Keeping it together: a phase field version of path-connectedness and its implementation

TL;DR

This paper introduces a topological constraint in phase field models to ensure path-connectedness in finite element simulations, demonstrated through applications in surface energy minimization and image segmentation.

Contribution

It presents a novel implementation of a topological constraint in phase field models, enabling control over connectedness in simulations.

Findings

Connected surfaces with the topological constraint

Disconnected surfaces without the constraint

Effective use of Dijkstra's algorithm for geodesic distance

Abstract

We describe the implementation of a topological constraint in finite element simulations of phase field models which ensures path-connectedness of preimages of intervals in the phase field variable. Two main applications of our method are presented. First, a discrete steepest decent of a phase field version of a bending energy with spontaneous curvature and additional surface area penalty is shown, which leads to disconnected surfaces without our topological constraint but connected surfaces with the constraint. The second application is the segmentation of an image into a connected component and its exterior. Numerically, our constraint is treated using a suitable geodesic distance function which is computed using Dijkstra's algorithm.