  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.
