# An Invitation to the Euler Characteristic Transform

arXiv: https://arxiv.org/abs/2310.10395 · HTML: https://mlsanity.com/paper/2310.10395/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2023.

## Built on by (most cited first, 9 of 9)

- [Topology meets Machine Learning: An Introduction using the Euler Characteristic Transform](https://mlsanity.com/paper/2410.17760/) (2024)
- [Point Cloud Synthesis Using Inner Product Transforms](https://mlsanity.com/paper/2410.18987/) (2024)
- [LEAP: Local ECT-Based Learnable Positional Encodings for Graphs](https://mlsanity.com/paper/2510.00757/) (2025)
- [Latent Manifold Reconstruction and Representation with Topological and Geometrical Regularization](https://mlsanity.com/paper/2505.04412/) (2025)
- [Molecular Machine Learning Using Euler Characteristic Transforms](https://mlsanity.com/paper/2507.03474/) (2025)
- [Tensor Computation of Euler Characteristic Functions and Transforms](https://mlsanity.com/paper/2511.03909/) (2025)
- [TopoGate: Quality-Aware Topology-Stabilized Gated Fusion for Longitudinal Low-Dose CT New-Lesion Prediction](https://mlsanity.com/paper/2602.17855/) (2026)
- [Geometry-Aware Simplicial Message Passing](https://mlsanity.com/paper/2605.06061/) (2026)
- [Encoding the Euler Characteristic Transform](https://mlsanity.com/paper/2606.10824/) (2026)

## Abstract (arXiv)

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape using sub-level sets of a a function defined based on a given direction, and then returning the Euler characteristics of these sublevel sets. Because the ECT has been shown to be injective on the space of embedded simplicial complexes, it has been used for applications spanning a range of disciplines, including plant morphology and protein structural analysis. In this survey article, we present a comprehensive overview of the Euler characteristic transform, highlighting the main idea on a simple leaf example, and surveying its its key concepts, theoretical foundations, and available applications.