# Representations, Metrics and Statistics For Shape Analysis of Elastic Graphs

arXiv: https://arxiv.org/abs/2003.00287 · HTML: https://mlsanity.com/paper/2003.00287/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Feb 2020.

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

- [Manifold Modeling in Quotient Space: Learning An Invariant Mapping with Decodability of Image Patches](https://mlsanity.com/paper/2203.05134/) (2022)
- [Topological Spatial Graph Coarsening](https://mlsanity.com/paper/2512.24327/) (2025)
- [Feature-based morphological analysis of shape graph data](https://mlsanity.com/paper/2602.16120/) (2026)
- [A shape-similarity latent space for fluid interfaces: invertible reduced-order modelling of droplet morphology](https://mlsanity.com/paper/2609.37947/) (2026)

## Abstract (arXiv)

Past approaches for statistical shape analysis of objects have focused mainly on objects within the same topological classes, e.g., scalar functions, Euclidean curves, or surfaces, etc. For objects that differ in more complex ways, the current literature offers only topological methods. This paper introduces a far-reaching geometric approach for analyzing shapes of graphical objects, such as road networks, blood vessels, brain fiber tracts, etc. It represents such objects, exhibiting differences in both geometries and topologies, as graphs made of curves with arbitrary shapes (edges) and connected at arbitrary junctions (nodes). To perform statistical analyses, one needs mathematical representations, metrics and other geometrical tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis and analytical statistical testing and modeling of graphical shapes. The efficacy of this framework is demonstrated using various simulated as well as the real data from neurons and brain arterial networks.