Keeping it together: a phase field version of path-connectedness and its implementation
This work provides a practical method for enforcing a topological constraint in phase field models, addressing a known limitation for applications requiring connected structures.
The paper implements a topological constraint in phase field simulations to enforce path-connectedness of preimages, demonstrating its necessity for maintaining connectivity in bending energy minimization and image segmentation. Without the constraint, surfaces become disconnected; with it, they remain connected.
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.