Representing Fields without Correspondences: the Lifted Euler Characteristic Transform

TL;DR

This paper introduces two novel topological transforms for representing fields in 3D space using Euler calculus, enabling shape analysis without correspondence maps and providing theoretical guarantees of injectivity.

Contribution

The paper generalizes topological transforms from shapes to real-valued fields using a lifting argument, establishing injectivity and bounds on directions needed for function determination.

Findings

Transforms are injective maps.

Lifting argument extends Euler calculus to fields.

Bound on directions needed for function identification.

Abstract

Topological transforms have been very useful in statistical analysis of shapes or surfaces without restrictions that the shapes are diffeomorphic and requiring the estimation of correspondence maps. In this paper we introduce two topological transforms that generalize from shapes to fields, $f:\mathbf{R}^3 \rightarrow \mathbf{R}$. Both transforms take a field and associate to each direction $v\in S^{d-1}$ a summary obtained by scanning the field in the direction $v$. The transforms we introduce are of interest for both applications as well as their theoretical properties. The topological transforms for shapes are based on an Euler calculus on sets. A key insight in this paper is that via a lifting argument one can develop an Euler calculus on real valued functions from the standard Euler calculus on sets, this idea is at the heart of the two transforms we introduce. We prove the transforms are injective maps. We show for particular moduli spaces of functions we can upper bound the number of directions needed determine any particular function.