Quadratic-Order Geodesics on Meshes

TL;DR

This method computes squared geodesic distances from point and curve sources using piecewise-quadratic elements, exactly reproducing flat distances regardless of mesh quality while improving accuracy over existing approaches on curved meshes.

Abstract

We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizations. Our method computes squared geodesic distances from point and curve sources using piecewise-quadratic elements, exactly reproducing flat distances regardless of mesh quality while improving accuracy over existing approaches on curved meshes. The formulation naturally supports sources placed anywhere on the mesh, not just at vertices.

Figures (17)

• Figure 1: Statistical analysis of RMSE and computation time. Left: Histograms of RMSE. Right: Distribution of computation time (in seconds). Our approach demonstrates a high numerical accuracy without considerable computational overhead.
• Figure 2: Histograms of $L_2$ error ratios per mesh computed for each mesh across methods. Ratio$<$1 (left of red line) is where our result is better.
• Figure 3: Our method naturally handles boundaries without special conditions.
• Figure 4: Convergence analysis on six meshes from Thingi10K (Numbers indicate mesh name). Each example displays the mesh, $L_2$ and $L_\infty$ error convergence plots, and angle histogram measuring the quality of the mesh (good-quality meshes are concentrated around $60^\circ$). The x-axis in the error diagrams represents the mean edge length $h$ of the subdivided mesh, and the error is measured against The reference FEG solution $u_{GT}$ on the finest level (three times subdivided). Note: To ensure a fair comparison, our method operates on one level of subdivision less than the linear methods. Our error profiles and convergence slopes are considerably better, especially for low-quality meshes, and generally better also in high-quality meshes.
• Figure 5: Convergence analysis on a unit sphere. Left: The unit sphere mesh used for evaluation. Middle and Right: $L_2$ and $L_\infty$ error convergence plots with respect to the mean edge length $h$. Errors are computed against the analytic closed-form spherical distance, with all vertices normalized to the unit sphere after each subdivision. Following the benchmark setup, our method (P2) operates on the coarse mesh levels, while all baseline methods (P1) operate on meshes subdivided once further. FEG is exact on the vertices, and thus has the lowest overall error for this semi-regular mesh; our method has a similar convergence rate.