The Euler characteristic transform (ECT) is an integral transform used widely in topological data analysis. Previous efforts by Curry et al. and Ghrist et al. have independently shown that the ECT is injective on all compact definable sets. In this work, we first study the injectivity of the ECT on definable sets that are not necessarily compact and prove a complete classification of constructible functions that the Euler characteristic transform is not injective on. We then introduce the quadric Euler characteristic transform (QECT) as a natural generalization of the ECT by detecting definable shapes with quadric hypersurfaces rather than hyperplanes. We also discuss some criteria for the injectivity of QECT.

翻译：欧拉特征变换（ECT）是拓扑数据分析中广泛使用的积分变换。Curry等人与Ghrist等人此前各自独立证明，ECT在所有紧致可定义集上具有单射性。本文首先研究ECT在非紧致可定义集上的单射性，并完整分类了使欧拉特征变换不具单射性的可构造函数。随后，我们引入二次欧拉特征变换（QECT）作为ECT的自然推广，通过二次超曲面而非超平面来探测可定义形状。此外，我们讨论了QECT单射性的若干判定准则。