Surface Minkowski tensors to characterize shapes on curved surfaces

TL;DR

This paper introduces surface Minkowski tensors for characterizing shapes on curved surfaces, providing a novel method that accounts for surface curvature and shape symmetries, with applications demonstrated in biological cell analysis.

Contribution

The paper develops a new surface Minkowski tensor framework using modified vector transport to analyze shapes on curved surfaces, extending shape classification methods to non-flat geometries.

Findings

The method effectively characterizes shape symmetries on curved surfaces.

Surface Minkowski tensors are stable under shape and surface perturbations.

Numerical experiments demonstrate applicability to biological cell shapes.

Abstract

We introduce surface Minkowski tensors to characterize rotational symmetries of shapes embedded in curved surfaces. The definition is based on a modified vector transport of the shapes boundary co-normal into a reference point which accounts for the angular defect that a classical parallel transport would introduce. This modified transport can be easily implemented for general surfaces and differently defined embedded shapes, and the associated irreducible surface Minkowski tensors give rise to the classification of shapes by their normalized eigenvalues, which are introduced as shape measures following the flat-space analog. We analyze different approximations of the embedded shapes, their influence on the surface Minkowski tensors, and the stability to perturbations of the shape and the surface. The work concludes with a series of numerical experiments showing the applicability of the approach on various surfaces and shape representations and an application in biology in which the characterization of cells in a curved monolayer of cells is considered.