity.com/paper/1810.03262/) (2018)

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

- [A Riemannian Approach for Spatiotemporal Analysis and Generation of 4D Tree-shaped Structures](https://mlsanity.com/paper/2408.12443/) (2024)
- [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)
- [A shape-similarity latent space for fluid interfaces: invertible reduced-order modelling of droplet morphology](https://mlsanity.com/paper/2609.37947/) (2026)
- [Autoregressive Frontier Expansion: Growing Trees with Graph Machine Learning](https://mlsanity.com/paper/2609.38506/) (2026)

## Abstract (arXiv)

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.