Solving variational problems and partial differential equations that map between manifolds via the closest point method

TL;DR

This paper introduces a novel closest point method for solving variational problems and PDEs that map between manifolds, simplifying complex constrained problems into manageable PDEs on the source manifold with projections, applicable to various manifolds and PDE types.

Contribution

The paper presents a new numerical framework using the closest point method to efficiently solve manifold-to-manifold PDEs, including harmonic and p-harmonic maps, with improved robustness over existing methods.

Findings

Efficient convergence compared to level set methods.

Successful computation of harmonic and p-harmonic maps in diverse examples.

Applications include texture denoising, map diffusion, and image enhancement.

Abstract

Maps from a source manifold $ {\mathcal M}$ to a target manifold ${\mathcal N}$ appear in liquid crystals, colour image enhancement, texture mapping, brain mapping, and many other areas. A numerical framework to solve variational problems and partial differential equations (PDEs) that map between manifolds is introduced within this paper. Our approach, the closest point method for manifold mapping, reduces the problem of solving a constrained PDE between manifolds ${\mathcal M}$ and ${\mathcal N}$ to the simpler problems of solving a PDE on ${\mathcal M}$ and projecting to the closest points on ${\mathcal N}.$ In our approach, an embedding PDE is formulated in the embedding space using closest point representations of ${\mathcal M}$ and ${\mathcal N}.$ This enables the use of standard Cartesian numerics for general manifolds that are open or closed, with or without orientation, and of any codimension. An algorithm is presented for the important example of harmonic maps and generalized to a broader class of PDEs, which includes $p$-harmonic maps. Improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods. Harmonic and $p$-harmonic maps are computed for a variety of numerical examples. In these examples, we denoise texture maps, diffuse random maps between general manifolds, and enhance colour images.