An Invitation to the Euler Characteristic Transform

Abstract

The Euler characteristic transform (ECT) is a simple to define yet powerful representation of shape. The idea is to encode an embedded shape by tracking how the Euler characteristic, a simple integer-valued topological invariant, changes as the shape is built up in a particular direction. 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 key concepts, theoretical foundations, and available applications.