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

Mattie Ji

TL;DR

This work extends the injectivity theory of the Euler characteristic transform (ECT) from compact definable sets to broader, non-compact definable sets by classifying all non-injective constructible pairs. It then introduces the quadric Euler characteristic transform (QECT), generalizing the ECT to detect shape via quadric hypersurfaces $x^T A x + x\cdot\nu \le t$ and develops a kernel-based framework to study its injectivity. The authors prove an injectivity result for QECT with fixed $v=0$ (up to sign) and provide an interpolation theorem showing injectivity when a norm constraint on $A$ is satisfied, connecting the hyperplane case to the quadric case. These results broaden the repertoire of topological descriptors in TDA and establish inversion-style formulas for ECT without compact support, enabling broader applicability in reconstructing shapes from Euler-characteristic summaries.

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.

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

TL;DR

This work extends the injectivity theory of the Euler characteristic transform (ECT) from compact definable sets to broader, non-compact definable sets by classifying all non-injective constructible pairs. It then introduces the quadric Euler characteristic transform (QECT), generalizing the ECT to detect shape via quadric hypersurfaces and develops a kernel-based framework to study its injectivity. The authors prove an injectivity result for QECT with fixed (up to sign) and provide an interpolation theorem showing injectivity when a norm constraint on is satisfied, connecting the hyperplane case to the quadric case. These results broaden the repertoire of topological descriptors in TDA and establish inversion-style formulas for ECT without compact support, enabling broader applicability in reconstructing shapes from Euler-characteristic summaries.

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.
Paper Structure (11 sections, 9 theorems, 31 equations)

This paper contains 11 sections, 9 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Background
  4. O-minimal structures
  5. Euler Calculus and the Euler Characteristic Transform
  6. Euler Characteristic Transform with Hyperplanes
  7. Euler Characteristic Transform with Quadric Hypersurfaces
  8. Generalized Kernel Spaces
  9. Quadric Euler Characteristic Transform
  10. Appendix
  11. Inversion Formula of $\mathop{\mathrm{ECT}}\nolimits$ without Compact Support

Key Result

Lemma 2.5

Let $S \subseteq \mathbb{R}^{m+n}$ be a definable set. For any $a\in\mathbb{R}^m$, define $S_a :=\{x\in\mathbb{R}^n\vert\, (a,x)\in S\}$. Then $\chi(S_a)$ takes only finitely many values as $a$ runs through $\mathbb R^m$, and for each integer $e$ the set $\{a \in \mathbb R^m: \chi(S_a) = e\}$ is def

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5: Rephrased from Proposition 2.10 of Chapter 4 of van1998tame
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Theorem 2.9: Fubini's Theorem for Euler integrals
  • Definition 2.10
  • ...and 13 more