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

Author

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  • Mattie Ji

    (Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA)

Abstract

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.

Suggested Citation

  • Mattie Ji, 2024. "On the Injectivity of Euler Integral Transforms with Hyperplanes and Quadric Hypersurfaces," Mathematics, MDPI, vol. 12(15), pages 1-13, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:15:p:2339-:d:1443661
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