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.

翻译：欧拉示性数变换（Euler Characteristic Transform, ECT）是一种定义简洁但功能强大的形状表示方法。其核心思想是：通过给定方向定义函数，利用该函数的子水平集对嵌入形状进行编码，再计算这些子水平集的欧拉示性数。由于ECT已被证明在嵌入单纯复形空间上具有单射性，它已被广泛应用于植物形态学、蛋白质结构分析等多个学科领域。本文作为综述文章，系统介绍了欧拉示性数变换，以简单叶片案例阐述核心思想，并全面梳理其关键概念、理论基础及现有应用。