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The Reeb Transform

Erin Chambers, Shankha Shubhra Mukherjee, Katharine Turner

arXiv:2610.03126v1cs.CGmath.AT

TL;DR

Shape transforms such as the ECT and PHT can uniquely determine o-minimal sets, prompting the question of whether Reeb graphs can do so. The paper studies the Reeb Transform for definable sets and proves injectivity in certain low-dimensional settings, while showing it fails for broader classes and above three dimensions.

  • Problem

    The paper asks whether the Reeb Transform is invertible, given that it records level-set connected components but not their homology.

    Source: §1 · p. 2

  • Method

    The paper studies Reeb transforms on o-minimal definable sets and establishes injectivity results for one-dimensional sets and compact surfaces in R3.

    Source: §1 · p. 3, §3 · pp. 13–14

  • Results

    The transform is injective for Set(d, 1) and compact surfaces in R3, but is not injective on Set(d) generally or above three dimensions.

    Source: §1 · p. 3, §2.2 · p. 12, §3 · pp. 13–14

  • Takeaways & Limitations

    Reeb transforms are discriminative and potentially useful for a wide class of low-dimensional settings, but are not complete shape invariants.

    Source: §1 · p. 3, §4 · p. 19

    The transform records level-set connectivity but not topology within individual components, so it is not injective on the full class Set(d).

    Source: §2.2 · p. 12

Abstract

from arXiv · show

Reeb Transforms offer a compact representation of how shapes in $\mathbb{R}^n$ change when sliced along varying directions. We define the Reeb Transform as the family of Reeb graphs induced by height functions along all directions in the unit sphere. In this paper, we develop a rigorous treatment of Reeb Transforms for o-minimal definable sets, with particular emphasis on one-dimensional stratified spaces embedded in $\mathbb{R}^d$ and on surfaces in $\mathbb{R}^3$. We establish injectivity of the Reeb Transform in multiple settings, including compact surfaces in $\mathbb{R}^3$, capturing the essential topological features needed to uniquely reconstruct such surfaces. However, in dimensions above three, the Reeb Transform ceases to be injective, indicating the limitations of this descriptor in higher-dimensional settings.

1 Introduction

The paper introduces the Reeb Transform as a directional shape descriptor based on Reeb graphs, which retain level-set connectivity but not higher-dimensional slice features. It asks whether this reduced information can nevertheless distinguish shapes in low-dimensional settings.

1 Introduction

  • Topological transforms such as the PHT and ECT are used to quantify shape geometry and topology, with applications including tumour progression and barley-seed morphology.Reeb graphs also have applications in shape matching, data visualization, and manifold learning; their extraction has the same asymptotic complexity as degree-zero persistent homology.Source: §1 · p. 2
  • The Reeb Transform replaces Euler curves and persistence modules with Reeb graphs, encoding how connected components evolve across slicing directions.Unlike signatures that capture higher-dimensional features, Reeb graphs retain connectivity information.Source: §1 · p. 2
  • The paper asks whether the Reeb Transform is invertible, noting that unlike the injective ECT and PHT, it records connected components but not level-set homology.The ECT and PHT uniquely determine o-minimal sets, though many directions may be needed.Source: §1 · p. 2
  • The transform is not universal: a solid cone and hollow cone can share a Reeb graph, as can a sphere, solid ball, and spherical shell.In these examples, each level set appears as a single connected component, regardless of its internal topology.Source: §1 · p. 2, Figure 1 · p. 3
  • The Reeb Transform is injective for certain low-dimensional classes, including surfaces in R3, but ceases to be injective above three dimensions.The paper presents it as an intermediate-level descriptor that loses information for more complex domains.Source: §1 · pp. 2–3

2 Prelims and Definitions

This section introduces the tame definable-set framework and height-based transforms, establishing what dense-direction Euler Characteristic Transform data determine and defining Reeb graphs and the Reeb Transform. It also discusses directional stability and finite sampling, while noting that agreement of Reeb graphs in every direction does not guarantee identical sets or internal topology.

2.1 Definable Sets, surfaces, and the Euler characteristic transform

  • O-minimal definability ensures tame topology: definable sets have finite cell decompositions and finitely many connected components, avoiding pathological oscillations.The section also establishes the availability of cylindrical definable cell decompositions compatible with sets and functions.Source: §2.1 · pp. 4–5
  • The paper targets Reeb Transform injectivity on Set(d, 1) and Surface(3), rather than all of Set(d), the class of compact definable sets.Set(d, 1) consists of sets stratified into zero-dimensional points and a possibly disconnected definable C1 embedded one-dimensional submanifold; Surface(3) consists of compact definable sets homeomorphic to closed embedded surfaces in R3.Source: §2.1 · p. 6
  • Agreement of Euler Characteristic Transform curves on a dense set of directions determines the transform, and agreement for two constructible sets implies the sets are equal.Continuity into L1 and the unique right-continuous representative allow values at every direction to be recovered from a dense subset.Source: §2.1 · p. 7

