Surface Simplification using Intrinsic Error Metrics

TL;DR

This paper introduces a fast surface mesh simplification method based on intrinsic error metrics, enabling efficient geometry processing tasks while preserving essential surface properties.

Contribution

It presents a novel intrinsic triangulation approach with error tracking and guarantees on element quality, improving upon extrinsic methods for solving equations on surfaces.

Findings

Enables decoupling mesh resolution from matrix size in computations.

Improves accuracy and guarantees in surface simplification.

Facilitates efficient solutions for geometric multigrid and geodesic computations.

Abstract

This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation - a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map.