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 study the invertibility of the ECT on definable sets that aren't necessarily compact, resulting in 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 invertibility 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等人及Ghrist等人此前已分别独立证明，ECT在全体紧致可定义集上具有单射性。本文研究ECT在非紧致可定义集上的可逆性，由此完整刻画了使欧拉特征变换失去单射性的可构造函数类。继而我们提出二次曲面欧拉特征变换（QECT），通过以二次超曲面替代超平面来检测可定义形状，作为ECT的自然推广。此外，本文还探讨了QECT可逆性的若干判据。