2.2 Introducing the Reeb Transform

  • The Reeb graph in a direction collapses each connected component of each height level to a point, yielding a finite graph-like quotient with an induced height function.The Reeb Transform assigns this graph, up to height-preserving isomorphism, to every direction on the unit sphere.Source: §2.2 · pp. 8–9
  • The sphere-indexed Reeb Transform motivates finite directional sampling, and Proposition 6 provides a directional stability bound for nonempty constructible sets.The proposition assumes a positive D satisfying the stated bounding condition; the supplied passage does not include the bound's formula.Source: §2.2 · pp. 9–10
  • Functional distortion distance compares Reeb graphs by minimizing the maximum height-function distortion over pairs of continuous maps between them.The distance measures discrepancies along paths and their images in the two graphs.Source: §2.2 · pp. 9–10
  • For non-empty A contained in B(0, D), rotating the slicing direction by θ changes its Reeb graph by at most 2D sin(θ/2) in functional distortion distance.Equivalently, the Reeb Transform is D-Lipschitz from the unit sphere with its Euclidean metric to Reeb graphs with functional distortion distance.Source: §2.2 · p. 11
  • The Reeb Transform declares two sets equivalent when their Reeb graphs are isomorphic in every slicing direction.This equivalence does not imply the sets are identical: for d ≥ 3, a ball and an annulus have isomorphic interval Reeb graphs in every direction despite differing in contractibility.Source: §2.2 · pp. 11–12
  • The Reeb Transform records how level-set connected components evolve across directions but misses topology within individual components, so it is not injective on Set(d).The ball and annulus example illustrates this limitation.Source: §2.2 · p. 12

3 Positive Injectivity Results

This section establishes positive injectivity results for Reeb Transforms of definable planar sets, embedded one-dimensional stratified spaces, and compact surfaces in R3. It examines how interior and exterior transforms encode component counts and level-set Euler characteristics, and proves uniqueness for surfaces from their transforms.

  • The Reeb Transform is injective when restricted to Set(2).Source: §3 · p. 12
  • For A in Set(d, 1), collapsing directions have measure zero, and equality of Reeb Transforms implies equality of Euler characteristic transforms.Outside the measure-zero union of collapsing directions, level-set components are contractible; the Euler characteristic transforms agree on this dense set of directions.Source: §3 · pp. 13–14
  • For compact surfaces in R3, χ(slice at t) = 2C(t) - C(t - ε) - C(t + ε) for sufficiently small ε, where C counts connected components.Source: §3 · p. 15
  • The Reeb Transform is injective on Surface(3), since its data determine the Euler characteristic of every level set in every direction.Source: §3 · p. 16
  • The proof does not directly extend to surfaces in R^d for d > 3, where slices can have components with nonzero genus.The paper conjectures that modifying collapsing directions could establish the needed slice formula on a dense set of directions.Source: §3 · p. 16
  • For a connected compact surface in R3, the pair of Reeb Transforms of its closed interior and exterior uniquely determines the surface.Its complement has one bounded and one unbounded path-connected component, defining the interior and exterior regions.Source: §3 · pp. 16–18

3 Positive Injectivity Results

  • C(A, w, s) = C(Aint, w, s) + C(Aext, w, s) − 1.The identity follows from Mayer–Vietoris for the open cover of the contractible plane and the stated homotopy equivalences.Source: §3 · p. 19
  • The interior and exterior Reeb graphs determine the component counts at each height, and matching transforms give matching interior and exterior counts.Applying the component-count identity gives C(A, w, s) = C(B, w, s) at t and its neighboring heights t − ε and t + ε.Source: §3 · p. 19
  • Matching interior and exterior transforms imply χ(LS(A, w, t)) = χ(LS(B, w, t)) for every t and every direction w.The level-set Euler characteristic is computed from component counts at t and at t − ε and t + ε.Source: §3 · p. 19

4 Conclusion and future work

The paper initiates the study of the Reeb Transform, finding it discriminative and potentially useful for low-dimensional settings, though it is not a complete invariant. The authors identify mapper transforms, applied evaluation, and alternative functions and representations as directions for future study.

4 Conclusion and future work

  • The Reeb Transform is not a complete invariant, but the results show it is discriminative and potentially useful in a wide class of low-dimensional settings.The authors describe this work as initiating the study of the transform.Source: §4 · p. 19
  • Mapper graphs converge to Reeb graphs in a limit [25] [12], but their instability and parameter choices make mapper transforms a distinct future question.The authors conjecture mapper transforms may share properties studied here, while noting that point-cloud data require different tools.Source: §4 · pp. 19–20
  • Applied study could evaluate Reeb graphs in shape-analysis pipelines and adapt dynamic-Reeb-graph methods to continuously varying height functions.The authors suggest such adaptations might yield fast algorithms and data structures.Source: §4 · p. 20
  • Future descriptors could investigate Reeb spaces or annotated Reeb graphs using intrinsic curvature or affine-Grassmannian functions instead of height functions.Examples of affine-Grassmannian functions include distance to a point, line, or hyperplane.Source: §4 · p. 20
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