Surface Maps via Adaptive Triangulations

Abstract

We present a new method to compute continuous and bijective maps (surface homeomorphisms) between two or more genus‐0 triangle meshes. In contrast to previous approaches, we decouple the resolution at which a map is represented from the resolution of the input meshes. We discretize maps via common triangulations that approximate the input meshes while remaining in bijective correspondence to them. Both the geometry and the connectivity of these triangulations are optimized with respect to a single objective function that simultaneously controls mapping distortion, triangulation quality, and approximation error. A discrete‐continuous optimization algorithm performs both energy‐based remeshing as well as global second‐order optimization of vertex positions, parametrized via the sphere. With this, we combine the disciplines of compatible remeshing and surface map optimization in a unified formulation and make a contribution in both fields. While existing compatible remeshing algorithms often operate on a fixed pre‐computed surface map, we can now globally update this correspondence during remeshing. On the other hand, bijective surface‐to‐surface map optimization previously required computing costly overlay meshes that are inherently tied to the input mesh resolution. We achieve significant complexity reduction by instead assessing distortion between the approximating triangulations. This new map representation is inherently more robust than previous overlay‐based approaches, is less intricate to implement, and naturally supports mapping between more than two surfaces. Moreover, it enables adaptive multi‐resolution schemes that, e.g., first align corresponding surface regions at coarse resolutions before refining the map where needed. We demonstrate significant speedups and increased flexibility over state‐of‐the art mapping algorithms at similar map quality, and also provide a reference implementation of the method.

Neural Semantic Surface Maps

“…Aggregation To assess the importance of the map distillation module, we present a qualitative comparison in Figure 10 with the method proposed by Surface Maps via Adaptive Triangulations (SMAT) [SPK23], where we replace manual correspondences with automatically extracted ones. As the original approach requires a set of bijective correspondences, we randomly subsample a set of N = 64 matches from the automatically extracted ones to ensure consistency, i.e ., no vertex appears twice.…”

“…We refer to it as Di-nov2+SMAT. Note SMAT [SPK23] optimize for isometric energy (Dirichlet), while we optimize only for smoothness, see Eq 9.…”

“…Figure 10: Qualitative comparison. Surface Maps via Adaptive Triangulations (SMAT)[SPK23] optimize for bijective and continuous maps, relying on manual annotations. We pair it with Dinov2, Dinov2+SMAT, by replacing these manual annotations with k = 64 automatically extracted ones, {φ i } with i = 1 : k, then we optimize the inter-surface map to construct an automatic inter-surface map.…”

A Scalable Combinatorial Solver for Elastic Geometrically Consistent 3D Shape Matching

“…Common triangulations between two shapes have been utilised by Schmidt et al (2020) or via intrinsic triangulation by Takayama (2022). Schmidt, Pieper, and Kobbelt (2023) propose to map shapes to spheres and then optimise for a homeomorphism.…”

“…While axiomatic methods can -in principle -take geometric consistency into account, most existing shape matching approaches disregard it (Van Kaick et al 2011;Ovsjanikov et al 2012;Tam et al 2012;Huang and Guibas 2013;Solomon et al 2016;Sahillioglu 2018;Bernard, Suri, and Theobalt 2020), mainly due to the resulting nonconvexity of the optimisation problem or hard-to-solve constraints. Although there are some exceptions, respective formulations can typically be solved only for small instances, such as the approach by Windheuser et al (2011a), or they rely on manual user initialisations to avoid poor local optima and to better constrain the problem (Schreiner et al 2004;Takayama 2022;Schmidt, Pieper, and Kobbelt 2023).…”