Representing Fields without Correspondences: the Lifted Euler Characteristic Transform

Understand

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.