# The Weighted Euler Curve Transform for Shape and Image Analysis

arXiv: https://arxiv.org/abs/2004.11128 · HTML: https://mlsanity.com/paper/2004.11128/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2020.

## Built on by (most cited first, 10 of 10)

- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [Euler Characteristic Transform Based Topological Loss for Reconstructing 3D Images from Single 2D Slices](https://mlsanity.com/paper/2303.05286/) (2023)
- [LEAP: Local ECT-Based Learnable Positional Encodings for Graphs](https://mlsanity.com/paper/2510.00757/) (2025)
- [Molecular Machine Learning Using Euler Characteristic Transforms](https://mlsanity.com/paper/2507.03474/) (2025)
- [Beyond Means: Topological Causal Effects under Persistent-Homology Ignorability](https://mlsanity.com/paper/2603.14169/) (2026)
- [Convolutional Persistence Transforms](https://mlsanity.com/paper/2208.02107/) (2022)
- [Invariant Representations of Embedded Simplicial Complexes](https://mlsanity.com/paper/2302.13565/) (2023)
- [Tensor Computation of Euler Characteristic Functions and Transforms](https://mlsanity.com/paper/2511.03909/) (2025)
- [Topological Ignorability for Structural Causal Effects Beyond Means](https://mlsanity.com/paper/2606.01184/) (2026)

## Abstract (arXiv)

The Euler Curve Transform (ECT) of Turner et al. is a complete invariant of an embedded simplicial complex, which is amenable to statistical analysis. We generalize the ECT to provide a similarly convenient representation for weighted simplicial complexes, objects which arise naturally, for example, in certain medical imaging applications. We leverage work of Ghrist et al. on Euler integral calculus to prove that this invariant---dubbed the Weighted Euler Curve Transform (WECT)---is also complete. We explain how to transform a segmented region of interest in a grayscale image into a weighted simplicial complex and then into a WECT representation. This WECT representation is applied to study Glioblastoma Multiforme brain tumor shape and texture data. We show that the WECT representation is effective at clustering tumors based on qualitative shape and texture features and that this clustering correlates with patient survival time.