Geodesic distance approximation using a surface finite element method for the $p$-Laplacian

TL;DR

This work addresses computing intrinsic geodesic distances to user-defined features on surfaces by solving a surface $p$-Poisson problem in the large-$p$ regime. It combines a surface finite element discretization with an ADMM solver to obtain distance fields that converge to the intrinsic geodesic distance as $p\to\infty$, while employing mixed Dirichlet/Neumann boundary conditions to handle open surfaces and boundaries. Numerical experiments on hemispheres, tori, and complex meshes demonstrate convergence to the true distance, preservation of the triangle inequality, and robustness to mesh perturbations, with comparisons to the heat method and polyhedral approaches. The method offers a PDE-based, potentially mesh-free framework that can be extended to other surface representations, though it incurs higher computational cost relative to some fast geometric methods for high-accuracy demands.

Abstract

We use the $p$-Laplacian with large $p$-values in order to approximate geodesic distances to features on surfaces. This differs from Fayolle and Belyaev's (2018) [1] computational results using the $p$-Laplacian for the distance-to-surface problem. Our approach appears to offer some distinct advantages over other popular PDE-based distance function approximation methods. We employ a surface finite element scheme and demonstrate numerical convergence to the true geodesic distance functions. We check that our numerical results adhere to the triangle inequality and examine robustness against geometric noise such as vertex perturbations. We also present comparisons of our method with the heat method from Crane et al. [2] and the classical polyhedral method from Mitchell et al. [3].

Figures (9)

• Figure 1: $S$ is the open hemisphere and $\Gamma_1$ (indicated in orange) represents the feature set. Top row: Distance-to-boundary problem ($\Gamma_1= \partial S, \Gamma_2 = \emptyset$). Bottom row: Distance-to-feature problem. $\Gamma_2$ is indicated in blue.
• Figure 2: One-dimensional example, with $S=(-1,1)$ and $\Gamma_1=\{0\}$. Left: the exact solution $u_p(x)$ (see \ref{["eq:a'"]}-\ref{["eq:c'"]} and \ref{['eq:1d-exact']}) converges to $|x|=\mathrm{dist} (x,\Gamma_1)$ as $p$ increases. Right: the corresponding pointwise errors $|x|-u_p(x)$, for various $p$-values.
• Figure 3: Computed distance approximations using three methods: our $p$-Poisson method (first and second columns, using $p=5$ and $p=100$, respectively), the heat method (third column), and the polyhedral method (fourth column). Top row: distance to a point on hemisphere. Second row: distance to an open curve and a point on hemisphere. Bottom row: distance to two closed curves on torus. For visualization near the feature set $\Gamma_1$, we highlight in green values falling below a chosen tolerance. The triangulations consists of $2307$ vertices and $791$ faces, $9484$ vertices and $3228$ faces, and $35019$ vertices and $105057$ faces respectively.
• Figure 4: Tracking of residuals and gradient norm for the example of distance to a point on the hemisphere using various $p$-values. Primal (left) and dual (right) residuals versus ADMM iterations. The algorithm is run on a hemisphere mesh with $10240$ tri cells and $\ell \approx 3.95 \times 10^{-2}$, $h \approx 6.29 \times 10^{-2}$.
• Figure 5: $p$-Poisson energy \ref{['eq:Ep']} versus ADMM iterations for distance to closed curves on torus using various $p$-values. The algorithm is run on a torus mesh with $196608$ tri cells and $\ell \approx 3.30 \times 10^{-2}$, $h \approx 5.49 \times 10 ^{-2}$.