Surface Simplification using Intrinsic Error Metrics
ACM Transactions on Graphics (SIGGRAPH 2023)
Hsueh-Ti Derek Liu University of Toronto & Roblox
Mark Gillespie Carnegie Mellon University
Benjamin Chislett University of Toronto
Nicholas Sharp University of Toronto & NVIDIA
Alec Jacobson University of Toronto & Adobe Research
Keenan Crane Carnegie Mellon University
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.
This work was funded in part by an NSF CAREER Award (IIS 1943123), NSF Award IIS 2212290, a Packard Fellowship, NSERC Discovery (RGPIN–2022–04680), the Ontario Early Research Award program, the Canada Research Chairs Program, a Sloan Research Fellowship, the DSI Catalyst Grant program, the Fields Institute for Mathematics, the Vector Institute for AI, and gifts from Adobe Inc., Facebook Reality Labs, and Google, Inc.
@article{Liu:2023:SSI, author = {Liu, Hsueh-Ti Derek and Gillespie, Mark and Chislett, Benjamin and Sharp, Nicholas and Jacobson, Alec and Crane, Keenan}, title = {Surface Simplification Using Intrinsic Error Metrics}, journal = {ACM Trans. Graph.}, volume = {42}, number = {4}, year = {2023}, publisher = {ACM}, address = {New York, NY, USA}, issn = {0730-0301}, url = {}, doi = {10.1145/3592403}, month = {jul}, articleno = {118}, }

The caption on Figure 19 has been fixed to refer to "intrinsic curvature" rather than "extrinsic curvature". The attribution has also been fixed to cite Rabinovich et al.'s [2018] paper on developable surfaces, rather than an unnamed "cloth simulation". Thanks to Olga Sorkine-Hornung for catching these issues!