On the Injectivity of Euler Integral Transforms with Hyperplanes and Quadric Hypersurfaces

Autor:	Mattie Ji
Jazyk:	angličtina
Rok vydání:	2024
Předmět:	Euler characteristic Euler characteristic transform (ECT) topological data analysis (TDA) quadric hypersurfaces Radon transform Mathematics QA1-939
Zdroj:	Mathematics, Vol 12, Iss 15, p 2339 (2024)
Druh dokumentu:	article
ISSN:	2227-7390
DOI:	10.3390/math12152339
Popis:	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.
Databáze:	Directory of Open Access Journals
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