The Reeb Transform
🤖 AI Summary
This study addresses the problem of compactly representing and uniquely reconstructing shapes in high-dimensional spaces based on their directional slice variations. The proposed method defines a family of Reeb graphs over all directions, integrating computational topology, o-minimal geometry, and height function analysis to establish a rigorous mathematical framework for one-dimensional stratified spaces and three-dimensional surfaces embedded in Euclidean space, along with injectivity guarantees across multiple settings. The key contribution is the proof that this descriptor captures topological features sufficient for the unique reconstruction of compact 3D surfaces. Furthermore, the work precisely delineates the boundaries of non-injectivity in four and higher dimensions, revealing intrinsic limitations inherent to high-dimensional generalizations.
📝 Abstract
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.
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